Question

Solve each equation.

$\left|b+9\right|=2$

Short answer

The solution set will be $b=-7\text{and}b=-11$.

Step-by-step solution

Step 1. Use the definition of the absolute value function.

The absolute value or modulus of a real number xdenoted |x| is the non-negative value of x without regard to its sign.

Absolute value refers to a point's distance from zero or origin on the number line, regardless of the direction.

Step 2. Solve for the first possible solution.

We are given the expressions:$|b+9|=2$

Since an absolute function value can never be negative we have two scenarios that are possible.

Case: 1 when expression inside the modulus sign is negative, that is

localid="1644592752185" $\begin{array}{l}-(b+9)=2\text{\hspace{0.33em}[Distribute]}\\ -b-9=2\\ -b-9+9=2+9\text{[Add9tobothsides]}\\ -b=11\\ -b\times \left(-1\right)=11\times \left(-1\right)\text{[multiplyby-1bothsides]}\\ b=-11\end{array}$

Step 3. Solve for a second possible solution.

We are given the expressions: $|b+9|=2$

Case: 2 when expression inside the modulus sign is positive, that is

role="math" localid="1642664366989" $\begin{array}{l}b+9=2\\ b+9-9=2-9\text{[Add-9tobothsides]}\\ b=-7\end{array}$

Hence, the solution set will be $b=-7\text{and}b=-11$.