Chapter 1.6, Problem 19PPS

To determine

To solve: the given inequality and graph the solution on number line.

Answer to Problem 19PPS

The solution is given by $-\frac{7}{8}\le t\le \frac{1}{8}$ .

Explanation of Solution

Given information:

Given inequality

  $\left|8t+3\right|\le 4$

Concept used:

Absolute value of a function a is given by

  $\left|a\right|=\{\begin{array}{l}aa>0\\ -aa<0\\ 0a=0\end{array}$

And

  $\begin{array}{l}\left|-a\right|=a\\ \left|a\right|=a\end{array}$

More over

  $\begin{array}{l}\left|x+a\right|\le b\\ \Rightarrow -b\le x+a\le b\end{array}$

Calculation:

Consider the given inequality,

  $\left|8t+3\right|\le 4$

Now, solve the inequality as shown,

  $\begin{array}{c}\left|8t+3\right|\le 4\\ -4\le 8t+3\le 4\\ -4-3\le 8t+3-3\le 4-3\\ -7\le 8t\le 1\\ -\frac{7}{8}\le t\le \frac{1}{8}\end{array}$

Hence, the solution of the inequality is $-\frac{7}{8}\le t\le \frac{1}{8}$ .

Now, graph the inequality as shown:

The closed ball suggests that the solution region is included the points $-\frac{7}{8}$ and $\frac{1}{8}$ .

The sign indicates the region is left side to the point $\frac{1}{8}$ and right side to the point $-\frac{7}{8}$ on number line.