Certainly! To solve the inequality [tex]\( |3x - 2| \geq 5 \)[/tex], we need to consider the definition of the absolute value function. The absolute value [tex]\( |A| \geq B \)[/tex] translates to two inequalities: [tex]\( A \geq B \)[/tex] or [tex]\( A \leq -B \)[/tex].
Given the inequality [tex]\( |3x - 2| \geq 5 \)[/tex], we set up two separate inequalities:
1. [tex]\(3x - 2 \geq 5\)[/tex]
2. [tex]\(3x - 2 \leq -5\)[/tex]
### Step-by-Step Solution:
#### Inequality 1: [tex]\(3x - 2 \geq 5\)[/tex]
1. Add 2 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
3x - 2 + 2 \geq 5 + 2
\][/tex]
[tex]\[
3x \geq 7
\][/tex]
2. Divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x \geq \frac{7}{3}
\][/tex]
#### Inequality 2: [tex]\(3x - 2 \leq -5\)[/tex]
1. Add 2 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
3x - 2 + 2 \leq -5 + 2
\][/tex]
[tex]\[
3x \leq -3
\][/tex]
2. Divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x \leq -1
\][/tex]
### Combining the Solutions:
The solutions to the inequalities are:
- From Inequality 1: [tex]\( x \geq \frac{7}{3} \)[/tex]
- From Inequality 2: [tex]\( x \leq -1 \)[/tex]
Thus, the combined solution set is:
[tex]\[
x \leq -1 \quad \text{or} \quad x \geq \frac{7}{3}
\][/tex]
### Writing the Solution in Interval Notation:
The intervals for the solution are:
[tex]\[
(-\infty, -1] \cup \left[\frac{7}{3}, \infty\right)
\][/tex]
So, the values of [tex]\( x \)[/tex] that satisfy the inequality [tex]\( |3x - 2| \geq 5 \)[/tex] are:
[tex]\[
(-\infty < x \leq -1) \cup (\frac{7}{3} \leq x < \infty)
\][/tex]
