To solve the inequality [tex]\(|7 + 8x| > 5\)[/tex], let's break it down step by step.
### Step 1: Understand Absolute Value Inequality
The inequality [tex]\(|7 + 8x| > 5\)[/tex] means that the expression inside the absolute value can take values either less than [tex]\(-5\)[/tex] or greater than [tex]\(5\)[/tex]. So, we need to consider two separate cases:
1. [tex]\(7 + 8x > 5\)[/tex]
2. [tex]\(7 + 8x < -5\)[/tex]
### Step 2: Solve Each Inequality
#### Case 1: [tex]\(7 + 8x > 5\)[/tex]
1. Subtract 7 from both sides:
[tex]\[
7 + 8x - 7 > 5 - 7
\][/tex]
simplifies to:
[tex]\[
8x > -2
\][/tex]
2. Divide both sides by 8:
[tex]\[
x > -\frac{2}{8}
\][/tex]
which simplifies to:
[tex]\[
x > -\frac{1}{4}
\][/tex]
#### Case 2: [tex]\(7 + 8x < -5\)[/tex]
1. Subtract 7 from both sides:
[tex]\[
7 + 8x - 7 < -5 - 7
\][/tex]
simplifies to:
[tex]\[
8x < -12
\][/tex]
2. Divide both sides by 8:
[tex]\[
x < \frac{-12}{8}
\][/tex]
which simplifies to:
[tex]\[
x < -\frac{3}{2}
\][/tex]
### Step 3: Combine the Results
The solution to the inequality [tex]\(|7 + 8x| > 5\)[/tex] includes values where [tex]\(x > -\frac{1}{4}\)[/tex] or [tex]\(x < -\frac{3}{2}\)[/tex]. Therefore, the solution set is:
[tex]\[
x < -\frac{3}{2} \text{ or } x > -\frac{1}{4}
\][/tex]
In conclusion, the inequality [tex]\(|7 + 8x| > 5\)[/tex] has the solution:
[tex]\[
x < -\frac{3}{2} \text{ or } x > -\frac{1}{4}
\][/tex]
