Chapter 1.6, Problem 21PPS

To determine

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

Answer to Problem 21PPS

The solution is given by $j\le -\frac{16}{5}\text{or}j\ge -\frac{8}{5}$ .

Explanation of Solution

Given information:

Given inequality

  $\left|-5j-4\right|\ge 12$

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

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

Calculation:

Consider the given inequality,

  $\left|-5j-4\right|\ge 12$

Now, solve the inequality as shown,

  $\begin{array}{c}\left|-5j-4\right|\ge 12\\ \left(-5j-4\right)\ge 12\text{or}\left(-5j-4\right)\le -12\\ \left(-5j-4+4\right)\ge 12+4\text{or}\left(-5j-4+4\right)\le -12+4\\ 5j\le -16\text{or}5j\ge -8\\ j\le -\frac{16}{5}\text{or}j\ge -\frac{8}{5}\end{array}$

Hence, the solution of the inequality is $j\le -\frac{16}{5}\text{or}j\ge -\frac{8}{5}$ .

Now, graph the inequality as shown:

  

The closed ball suggests that the solution region is included the points $-\frac{16}{5}$ and $8$ .

The sign indicates the region is left side to the point $-\frac{16}{5}$ and right side to the point $-\frac{8}{5}$ on number line.