Q. 30PT - Algebra 2

Question

Solve each inequality. Describe the solution set using set builder or interval notation. Then graph the solution set on a number line.

 $\left|\mathbf{3}\mathbf{y}\mathbf{-}\mathbf{1}\right|>\mathbf{5}$

Step-by-Step Solution

Answer

Solution set of inequality is $\left\{\mathbf{y}\mathbf{|}\mathbf{y}<\mathbf{-}\frac{\mathbf{4}}{\mathbf{3}}\mathbf{ory}>\mathbf{2}\right\}$.

Solution set on number line is 

Step 1. Write properties of inequality.

Properties of inequalities are written in following table:

Step 2. Description of step.

For all real numbers p] and q, $q>0$,

Case1) If $\left|p\right|<q$ then $-q<p<q$.

Case2) If $\left|p\right|>q$ then $p>q\text{ or }p<-q$.

Step 3. Description of step.

Apply case 1 to $\left|3y-1\right|>5$ which implies $3y-1>5$ or $3y-1<-5$.

Step 4. Description of step.

To solve given inequality, use subtraction and division properties of inequality. Add 1 on each side then divide each side by 3 and simplify.

$\begin{array}{c}3y-1+1>5+1\\ \frac{3y}{3}>\frac{6}{3}\\ y>2\end{array}$

Step 5. Description of step.

Add 1 on each side then divide each side by 3 and simplify.

$\begin{array}{c}3y-1+1<-5+1\\ \frac{3y}{3}<\frac{-4}{3}\\ y<\frac{-4}{3}\end{array}$

Thus, solution is $y>2$ or $y<-\frac{4}{3}$.

Step 6. Describe solution set in interval notation.

As solution set of the inequality is all real numbers less than $-\frac{4}{3}$ or greater than 2.

Write the solution set builder notation.

$\left\{y|y<-\frac{4}{3}\text{ or }y>2\right\}$

Step 7. Graph the solution set.

Open circle on 2 and $-\frac{4}{3}$ shows these points are excluded in the solution set.