- Example 1 - Solve the inequality \(2^{x-1} > 8\)
Solution - \begin{eqnarray}2^{x-1} &>& 8\\\nonumber 2^{x-1} &>& 2^3 \\\nonumber x-1 &>& 3 \\\nonumber x &>& 4 \Rightarrow x \in \langle 4, \infty \rangle\end{eqnarray} - Example 2 Solve the inequality \(0.4^{2x-3} \leq \frac{5}{2}\)
Solution - \begin{eqnarray}0.4^{2x-3} &\leq& \frac{5}{2}\\\nonumber \left(\frac{4}{10}\right)^{2x-3} &\leq& \left(\frac{5}{2}\right) \\\nonumber \left(\frac{5}{2}\right)^{-2x + 3} &\leq& \left(\frac{5}{2}\right)\\\nonumber -2x + 3 &\leq& 1 \\\nonumber -2x &\leq& -2 /:(-2) \\\nonumber x &\geq& 1\Rightarrow x \in [1, +\infty\rangle\end{eqnarray} - Example 3 Solve the inequality \(3^{x-3} < 4 \)
Solution - \begin{eqnarray}3^{x-3} &<& 4/\log \\\nonumber (x-3) \log3 &<& \log 4/:\log 3 \\\nonumber x &<& \frac{\log 4}{\log 3} + 3 \\\nonumber x &<& 4.26 \end{eqnarray} - Example 4 Solve the inequality \(0.5^x > 32\)
Solution - \begin{eqnarray}0.5^x &>& 32\\\nonumber \left(\frac{1}{2}\right)^{x} &>& 2^5 \\\nonumber 2^{-x} &>& 2^5\\\nonumber -x &>& 5/:(-1) \\\nonumber x &<& -5\end{eqnarray} - Example 5 Solve the inequality \(\left(\frac{1}{2}\right)^{x^2 + 3x} \leq 0.125 \cdot 2^{-x}\)
Solution - \begin{eqnarray}\left(\frac{1}{2}\right)^{x^2 + 3x} &\leq& 0.125 \cdot 2^{-x} \\\nonumber 2^{-x^2 - 3x} &\leq& 2^{-x-3}\\\nonumber -x^2 -3x &\leq& -x -3 \\\nonumber -x^2 - 2x - 3 &\leq& 0 \end{eqnarray} Solve quadratic inequaolity \begin{eqnarray}x_{1,2} &=& \frac{2 \pm \sqrt{4+12}}{-2}\\\nonumber x_{1,2} &=& \frac{2\pm 4}{-2} \\\nonumber x_1 &=& -3 \\\nonumber x_2 &=& 1\end{eqnarray} The zero points or roots of quadratic function are determined. Since the quadratic function has leading coefficient (\(a = -1)\)) negative the function is turned downwards. The solution of the inequaltiy can be written as: $$x \in \langle -\infty, -3 ] \cup [ 1, +\infty \rangle $$ - Example 6 Solve the inequality \(4^x < 5\)
Solution - \begin{eqnarray}4^x &<& 5\\\nonumber 2^2x &<& 5 /\log_2 \\\nonumber 2x \log_2 2&<& \log_2 5 \\\nonumber x &<& \frac{1}{2} \log_2 5\end{eqnarray} - Example 7 Solve the inequality \(-3^{x^2} > 3^{-|x|}\)
Solution - \begin{eqnarray}-3^{x^2} &>& 3^{-|x|}\\\nonumber -x^2 + |x|&>& 0/:(-1) \\\nonumber x^2 - |x| &<& 0 \end{eqnarray} The solution to the first inequality \begin{eqnarray} \\\nonumber x^2 - x&<& 0 \\\nonumber x_{1,2} &=& \frac{1\pm\sqrt{1}}{2} \\\nonumber x_1 &=& 1 \\\nonumber x_2 &=& 0\end{eqnarray} The solution of the second inequality \begin{eqnarray}x^2 + x &>& 0 \\\nonumber x_{1,2} &=& \frac{-1\pm 1}{2} \\\nonumber x_1 &=& -1 \\\nonumber x_2 &=& 0\end{eqnarray} Solution of the initial inequality can be written in the following form $$ x \in \langle -1, 1 \rangle \ {0}$$ - Example 8 Solve the inequality \(6^{2x+3} < 2^{x+7} \cdot 3^{3x-1}\)
Solution - \begin{eqnarray}6^{2x+3} &<& 2^{x+7} \cdot 3^{3x-1}/:6^{2x}\\\nonumber 6^3 &<& \frac{2^x \cdot 3^{2x}}{6^2x} \cdot 2^7 \cdot 3^{-1} \\\nonumber 2^{x-2x} \cdot 3^{3x-2x} \cdot 128 \cdot \frac{1}{3} &>& 216 \\\nonumber \left(\frac{3}{2}\right)^x \cdot \frac{128}{3} &>& 216/\cdot \frac{3}{128} \\\nonumber \left(\frac{3}{2} \right)^x &>& \frac{648}{128}\\\nonumber \left(\frac{3}{2}\right)^x &>& \left(\frac{3}{2}\right)^4 \\\nonumber x&>& 4\\\nonumber x \in \langle 4 ,+\infty \rangle\end{eqnarray} - Example 9 Solve the inequality \(9^x - 3^{x+1} + 2\geq 0\)
Solution - \begin{eqnarray}9^x - 3^{x+1} + 2&\geq& 0 \Rightarrow 3^x =t\\\nonumber t^2-3t + 2 &geq& 0 \\\nonumber t_{1,2} &=& \frac{3 \pm \sqrt{9 - 8}}{2} \\\nonumber t_1 &=& \frac{4}{2} = 2 \\\nonumber t_2 &=& \frac{3-1}{2} = 1\\\nonumber 3^{x_1} &\geq& t_1 /\log_3\\\nonumber x_1 &\geq& \log_3 2 \\\nonumber 3^{x_2} &\leq& t_2/\log_3 \\\nonumber x_2 &\leq& 0\end{eqnarray} Solution of the inequality can be written as: $$ \langle -\infty, 0 ] \cup [\log_3 2, +\infty\rangle$$ - Example 10 Solve the inequality \(2^{\log(2x-3)} > \frac{1}{2}\)
Solution - \begin{eqnarray}2^{\log(2x-3)} &>& \frac{1}{2} \\\nonumber 2^{\log(2x-3)} &>& 2^{-1} \\\nonumber \log(2x-3) &>& -1 \\\nonumber 2x-3 &>& \frac{1}{10} \\\nonumber 2x &>& 3 + \frac{1}{10} \\\nonumber 2x &>& \frac{30+1}{10}/:2 \\\nonumber x &>& \frac{31}{20} \end{eqnarray} - Example 11
Solution - \begin{eqnarray}\end{eqnarray} - Example 12 Solve the inequality \(0.7^{\frac{x^2+9}{x+3}} > 1\)
Solution - This inequality is only valid if denominator of the fraction in the exponent is strictly greater than 0. $$ x + 3 >0 \Rightarrow x > -3 \Rightarrow x \in \langle -\infty, -3 \rangle$$ - Example 13 Solve the inequaltiy \(\frac{10^{\frac{1}{3}} \cdot 1000}{10^x} < 0.01 \)
Solution - \begin{eqnarray}\frac{10^{\frac{1}{3}} \cdot 1000}{10^x} &<& 0.01\\\nonumber 10^{\frac{1}{3}}\cdot 10^3 \cdot 10^{-x} &<& 10^{-2}\\\nonumber 10^{\frac{1+9}{3}} \cdot 10^{-x} &<& 10^{-2}\\\nonumber 10^{frac{10}{3}} \cdot 10^{-x} &<& 10^{-2} /\cdot 10^{-\frac{10}{3}} \\\nonumber 10^{-x} &<& 10^{\frac{-10-6}{3}}\\\nonumber -x &<& -\frac{16}{3}/\cdot (-1) \\\nonumber x&>& \frac{16}{3} \Rightarrow x \in \langle \frac{16}{3}, +\infty\rangle\end{eqnarray} - Example 14 Solve the inequality \(\left(\frac{3}{5}\right)^x \geq \left(1\frac{2}{3}\right)^3\)
Solution - \begin{eqnarray}\left(\frac{3}{5}\right)^x &\geq& \left(\frac{5}{3}\right)^3 \\\nonumber \left(\frac{3}{5}\right)^x &\leq& \left(\frac{3}{5}\right)^{-3} \\\nonumber x &\leq& -3 \\\nonumber x &\in& \langle -\infty, -3]\end{eqnarray} - Example 15 Solve the inequality \(10^x\leq 0.1 \cdot 1000^{x-1}\)
Solution - \begin{eqnarray}10^x &\leq& 0.1 \cdot 1000^{x-1}\\\nonumber 10^x &\leq& 10^{-1} \cdot 10^{3x-3}\\\nonumber 10^x &\leq& 10^{3x-4} \\\nonumber x &\leq& 3x -4 \\\nonumber -2x &\leq& -4 /\cdot (-\frac{1}{2}) \\\nonumber x &\geq& 2 \\\nonumber x &\in& [2, +\infty\rangle\end{eqnarray} - Example 16 Solve the inequality \(4^{x-2} > 0.125\)
Solution - \begin{eqnarray}4^{x-2} &>& 0.125\\\nonumber 2^{2x-4} &>& \frac{1}{8}\\\nonumber 2^{2x-4} &>& 2^{-3} \\\nonumber 2x-4 &>& -3 \\\nonumber 2x &>& 1/:2 \\\nonumber x &>& \frac{1}{2} \\\nonumber x &\in& \langle \frac{1}{2}, +\infty\rangle\end{eqnarray} - Example 17 Solve the inequality \(\frac{1}{5^{2x-4}}< 125 \)
Solution - \begin{eqnarray}\frac{1}{5^{2x-4}} &<& 125\\\nonumber 5^{-2x+4} &<& 5^3 \\\nonumber -2x + 4 &<& 3 \\\nonumber -2x &<& -1/:(-1) \\\nonumber x &>& \frac{1}{2}\\\nonumber x &\in& \Big\langle \frac{1}{2}, +\infty\Big\rangle\end{eqnarray} - Example 18 Solve the inequality \(0.5^{2x^2 + 2} \geq 0.5^{5x}\)
Solution - \begin{eqnarray}0.5^{2x^2 + 2} &\geq& 0.5^{5x}\\\nonumber \left(\frac{1}{2}\right)^{2x^2 +2} &\geq& \left(\frac{1}{2}\right)^{5x}\\\nonumber 2^{-2x^2 -2} &\geq& -5x /\cdot(-1) \\\nonumber -2x^2 - 2 &\geq& -5x \\\nonumber -2x^2 +5x -2 &\geq& 0 /\cdot(-1) \\\nonumber 2x^2 -5x +2 &\leq& 0 \\\nonumber x_{1,2} &=& \frac{5 \pm\sqrt{25 - 4\cdot 2 \cdot 2}}{4}\\\nonumber x_{1,2} &=& \frac{5 \pm 3}{4}\\\nonumber x_1 &=& 2 \\\nonumber x_2 &=& \frac{1}{2} \\\nonumber x &\in& \Big\langle \frac{1}{2}, 2 \Big\rangle\end{eqnarray} - Example 19 Solve the inequality \(4^{2x-3} \leq \left(\frac{1}{2}\right)^3\)
Solution - \begin{eqnarray}4^{2x-3} &\leq& \left(\frac{1}{2}\right)^3\\\nonumber 2^{4x-6} &\leq& 2^{-3}\\\nonumber 4x -6 &\leq& -3 \\\nonumber 4x &\leq& 3 /: 4 \\\nonumber x&\leq& \frac{3}{4} \\\nonumber x &\in& \Big\langle -\infty, \frac{4}{3}\Big]\end{eqnarray} - Example 20 Solve the inequality \(2^x\cdot 5^x > 0.1(10^{x-1})^5\)
Solution - \begin{eqnarray}2^x\cdot 5^x &>& 0.1(10^{x-1})^5 \\\nonumber 10^x &>& \frac{1}{10} 10^{5x-5} \\\nonumber 10^x &>& 5x- 6 \\\nonumber 10^{-4x} &>& 10^{-6}q \\\nonumber -4x &>& - 6/:(-4)\\\nonumber x &<& \frac{3}{2}\\\nonumber x&\in& \Big\langle -\infty, \frac{3}{2}\Big\rangle \end{eqnarray} - Example 21 Solve the inequality \(3^{2x-3} \geq \left(\frac{1}{2}\right)^{-2}\)
Solution - \begin{eqnarray}3^{2x-3} &\geq&\left(\frac{1}{2}\right)^{-2}\\\nonumber 3^{2x-3} &\geq& 2^2\\\nonumber 2x-3 &\geq& 2\\\nonumber 2x &\geq& 5/:2 \\\nonumber x&\geq& \frac{5}{2}\\\nonumber x&\in& \Big[\frac{5}{2}, +\infty \Big \rangle\end{eqnarray} - Example 22 Solve the inequality \(27^{2x-2} > 81^{3x-5}\)
Solution - \begin{eqnarray}27^{2x-2} &>& 81^{3x-5} \\\nonumber 3^{6x-6} &>& 3^{12x-20} \\\nonumber 6x-6 &>& 12x -20 \\\nonumber -6x &>& -14 /: (-6) \\\nonumber x &<& \frac{7}{3} \\\nonumber x &\in& \Big\langle-\infty, \frac{7}{3}\Big\rangle\end{eqnarray}
How to solve exponential inequality ?
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