Chapter 4, Problem 7STP

To determine

Tofind:the solution of the inequality equation and choose the correct option given below.

Answer to Problem 7STP

The solution of the given equation lies in the interval of $\left(-\infty ,-15\right]\cup \left(1,\infty \right]$ .

Explanation of Solution

Given:

  $6-\left|\text{x+7}\right|\le -2$ .

Concept used:

In equality the symbol of less than or greater than will change if both sides cancel the negative sign.

If $\text{-x}\ge -\text{2}$

Then

  $\text{x}\le \text{2}$ .

Calculation:

  $6-\left|\text{x+7}\right|\le -2$ .

  $-\left|\text{x+7}\right|\le -8$ .

  $\left|\text{x+7}\right|\ge 8$ .

To solve the inequality first solve the equation.

  $\text{x+7=8}$ and $\text{x+7=-8}$ .

  $\text{x=1 and x=-15}$ .

Now consider the graph of $\text{f}\left(\text{x}\right)\text{=}\left|\text{x+7}\right|\text{-8}$ .

  

Since $\left|\text{x+7}\right|\ge 8$ . Then the graph of the function must be greater than $\text{y=0}$ and the two intervals of the given inequality are:

  $\left(-\infty ,-15\right]\cup \left(1,\infty \right]$ .

Hence, the solution of the given equation lies in the interval of $\left(-\infty ,-15\right]\cup \left(1,\infty \right]$ .