Chapter 3, Problem 3CCSR

Solution

To identify: The graph that represents the solution of the inequality $\left|3x+12\right|\ge 3$ .

The given options are:

  

Option A is the correct option.

Given information:

The given inequality is $\left|3x+12\right|\ge 3$ .

Explanation:

Consider the inequality:

  $\left|3x+12\right|\ge 3$

The inequality is equivalent to:

  $3x+12\ge 3\text{and}3x+12\ge -3$

Now solve each inequality,

  $3x+12\ge 3$

Subtract 12 from each side,

  $\begin{array}{c}3x+12-12\ge 3-12\\ 3x\ge -9\end{array}$

Divide each side by 3,

  $\begin{array}{c}\frac{3x}{3}\ge \frac{-9}{3}\\ x\ge -3\end{array}$

Now solve the inequality,

  $3x+12\ge -3$

Subtract 12 from each side,

  $\begin{array}{c}3x+12-12\ge -3-12\\ 3x\ge -15\end{array}$

Divide each side by 3,

  $\begin{array}{c}\frac{3x}{3}\ge \frac{-15}{3}\\ x\ge -5\end{array}$

The inequality will be satisfied if $-5\le x\ge -3$ .

Now draw the graph for the inequality $\left|3x+12\right|\ge 3$:

  

Therefore, the correct option is A .