Chapter 1, Problem 11PT

To determine

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

Answer to Problem 11PT

The solution is given by $-7\le p\le 15$ .

Explanation of Solution

Given information:

Given inequality

  $\left|p-4\right|\le 11$

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,

  $\left|p-4\right|\le 11$

Now, solve the inequality as shown,

  $\begin{array}{c}\left|p-4\right|\le 11\\ -11\le p-4\le 11\\ -11+4\le p-4+4\le 11+4\\ -7\le p\le 15\end{array}$

Hence, the solution of the inequality is $-7\le p\le 15$ .

Now, graph the inequality as shown:

The closed ball suggests that the solution region is included the points $-7$ and $15$ .

The sign indicates the region is left side to the point $15$ and right side to the point $-7$ on number line.