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Solve [tex]$4|x+5|=16$[/tex]

A. [tex]$x=-1$[/tex] and [tex][tex]$x=-9$[/tex][/tex]
B. [tex]$x=1$[/tex] and [tex]$x=-9$[/tex]
C. [tex][tex]$x=-1$[/tex][/tex] and [tex]$x=9$[/tex]
D. [tex]$x=1$[/tex] and [tex][tex]$x=-1$[/tex][/tex]

Answer :

To solve the equation [tex]\(4|x+5|=16\)[/tex], follow these steps:

1. Isolate the absolute value:
Divide both sides of the equation by 4 to simplify:
[tex]\[
4|x+5| = 16 \implies |x+5| = \frac{16}{4} \implies |x+5| = 4
\][/tex]

2. Set up the equations:
Since we have an absolute value equation, it means [tex]\(x+5\)[/tex] can be either 4 or -4. This gives us two separate linear equations to solve:
[tex]\[
x+5 = 4 \quad \text{and} \quad x+5 = -4
\][/tex]

3. Solve each equation:
- For [tex]\(x+5 = 4\)[/tex]:
[tex]\[
x+5 = 4 \implies x = 4-5 \implies x = -1
\][/tex]
- For [tex]\(x+5 = -4\)[/tex]:
[tex]\[
x+5 = -4 \implies x = -4-5 \implies x = -9
\][/tex]

4. Verify the solutions:
To ensure our solutions are correct, substitute [tex]\(x\)[/tex] back into the original equation:
- For [tex]\(x = -1\)[/tex]:
[tex]\[
4|x+5| = 4|-1+5| = 4|4| = 16
\][/tex]
- For [tex]\(x = -9\)[/tex]:
[tex]\[
4|x+5| = 4|-9+5| = 4|-4| = 16
\][/tex]
Both solutions satisfy the original equation.

Therefore, the solutions to the equation [tex]\(4|x+5| = 16\)[/tex] are [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].

The correct answer is:
A. [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex]

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