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Select the correct answer.

Solve:

[tex]5|x+9|=80[/tex]

A. [tex]x=-25[/tex] or [tex]x=9[/tex]
B. [tex]x=-25[/tex] or [tex]x=7[/tex]
C. [tex]x=7[/tex] or [tex]x=16[/tex]
D. [tex]x=-7[/tex] or [tex]x=25[/tex]

Answer :

To solve the equation [tex]\(5|x+9|=80\)[/tex], we need to consider the properties of absolute values. An absolute value equation like this can be broken down into two separate cases since the absolute value [tex]\(|x+9|\)[/tex] can either be positive or negative in terms of its expression. Here's how we can solve it step by step:

1. Isolate the absolute value:
[tex]\[
5|x+9|=80
\][/tex]
First, divide both sides by 5 to isolate the absolute value expression:
[tex]\[
|x+9|=16
\][/tex]

2. Consider the two possible cases for [tex]\(|x+9| = 16\)[/tex]:
- Case 1: The expression inside the absolute value is positive or zero.
[tex]\[
x + 9 = 16
\][/tex]
Solve for [tex]\(x\)[/tex] by subtracting 9 from both sides:
[tex]\[
x = 16 - 9 = 7
\][/tex]

- Case 2: The expression inside the absolute value is negative.
[tex]\[
x + 9 = -16
\][/tex]
Solve for [tex]\(x\)[/tex] by subtracting 9 from both sides:
[tex]\[
x = -16 - 9 = -25
\][/tex]

Thus, the solutions to the equation are [tex]\(x = 7\)[/tex] or [tex]\(x = -25\)[/tex].

So the correct answer is [tex]\(B. \, x = -25 \text{ or } x = 7\)[/tex].

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