>Logarithmic Equations>How to Solve Exponential Inequalities

How to Solve Exponential Inequalities

When solving exponential inequalities it is very important that you look at the value of the argument $a$ in the logarithm. That value determines whether you have to turn the inequality sign when you multiply or divide by $\ln a$ or $\log a$ .

Rule

Solving exponential inequalities

When

a > 1

\ln (a) > 0

and you can solve the inequality as usual.

\begin{array}{l} a^{x} & > b & a^{x} & > b \\ \ln a^{x} & > \ln b & \log a^{x} & > \log b \\ x \ln a & > \ln b & x \log a & > \log b \\ x & > \frac{\ln b}{\ln a} & x & > \frac{\log b}{\log a} \end{array}

When $0 < a < 1$ , $\ln (a) < 0$ , you will have to turn the inequality sign as you end up dividing or multiplying by a negative number!

\begin{array}{l} a^{x} & > b & a^{x} & > b \\ \ln a^{x} & > \ln b & \log a^{x} & > \log b \\ x \ln a & > \ln b & x \log a & > \log b \\ x & < \frac{\ln b}{\ln a} & x & < \frac{\log b}{\log a} \end{array}

Example 1

Solve the inequality ${3.5}^{x} > 439$

\begin{array}{l} 3.5 x > 439 \ln {3.5}^{x} & > \ln 439 x \ln 3.5 & > \ln 439 & | \div \ln 3.5 x & > \frac{\ln 439}{\ln 3.5} \approx 4.9Example 2 Solve the inequality 50 \cdot {1.05}^{x} > 300 \begin{array}{l} 50 \cdot {1.05}^{x} > 300 | \div 50 {1.05}^{x} & > 6 \ln {1.05}^{x} & > \ln 6 x \ln 1.05 & > \ln 6 & | \div \ln 1.05 x & > \frac{\ln 6}{\ln 1.05} \approx 36.7Example 3 Solve the inequality 3 \cdot {0.25}^{x} > 27 \begin{array}{l} 3 \cdot {0.25}^{x} > 27 | \div 3 {0.25}^{x} & > 9 \lg {0.25}^{x} & > \lg 9 x \lg 0.25 & > \lg 9 & | \div \log 0.25 x & < \frac{\lg 9}{\log 0.25} \approx - 1.6Previous entry What Are Logarithmic Equations? Next entry How to Solve Logarithmic InequalitiesGo backHome Curriculum How can I help you? Send Stats LeagueAbout Us About House of Math Employees Career Media Lectures Blog Contact support@houseofmath.com Address House of Math 470 Ramona Street Palo Alto, CA 94301 USA FAQ Links WeTakeAction Math Magic for Ukraine Tutoring Bootcamps Junior Math Pen & Paper exercises Encyclopedia Language/Curriculum Terms Privacy Cookies Download our app{"props":{"pageProps":{"campaign":false,"articleBody":" When solving exponential inequalities it is very important that you look at the value of the argument a in the logarithm. That value determines whether you have to turn the inequality sign when you multiply or divide by ln ⁡ a or log ⁡ a. Rule Solving exponential inequalities When a \u0026#x003E; 1, ln ⁡ (a) \u0026#x003E; 0 and you can solve the inequality as usual. ax \u0026#x003E; b ax \u0026#x003E; b ln ⁡ ax \u0026#x003E; ln ⁡ b log ⁡ ax \u0026#x003E; log ⁡ b x ln ⁡ a \u0026#x003E; ln ⁡ b x log ⁡ a \u0026#x003E; log ⁡ b x \u0026#x003E; ln ⁡ b ln ⁡ a x \u0026#x003E; log ⁡ b log ⁡ a When 0 \u0026#x003C; a \u0026#x003C; 1, ln ⁡ (a) \u0026#x003C; 0, you will have to turn the inequality sign as you end up dividing or multiplying by a negative number! ax \u0026#x003E; b ax \u0026#x003E; b ln ⁡ ax \u0026#x003E; ln ⁡ b log ⁡ ax \u0026#x003E; log ⁡ b x ln ⁡ a \u0026#x003E; ln ⁡ b x log ⁡ a \u0026#x003E; log ⁡ b x \u0026#x003C; ln ⁡ b ln ⁡ a x \u0026#x003C; log ⁡ b log ⁡ a Example 1 Solve the inequality 3.5x \u0026#x003E; 439 3.5x \u0026#x003E; 439 ln ⁡ 3.5x \u0026#x003E; ln ⁡ 439 x ln ⁡ 3.5 \u0026#x003E; ln ⁡ 439 | ÷ ln ⁡ 3.5 x \u0026#x003E; ln ⁡ 439 ln ⁡ 3.5 ≈4.9 Example 2 Solve the inequality 50 ⋅1.05x \u0026#x003E; 300 50 ⋅1.05x \u0026#x003E; 300 | ÷ 50 1.05x \u0026#x003E; 6 ln ⁡ 1.05x \u0026#x003E; ln ⁡ 6 x ln ⁡ 1.05 \u0026#x003E; ln ⁡ 6 | ÷ ln ⁡ 1.05 x \u0026#x003E; ln ⁡ 6 ln ⁡ 1.05 ≈36.7 Example 3 Solve the inequality 3 ⋅0.25x \u0026#x003E; 27 3 ⋅0.25x \u0026#x003E; 27 | ÷3 0.25x \u0026#x003E; 9 lg ⁡ 0.25x \u0026#x003E; lg ⁡ 9 xlg ⁡ 0.25 \u0026#x003E; lg ⁡ 9 | ÷log ⁡ 0.25 x \u0026#x003C; lg ⁡ 9 log ⁡ 0.25 ≈−1.6 ","topic":{"breadcrumbs":[{"_key":"c0cf1a4b-5f7d-5390-9a52-e6ae2428b50c","h1":"Encyclopedia","path":"/encyclopedia"},{"_key":"e9327e4b-42fe-571b-bcb9-83fdb63d5f21","h1":"Algebra","path":"/encyclopedia/algebra"},{"_key":"5746e0ab-24ae-5841-af0f-f1b35cfab6c9","h1":"Logarithms","path":"/encyclopedia/algebra/logarithms"},{"_key":"db5be093-ee70-5f75-bdd2-f3d4e77d9fda","h1":"Logarithmic Equations","path":"/encyclopedia/algebra/logarithms/logarithmic-equations"},{"_key":"22deaf56-6168-575e-8a39-4a3d360193ac","h1":"How to Solve Exponential Inequalities","path":"/encyclopedia/algebra/logarithms/logarithmic-equations/how-to-solve-exponential-inequalities"}],"cssUrl":"/c47dbd75-910f-42be-a3fb-3d98b0d0fbb6-en-1.css","h1":"How to Solve Exponential Inequalities","htmlContent":{"html":" \u003c/p\u003e\u003cp class=\"indent\" \u003e     When solving exponential inequalities it is very important that you look at the value of the argument \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e in the logarithm. That value determines whether you have to turn the inequality sign when you multiply or divide by \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e or \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/math type=\"punct\"\u003e. \u003c/p\u003e      \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-701\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Rule\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"rule-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eSolving\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eexponential\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003einequalities\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003eWhen \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003c/math\u003e, \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/math\u003e and you can solve the inequality as usual. \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                         \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace 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class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                        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mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003c/math\u003e, \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/math\u003e, you will \u003cspan  class=\"ec-lmri-12x-x-144\"\u003ehave to \u003c/span\u003eturn the inequality sign as you end up dividing or multiplying by a negative number! \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                         \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                       \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                        \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                              \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-702\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the inequality \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                          \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-label\"\u003e                          \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                 \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-label\"\u003e                          \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                 \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e                                \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e   \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003c/mfrac\u003e  \u003cmo  class=\"MathClass-rel\"\u003e≈\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e   \u003c/div\u003e  \u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                                                                                                                                                          \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-703\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the inequality \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                        \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                           \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                       \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e                        \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e                        \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e                        \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                            \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                            \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-label\"\u003e                             \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e  \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-rel\"\u003e≈\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                            \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                            \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e   \u003c/div\u003e  \u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                                                                                                                                      \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-704\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 3\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the inequality \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                          \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                               \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                 \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e lg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e lg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                              \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003elg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e lg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                              \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e\u003cmi class=\"qopname\"\u003elog\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  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ready","title.other":"Your personal learning analysis is ready","title.parent":"We are ready to map your children's math skills","title.teacher":"You are now ready to check out the system where all students can learn math every time.","subtitle.student":"You are now ready to map your first math skills in 90 seconds without doing math! You heard right - without doing math!","subtitle.other":"You are now ready to map your first math skills in 90 seconds without doing math! You heard right - without doing math!","subtitle.parent":"You are now ready to map your first math skills in 90 seconds without doing math! You heard right - without doing math!","subtitle.teacher":"It is built to optimize student learning, so the teacher panel is not complete yet."},"math-practice-challenges":{"title":{"student":"My math practice could be more challenging","parent":"I see untapped potential in my children","teacher":"I see untapped potential in many of my students","other":"My math practice could be more challenging"}},"math-routine":{"title":{"student":"Mastering math makes me feel smart","parent":"My support makes a real difference in my children's math progress","teacher":"I have time to make all my students feel seen","other":"Mastering math makes me feel smart"}},"succeed-in-something":{"title.student":"When you get stuck, what do you do first?","title.teacher":"How do you usually celebrate student accomplishments?","title.other":"When you succeed in something, you'd rather...","title.parent":"How do you celebrate your children's accomplishments?","subtitle":"","student":{"think-of-what-could-be-done-better":{"title":"Give up pretty fast"},"celebrate-the-win":{"title":"Search for help"}},"parent":{"think-of-what-could-be-done-better":{"title":"Think about the next step"},"celebrate-the-win":{"title":"Celebrate the win"}},"teacher":{"think-of-what-could-be-done-better":{"title":"Think about the next step"},"celebrate-the-win":{"title":"Celebrate the win"}}},"time-you-commit":{"title.student":"How much time can you commit?","title.teacher":"Do you feel you catch learning gaps in time?","title.other":"How much time can you commit?","title.parent":"Rewards help motivate my children to practice math","subtitle":"","student":{"5-min":{"title":"5 min/day","subtitle":"quick daily practice"},"15-min":{"title":"15 min/day","subtitle":"steady progress"},"30-min":{"title":"30 min/day","subtitle":"Fast improvement"},"60-min":{"title":"60+ min/day","subtitle":"Maximum results"}},"parent":{"5-min":{"title":"Works like a charm","subtitle":"Finally self-going"},"15-min":{"title":"Increases stamina","subtitle":"They do more math"},"30-min":{"title":"Not sure","subtitle":"Does it have an effect?"},"60-min":{"title":"It is noise","subtitle":"They lose focus"}},"teacher":{"5-min":{"title":"Yes, full control","subtitle":"Impressive!"},"15-min":{"title":"Most of the time","subtitle":"We'll take you the last stretch"},"30-min":{"title":"Now and then","subtitle":"We'll help you see all"},"60-min":{"title":"Not really","subtitle":"No worries, we'll fix that"}}},"find-challenging-dragon":{"title":{"parent":{"1":"Helping with math feels difficult and uncertain","2":"You can help sometimes, but it doesn't always feel easy","3":"You generally feel comfortable helping your children with math","4":"Your confidence is a real advantage for your children"},"student":{"1":"Your future isn't decided — still much time to shape it into what you want","2":"Growth often starts with small steps and support","3":"Progress comes from showing up and trying – skills for life!","4":"Effort opens doors to math and a bright future"},"teacher":{"1":"One-size-fits-all homework often falls short","2":"Designing homework for mixed levels is hard","3":"Homework mostly works, but edge cases remain","4":"You already differentiate effectively"},"other":{"1":"Your future isn't decided — still much time to shape it into what you want","2":"Growth often starts with small steps and support","3":"Progress comes from showing up and trying – skills for life!","4":"Effort opens doors to math and a bright future"}},"subtitle":{"parent":{"1":"We'll guide you step by step so you know exactly how to support your children — even if math wasn't your strong subject.","2":"We'll strengthen your understanding and give you clear guidance so you feel more confident when your children need support.","3":"We'll help you stay aligned with what they need most to keep their progress — and grades — moving up.","4":"We'll help you use that strength strategically to support top performance and strong test results."},"student":{"1":"Our founder flunked math in high school and became a mathematician now helping millions master math. We know how to help you. Let's go!","2":"Even a little effort can unlock understanding and new possibilities over time. We are here to get you going and guide you on your way.","3":"Steady effort leads to improvement and prepares you for bigger challenges ahead. We'll take you even one step further.","4":"Working hard builds skills, confidence, and habits that create opportunities in school and in life. We are here to make your journey fast, easy and fun!"},"teacher":{"1":"We'll map individual gaps and generate personalized practice that supports every learner so they progress between every class","2":"We'll identify where students diverge and create targeted paths that meet them where they are. That way all students progress between classes.","3":"We'll help you spot where adjustments can unlock better outcomes for more students. That way all students progress between classes.","4":"You stay in control, with clear insight into how to support each student and maintain progress between classes."},"other":{"1":"Our founder flunked math in high school and became a mathematician now helping millions master math. We know how to help you. Let's go!","2":"Even a little effort can unlock understanding and new possibilities over time. We are her to get you going and guide you on your way.","3":"Steady effort leads to improvement and prepares you for bigger challenges ahead. We'll take you even one step further.","4":"Working hard builds skills, confidence, and habits that create opportunities in school and in life. We are here to make your journey fast, easy and fun!"}}},"personal-math-recipe":{"title.student":{"1":"Your personal learning recipe is 72% ready!","2":"Your personal learning recipe is 72% ready!","3":"Your personal learning recipe is 72% ready!","4":"Your personal learning recipe is 72% ready!"},"title.other":{"1":"Your personal learning recipe is 72% ready!","2":"Your personal learning recipe is 72% ready!","3":"Your personal learning recipe is 72% ready!","4":"Your personal learning recipe is 72% ready!"},"title.parent":{"1":"Your learning analysis is 72% ready!","2":"Your learning analysis is 72% ready!","3":"Your learning analysis is 72% ready!","4":"Your learning analysis is 72% ready!"},"title.teacher":{"1":"Your learning analysis is 72% ready!","2":"Your learning analysis is 72% ready!","3":"Your learning analysis is 72% ready!","4":"Your learning analysis is 72% ready!"},"subtitle.parent":{"1":"Just a little more, and you are ready to give your children a new math start","2":"Just a little more, and you are ready to give your children a new math start","3":"Just a little more, and you are ready to give your children a new math start","4":"Just a little more, and you are ready to give your children a new math start"},"subtitle.teacher":{"1":"Just a little more, and you are ready to give your students a new math start","2":"Just a little more, and you are ready to give your students a new math start","3":"Just a little more, and you are ready to give your students a new math start","4":"Just a little more, and you are ready to give your students a new math start"},"subtitle.other":{"1":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","2":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","3":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","4":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe."},"subtitle.student":{"1":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","2":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","3":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe.","4":"Just a little more and we're set. Tell us a bit more about what you like to learn, and we'll fine-tune your recipe."}},"friendly-push-dragon":{"title":{"student":{"1":"You rarely need external encouragement","2":"An occasional nudge can be helpful","3":"A friendly push is a powerful motivator","4":"Support and encouragement make a big difference for you"},"other":{"1":"You rarely need external encouragement","2":"An occasional nudge can be helpful","3":"A friendly push is a powerful motivator","4":"Support and encouragement make a big difference for you"},"parent":{"1":"Stressful and hard to manage","2":"Often challenging","3":"Mostly manageable","4":"Smooth and stress-free"},"teacher":{"1":"You're not alone! It can be hard to know if the basics are truly sticking","2":"Foundations can look solid until the content ramps up","3":"You're in a good place","4":"It sounds like your students are on a strong track"}},"subtitle":{"student":{"1":"You're highly self-driven. You're confident in staying motivated and moving forward on your own. Excellent trait to have to master math. But being seen is always nice, so we'll be here to cheer you on.","2":"You're generally self-motivated. You usually stay on track without much outside support. We'll be your little friendly reminder to keep going steadily.","3":"We'll tailor practice, explanations, and recommendations to help you keep progressing toward mastery.","4":"We'll proactively guide and motivate you with timely nudges to help you stay focused and make steady progress with your math."},"other":{"1":"You're highly self-driven. You're confident in staying motivated and moving forward on your own. Excellent trait to have to master math. But being seen is always nice, so we'll be here to cheer you on.","2":"You're generally self-motivated. You usually stay on track without much outside support. We'll be your little friendly reminder to keep going steadily.","3":"We'll tailor practice, explanations, and recommendations to help you keep progressing toward mastery.","4":"We'll proactively guide and motivate you with timely nudges to help you stay focused and make steady progress with your math."},"parent":{"1":"Homework at home is usually stressful and difficult for our family to handle.","2":"Homework time can be tough, and my children often get stuck or frustrated.","3":"Most days it goes fine, but sometimes my children need extra time or a bit of help from me.","4":"Homework time at home is usually calm, and my children can finish without much stress."},"teacher":{"1":"A structured baseline check can pinpoint missing prerequisites and give you a clear plan for strengthening their foundation step by step.","2":"A targeted diagnostic can quickly show which core skills need reinforcing so you can focus your time where it matters most","3":"A short skills check can validate the foundation and reveal any small gaps before they become barriers in higher-level topics","4":"A quick mastery snapshot can help confirm consistency across the class and highlight who's ready for stretch challenges"}}},"personal-math-recipe-2":{"title.student":{"1":"Your personal learning recipe is 94% ready!","2":"Your personal learning recipe is 94% ready!","3":"Your personal learning recipe is 94% ready!","4":"Your personal learning recipe is 94% ready!"},"title.other":{"1":"Your personal learning recipe is 94% ready!","2":"Your personal learning recipe is 94% ready!","3":"Your personal learning recipe is 94% ready!","4":"Your personal learning recipe is 94% ready!"},"title.parent":{"1":"Your learning analysis is 94% ready!","2":"Your learning analysis is 94% ready!","3":"Your learning analysis is 94% ready!","4":"Your learning analysis is 94% ready!"},"title.teacher":{"1":"Your learning analysis is 94% ready!","2":"Your learning analysis is 94% ready!","3":"Your learning analysis is 94% ready!","4":"Your learning analysis is 94% ready!"},"subtitle.student":{"1":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","2":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","3":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","4":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps."},"subtitle.other":{"1":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","2":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","3":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps.","4":"Just a few more answers, and we'll tailor your personal learning experience with the right starting level and next steps."},"subtitle.parent":{"1":"You are nearly there now!","2":"You are nearly there now!","3":"You are nearly there now!","4":"You are nearly there now!"},"subtitle.teacher":{"1":"You are nearly there now!","2":"You are nearly there now!","3":"You are nearly there now!","4":"You are nearly there now!"}},"succeed-in-something-dragon":{"title":{"student":{"1":"Getting stuck can feel frustrating and overwhelming","2":"Not giving up is a winning strategy in math and life"},"other":{"1":"Getting stuck can feel frustrating and overwhelming","2":"Not giving up is a winning strategy in math and life"},"parent":{"1":"You're a math improver","2":"You balance progress and celebration"},"teacher":{"1":"You're forward-looking by nature","2":"You balance progress and celebration"}},"subtitle":{"parent":{"1":"You like refining your skills and spotting what to fix next. 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We'll be your No. 1 resource in these moments."}}},"like-math-dragon":{"title":{"parent":{"1":"Math feels challenging and not very enjoyable for you","2":"Math is not your favorite subject, but you're willing to work with it ","3":"You generally like math and enjoy working with it","4":"Your enthusiasm is key to achieve success"},"student":{"1":"Math feels challenging and not very enjoyable for you","2":"Math is not your favorite subject, but you're willing to work with it","3":"You generally like math and enjoy working with it","4":"Your enthusiasm is key to achieve success"},"teacher":{"1":"You handle mixed-ability classrooms with confidence.","2":"You've found ways to manage differences in ability","3":"Different ability levels is a challenge in some lessons","4":"Teaching wide ability ranges can feel overwhelming"},"other":{"1":"Math feels challenging and not very enjoyable for you","2":"Math is not your favorite subject, but you're willing to work with it","3":"You generally like math and enjoy working with it","4":"Your enthusiasm is key to achieve success"}},"subtitle":{"parent":{"1":"Don't worry! 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We help you build real understanding to keep your momentum going."},"teacher":{"1":"We'll help you prioritize and use small moments more effectively so every student feels noticed and supported","2":"We'll help you spot where small adjustments can make a big difference","3":"We'll help you turn those moments into consistent, meaningful support for everyone","4":"We'll help you scale that care so no one slips through the cracks"},"other":{"1":"Progress isn't always loud, but the work you put in matters and deserves recognition.","2":"Even if it doesn't excite you yet, we'll help make progress feel clearer—and worth noticing.","3":"With the right guidance and practice, we help turn steady progress into moments where math really clicks.","4":"That feeling matters. We help you build real understanding to keep your momentum going."}}},"feel-math-skills-dragon":{"title":{"parent":{"1":"Motivation is a real challenge right now","2":"Motivation comes and goes","3":"Your children are generally motivated.","4":"Your children are highly motivated"},"student":{"1":"We'll make you feel less nervous and more confident before your math tests","2":"We'll make you feel less nervous before your math tests","3":"Math tests is a moment of  happiness for you","4":"You love taking math tests and we understand why!"},"teacher":{"1":"Motivation is often a challenge","2":"Motivation comes and goes","3":"Most students are generally motivated","4":"Your students are highly motivated to learn math"},"other":{"1":"We'll make you feel less nervous and more confident before your math tests","2":"We'll make you feel less nervous before your math tests","3":"Math tests is a moment of  happiness for you","4":"You love taking math tests and we understand why!"}},"subtitle":{"parent":{"1":"We'll help make progress visible and rewarding, so your children stay engaged and confident","2":"We'll support you with strategies that keep your children focused and moving forward — especially when things get hard.","3":"We'll help maintain that momentum with clear goals and progress tracking","4":"We'll help channel that drive into consistent improvement and strong test performance."},"student":{"1":"That's okay. 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And we will get you on the right track."},"teacher":{"1":"We'll help you focus on practical strategies that make progress possible, for all students","2":"We'll support you in using our platform to expand your reach without adding pressure","3":"We'll help you turn that belief into consistent, classroom-wide impact","4":"We'll help you put them to work so all learners can succeed in your classroom"},"other":{"1":"That's a powerful mindset. You believe math is learnable, and with effort and practice you can keep improving. We'll help you strengthen your foundation, close any gaps, and move forward with confidence.","2":"Even if it feels challenging at times, we'll help you strengthen confidence as your skills grow.","3":"We'll help you build understanding step by step and show that math skills can be learned.","4":"Millions of students feel like this BUT it is a misconception. The reason for this feeling is that you are trying ot learn math while having too many knowledge gaps. 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","enterItHere":"Enter it here","reactivatePlan":"Reactivate subscription","discountCode":"Discount code","discountCodePlaceholder":"Code","applyDiscount":"Apply","dismissDiscount":"Remove discount code","discountApplied":"Discount applied","maxLicenses":"Maximum 10 licenses","partnerCoupon":{"appliedLabel":"APPLIED PARTNER DISCOUNT","dnbTitle":"DNB Partner","knowitTitle":"Knowit Partner","dnbAppliedVia":"Applied via DnB partnership","knowitAppliedVia":"Applied via Knowit partnership","badgeLabel":"Partner pricing","codeLabel":"Partner code","description":"Your partner discount is already applied. 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