Q54. - Algebra 1

Question

Solve each equation. Then graph the solution set.

$\left|\frac{2b}{3}+8\right|=20$

Step-by-Step Solution

Answer

The Solution set is $\{-42,18\}$.

Step 1. Case when the modulus expression is positive.

The absolute value or modulus of a real number x, denoted $\left|x\right|$, it is the non-negative value of $x$ without regard to its sign.

$\left|x\right|=x$ if $x$ is positive,

$\left|x\right|=-x$ if $x$ is negative,

$\left|0\right|=0$.

For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5.

 

We are given the expression $\left|\frac{2b}{3}+8\right|=20$ assuming a positive value for the modulus expression we get:

$\begin{array}{l}\frac{2b}{3}+8=20\\ \frac{2b}{3}=20-8\\ \frac{2b}{3}=12\\ 2b=12\left(3\right)\\ b=\frac{36}{2}\\ b=18\end{array}$

Thus $b=18$ is one of the solutions.

Step 2. Case when the modulus expression is negative.

$\begin{array}{l}\frac{2b}{3}+8=-20\\ \frac{2b}{3}=-20-8\\ \frac{2b}{3}=-28\\ 2b=-28\left(3\right)\\ b=\frac{-84}{2}\\ b=-42\end{array}$ 

Thus $b=-42$ is one of the solutions.

Step 3. Graphing the solution set.

The solution set is $\{-42,18\}$, we graph these values on the number line as shown below: