To solve the inequality [tex]\( 9|x+8| > 45 \)[/tex], let’s follow these steps:
### Step 1: Isolate the absolute value
First, divide both sides of the inequality by 9 to isolate the absolute value:
[tex]\[
|x + 8| > \frac{45}{9}
\][/tex]
This simplifies to:
[tex]\[
|x + 8| > 5
\][/tex]
### Step 2: Remove the absolute value
We know that for any real number [tex]\( a \)[/tex]:
[tex]\[
|a| > b \quad \text{is equivalent to} \quad a > b \quad \text{or} \quad a < -b
\][/tex]
Applying this to our inequality:
[tex]\[
x + 8 > 5 \quad \text{or} \quad x + 8 < -5
\][/tex]
### Step 3: Solve the inequalities
1. First Inequality:
[tex]\[
x + 8 > 5
\][/tex]
Subtract 8 from both sides:
[tex]\[
x > 5 - 8
\][/tex]
[tex]\[
x > -3
\][/tex]
2. Second Inequality:
[tex]\[
x + 8 < -5
\][/tex]
Subtract 8 from both sides:
[tex]\[
x < -5 - 8
\][/tex]
[tex]\[
x < -13
\][/tex]
### Step 4: Combine the solutions
The combined solution from both inequalities is:
[tex]\[
x > -3 \quad \text{or} \quad x < -13
\][/tex]
### Step 5: Match with the provided choices
Looking at the choices given:
- [tex]\( x < -3 \)[/tex] or [tex]\( x > 13 \)[/tex]
- [tex]\( -3 < x < 13 \)[/tex]
- [tex]\( x < -13 \)[/tex] or [tex]\( x > -3 \)[/tex]
- [tex]\( -13 < x < -3 \)[/tex]
The correct answer that matches our solution ([tex]\( x > -3 \)[/tex] or [tex]\( x < -13 \)[/tex]) is:
[tex]\[
\boxed{x < -13 \text{ or } x > -3}
\][/tex]
