To solve the inequality [tex]\( |7 + 8x| > 5 \)[/tex], we need to consider the nature of the absolute value function. An inequality of the form [tex]\( |A| > B \)[/tex] can be broken down into two separate inequalities:
1. [tex]\( A > B \)[/tex]
2. [tex]\( A < -B \)[/tex]
Let's apply this to our specific problem:
1. [tex]\( 7 + 8x > 5 \)[/tex]
2. [tex]\( 7 + 8x < -5 \)[/tex]
We'll solve each inequality separately.
### Solving [tex]\( 7 + 8x > 5 \)[/tex]
Subtract 7 from both sides:
[tex]\[ 8x > -2 \][/tex]
Now divide both sides by 8:
[tex]\[ x > -\frac{1}{4} \][/tex]
### Solving [tex]\( 7 + 8x < -5 \)[/tex]
Subtract 7 from both sides:
[tex]\[ 8x < -12 \][/tex]
Now divide both sides by 8:
[tex]\[ x < -\frac{3}{2} \][/tex]
### Combine the two results
The solution to the inequality [tex]\( |7 + 8x| > 5 \)[/tex] is a combination of the two solutions we derived. Therefore, we have two intervals:
[tex]\[ x < -\frac{3}{2} \quad \text{or} \quad x > -\frac{1}{4} \][/tex]
### Final Solution
Putting it all together, the answer to the inequality [tex]\( |7 + 8x| > 5 \)[/tex] is:
[tex]\[ x < -\frac{3}{2} \quad \text{or} \quad x > -\frac{1}{4} \][/tex]
