Question

Solve the following equation: $-8 < -\left( 3x-5 \right) < 13$?

Answer 1

Hint: We start solving the problem by considering one of the inequalities given in the equation $-8 < -\left( 3x-5 \right)$ and making the necessary calculations to get the solution set for x which satisfies this inequality. We then consider the next inequality $-\left( 3x-5 \right) < 13$ and make the necessary calculations to get the solution set for x which satisfies this inequality. We then check the both obtained solution sets and take the common solution set for x in order to get a solution set for the given solution.

Complete step by step answer:
According to the problem, we need to solve the given equation: $-8 < -\left( 3x-5 \right) < 13$.
Let us consider $-8 < -\left( 3x-5 \right)$.
$\Rightarrow -8 < -3x+5$.
$\Rightarrow 3x < 8+5$.
$\Rightarrow 3x < 13$.
$\Rightarrow x < \dfrac{13}{3}$ ---(1).
Now, let us consider $-\left( 3x-5 \right) < 13$.
$\Rightarrow -3x+5 < 13$.
$\Rightarrow 5-13 < 3x$.
$\Rightarrow 3x > -8$.
$\Rightarrow x > \dfrac{-8}{3}$ ---(2).
From equations (1) and (2), we get the interval of values of x as $\dfrac{-8}{3} < x < \dfrac{13}{3}$.

So, we have found the solution set for the given equation $-8 < -\left( 3x-5 \right) < 13$ is $\dfrac{-8}{3} < x < \dfrac{13}{3}$.

Note: We can also the report the obtained solution set of x for the given equation as $\left( \dfrac{-8}{3},\dfrac{13}{3} \right)$. We can also solve the given problem as shown below:
We have given $-8 < -\left( 3x-5 \right) < 13$ ---(1).
Let us multiply the equation (1) with -1 on both sides.
So, we get $8 > \left( 3x-5 \right) > -13$ ---(2).
Now, let us add the equation (2) with 5.
So, we get $8+5 > 3x-5+5 > -13+5$.
$\Rightarrow 13 > 3x > -8$ ---(3).
Now, let us divide the equation (3) with 3.
$\Rightarrow \dfrac{13}{3} > \dfrac{3x}{3} > \dfrac{-8}{3}$.
$\Rightarrow \dfrac{-8}{3} < x < \dfrac{13}{3}$.
So, the solution set for x to satisfy the equation $-8 < -\left( 3x-5 \right) < 13$ is $\dfrac{-8}{3} < x < \dfrac{13}{3}$.