Question

Solve each equation. Check the solution.

If $8\left|4\text{x}-3\right|=64$

Short answer

The value of $\text{x}$ is$\mathbf{2}.\mathbf{75}$ and $-\mathbf{1}.\mathbf{25}$.

Step-by-step solution

- State the definition of absolute value.

The distance of a number from 0 in the number line is called the absolute value and since distance cannot be negative so absolute value is always positive.

- Solve the inequality.

If x is a number, then absolute value of x is written as $\left|x\right|$.

Here, the inequality is:

$8\left|4x-3\right|=64$

Case 1:

$\begin{array}{c}a=b\\ 8\left(4x-3\right)=64\\ 32x-24=64\\ 32x-24+24=64+24\\ 32x=88\\ x=2.75\end{array}$

Or Case 2:

$\begin{array}{c}a=-b\\ 8\left(4x-3\right)=-64\\ 32x-24=-64\\ 32x-24+24=-64+24\\ 32x=-40\\ x=-1.25\end{array}$

The solution is $2.75$and $-1.25$

- Check the solution.

For case 1: $x=2.75$.

Check:

$\begin{array}{r}8\left|4x-3\right|=64\\ 8\left|\left(4\times 2.75\right)-3\right|=64\\ 8\left|11-3\right|=64\\ 8\left|8\right|=64\\ 8\times 8=64\\ 64=64\end{array}$

Thus, it is correct.

For case 2: $x=-1.25$.

Check:

$\begin{array}{r}8\left|4x-3\right|=64\\ 8\left|\left(4\times -1.25\right)-3\right|=64\\ 8\left|-5-3\right|=64\\ 8\left|-8\right|=64\\ 8\times 8=64\\ 64=64\end{array}$

Thus, it is also correct.

- State the conclusion.

Therefore, the solution set for x is $\left\{2.75,-1.25\right\}$.