Question

Solve each equation. Check the solution.

$\left|\text{x}+4\right|+3=17$

Short answer

The value of $\text{x}$ is$\mathbf{10}$.

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:

$\left|x+4\right|+3=17$

Case 1:

$\begin{array}{c}a=b\\ x+4+3=17\\ x+7=17\\ x+7-7=17-7\\ x=10\end{array}$

Or Case 2:

$\begin{array}{c}a=-b\\ x+4+3=-17\\ x+7=-17\\ x+7-7=-17-7\\ x=-24\end{array}$

The solution is $10$and $-24$.

- Check the solution.

For case 1: $x=10$.

Check:

$\begin{array}{r}\left|x+4\right|+3=17\\ \left|10+4\right|+3=17\\ \left|14\right|+3=17\\ 14+3=17\\ 17=17\end{array}$

Thus, it is correct.

For case 2: $x=-24$.

Check:

$\begin{array}{r}\left|x+4\right|+3=17\\ \left|-24+4\right|+3=17\\ \left|-20\right|+3=17\\ 20+3=17\\ \xcancel{23=17}\end{array}$

Thus, it is not correct.

For $x=-24$, the soultion is not correct.

- State the conclusion.

Since, $23\ne 17$that is for $x=-24$, the soultion is not correct, then there is only one that is 10. Therefore, the solution set for x is $\left\{10\right\}$.