Q. 28 - 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.

 $\left|\mathbf{5}\mathbf{+}\mathbf{k}\right|\le \mathbf{8}$

Step-by-Step Solution

Answer

Solution set of inequality is $[\mathbf{-}\mathbf{13}\mathbf{,}\mathbf{3}]$.

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 $\left|5+k\right|\le 8$ which implies $5+k\le 8$ or $5+k\ge -8$.

Step 4. Description of step.

To solve given inequality, use subtraction property of inequality. Subtract 5 from each side and simplify.

 $\begin{array}{c}5+k-5\le 8-5\\ k\le 3\end{array}$

Step 5. Description of step.

Subtract 5 from each side and simplify.

$\begin{array}{c}5+k-5\ge -8-5\\ k\ge -13\end{array}$

Thus, solution is $k\le 3$ or $k\ge -13$.

Step 6. Describe solution set in interval notation.

As solution set of the inequality is all real numbers less than or equal to 3 or greater than or equal to $-13$.

Write the solution set in interval notation.

$[-13,3]$

Step 7. Graph the solution set.

Closed circle on 3 and $-13$ shows these points are included in the solution set.