Sure, let's solve the inequality [tex]\(|3x - 5| < x + 2\)[/tex] step by step.
### Understanding the Inequality
First, recall that the absolute value inequality [tex]\(|A| < B\)[/tex] can be split into two separate inequalities:
[tex]\[ -B < A < B \][/tex]
In our case, [tex]\(A = 3x - 5\)[/tex] and [tex]\(B = x + 2\)[/tex]. Thus, we have:
[tex]\[ -(x + 2) < 3x - 5 < x + 2 \][/tex]
### Splitting into Two Inequalities
Let's solve each part of the inequality separately.
#### 1. Solving [tex]\(-(x + 2) < 3x - 5\)[/tex]
Starting with the left part:
[tex]\[ -(x + 2) < 3x - 5 \][/tex]
[tex]\[ -x - 2 < 3x - 5 \][/tex]
Add [tex]\(x\)[/tex] to both sides:
[tex]\[ -2 < 4x - 5 \][/tex]
Add 5 to both sides:
[tex]\[ 3 < 4x \][/tex]
Divide both sides by 4:
[tex]\[ \frac{3}{4} < x \][/tex]
Or:
[tex]\[ x > \frac{3}{4} \][/tex]
#### 2. Solving [tex]\(3x - 5 < x + 2\)[/tex]
Now for the right part:
[tex]\[ 3x - 5 < x + 2 \][/tex]
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[ 2x - 5 < 2 \][/tex]
Add 5 to both sides:
[tex]\[ 2x < 7 \][/tex]
Divide both sides by 2:
[tex]\[ x < \frac{7}{2} \][/tex]
### Combining the Solutions
We need [tex]\(x\)[/tex] to satisfy both inequalities simultaneously:
[tex]\[ \frac{3}{4} < x < \frac{7}{2} \][/tex]
### Solution in Set Builder Notation
So, the solution to the inequality [tex]\(|3x - 5| < x + 2\)[/tex] is:
[tex]\[ \left\{ x \in \mathbb{R} \; | \; \frac{3}{4} < x < \frac{7}{2} \right\} \][/tex]
