Chapter 2.4, Problem 15E

To determine

Tosolve:The inequality $6\ge -6\left(a+2\right)$ and graph the solution.

Explanation of Solution

Given information:

The given inequality is $6\ge -6\left(a+2\right)$ .

Formulaused:

The inequality $\frac{x+a}{c}\ge b\Rightarrow x\ge bc-a$ ,

Calculations:

The given inequality is $6\ge -6\left(a+2\right)$ .

Now, $6\ge -6\left(a+2\right)$

  $\Rightarrow \frac{6}{6}\ge -\frac{6\left(a+2\right)}{6}$

  $\Rightarrow 1\ge -\left(a+2\right)$

$\Rightarrow 1\times -1\le -\left(a+2\right)\times -1$$\because a\le b\Rightarrow -a\ge -b$

  $\Rightarrow -1\le a+2$

  $\Rightarrow -1-2\le a+2-2$

  $\Rightarrow -3\le a$ .

Thus, the solution o the inequality $6\ge -6\left(a+2\right)$ is $-3\le a$

The graph of inequality $6\ge -6\left(a+2\right)$ taking variable “ a ” along horizontal axis is shown in Figure-1 here.

Now, $d+9<-3\Rightarrow d<-3-9=-12$ and $d+9>3\Rightarrow d>3-9=-6$

Thus, the solution of given inequality is $d<-12$ and $d>-6$ .

The graph of the solution is shaded region shown in Figure-1.

Interpretation:

From the graph, we see that the solution of inequality region of xy -plane defined by inequalities $d<-12$ and $d>-6$ .