Q39. - Algebra 2

Question

Solve each inequality. Graph the solution set on a number line.

$\left|3w+2\right|\le 5$

Step-by-Step Solution

Answer

Solution of inequality is $-\frac{7}{3}\le w\text{ and }w\le 1$

Solution set in number line is

Step 1 - Definition of absolute value inequality

For real numbers $a\text{ and }b$ with b>0

Absolute value inequality is $\left|a\right|<b$ when $-b<a\text{ and }a<b$, and

Absolute value inequality is $\left|a\right|>b$ when $a<-b\text{ or }a>b$ 

Step 2 - Write properties of inequality

Properties of inequalities are written in following table:

Step 3 - Simplify 3 w + 2 &#8804; 5

From definition of absolute value inequality, given expression is simplified to 

$$" width="9" height="19" style="max-width: none; vertical-align: -4px;" >$-5\le 3w+2\text{ and }3w+2\le 5$

Use subtraction and division property of inequality, subtract with same number does not change the sign of inequality. Also division by same positive number does not reverse the sign of inequality as shown below

$\begin{array}{c}-5-2\le 3w+2-2\text{ and }3w+2-2\le 5-2\\ -7\le 3w\text{ and }3w\le 3\\ -\frac{7}{3}\le \frac{3w}{3}\text{ and }\frac{3w}{3}\le \frac{3}{3}\\ -\frac{7}{3}\le w\text{ and }w\le 1\end{array}$

Step 4 - Graph the solution set

For $-\frac{7}{3}\le w\text{ and }w\le 1$, number line starts from $-\frac{7}{3}$ to 1

Dot on $-\frac{7}{3}\text{ and 1}$ shows these points are included in the solution.