Question

How do you solve the exponential inequality ${{5}^{x-4}}\le {{25}^{x-6}}$?

Answer 1

To solve the exponential inequality first we will make the base on both sides of the inequality same. Then if we get the same base on both sides we will compare the powers by using the same inequality sign. By simplifying the obtained equation we will get the desired answer.

Complete step-by-step solution:
We have been given the exponential inequality ${{5}^{x-4}}\le {{25}^{x-6}}$.
We have to solve the given equation.
Now, we know that we can write 25 as $25={{5}^{2}}$
Substituting the value in the given equation we will get
$\Rightarrow {{5}^{x-4}}\le {{5}^{2}}^{\left( x-6 \right)}$
Now, as the bases on both sides of the equation are the same so we can compare the powers. Then we will get
$\Rightarrow x-4\le 2\left( x-6 \right)$
Now, simplifying the obtained equation we will get
$\begin{aligned} & \Rightarrow x-4\le 2x-12 \\\ & \Rightarrow x-2x\le -12+4 \\\ & \Rightarrow -x\le -8 \\\ & \Rightarrow x\ge 8 \\\ \end{aligned}$
Hence by solving the given exponential inequality we get the value as x must be greater than or equal to 8.

Note: The point to be noted is that if we multiply or divide both sides by a negative number or when we swap values from LHS to RHS or vice versa, the direction of inequality changes. Alternative way to solve the exponential inequality is to apply logarithm on both sides of the equation. Then by applying the logarithm properties we can solve the equation and get the same value as obtained in the solution.