To find the solution set for the inequality [tex]\(4 \leq 3x - 2 < 13\)[/tex], follow these steps:
1. Break the compound inequality into two separate inequalities:
- [tex]\(4 \leq 3x - 2\)[/tex]
- [tex]\(3x - 2 < 13\)[/tex]
2. Solve each inequality separately:
First Inequality: [tex]\(4 \leq 3x - 2\)[/tex]
a. Add 2 to both sides of the inequality:
[tex]\[
4 + 2 \leq 3x - 2 + 2
\][/tex]
This simplifies to:
[tex]\[
6 \leq 3x
\][/tex]
b. Divide both sides by 3:
[tex]\[
\frac{6}{3} \leq \frac{3x}{3}
\][/tex]
This simplifies to:
[tex]\[
2 \leq x
\][/tex]
or equivalently:
[tex]\[
x \geq 2
\][/tex]
Second Inequality: [tex]\(3x - 2 < 13\)[/tex]
a. Add 2 to both sides of the inequality:
[tex]\[
3x - 2 + 2 < 13 + 2
\][/tex]
This simplifies to:
[tex]\[
3x < 15
\][/tex]
b. Divide both sides by 3:
[tex]\[
\frac{3x}{3} < \frac{15}{3}
\][/tex]
This simplifies to:
[tex]\[
x < 5
\][/tex]
3. Combine the solutions:
- From the first inequality, we have [tex]\(x \geq 2\)[/tex].
- From the second inequality, we have [tex]\(x < 5\)[/tex].
Combining these two results, we get:
[tex]\[
2 \leq x < 5
\][/tex]
4. Solution Set:
The solution set for the inequality [tex]\(4 \leq 3x - 2 < 13\)[/tex] is:
[tex]\[
[2, 5)
\][/tex]
This means that [tex]\(x\)[/tex] can take any real value from 2 to just less than 5, inclusive of 2 but not including 5.
