Certainly! Let's solve the given inequalities step by step.
### Inequality 1: [tex]\(|x-13| \leq 4\)[/tex]
The absolute value inequality [tex]\(|x-13| \leq 4\)[/tex] can be broken down into a compound inequality:
[tex]\[
-4 \leq x - 13 \leq 4
\][/tex]
To isolate [tex]\(x\)[/tex], we add 13 to all parts of the inequality:
[tex]\[
-4 + 13 \leq x - 13 + 13 \leq 4 + 13
\][/tex]
Simplifying this, we get:
[tex]\[
9 \leq x \leq 17
\][/tex]
So, the solution to the first inequality [tex]\(|x-13| \leq 4\)[/tex] is:
[tex]\[
x \in [9, 17]
\][/tex]
### Inequality 2: [tex]\(|3x - 1| > 4\)[/tex]
The absolute value inequality [tex]\(|3x - 1| > 4\)[/tex] can be split into two separate inequalities:
1. [tex]\(3x - 1 > 4\)[/tex]
2. [tex]\(3x - 1 < -4\)[/tex]
#### Case 1: [tex]\(3x - 1 > 4\)[/tex]
Solving for [tex]\(x\)[/tex], we add 1 to both sides:
[tex]\[
3x - 1 + 1 > 4 + 1
\][/tex]
Simplifying:
[tex]\[
3x > 5
\][/tex]
Dividing by 3:
[tex]\[
x > \frac{5}{3}
\][/tex]
This simplifies to:
[tex]\[
x > 1.\overline{6}
\][/tex]
#### Case 2: [tex]\(3x - 1 < -4\)[/tex]
Solving for [tex]\(x\)[/tex], we add 1 to both sides:
[tex]\[
3x - 1 + 1 < -4 + 1
\][/tex]
Simplifying:
[tex]\[
3x < -3
\][/tex]
Dividing by 3:
[tex]\[
x < -1
\][/tex]
So, the solution to the second inequality [tex]\(|3x - 1| > 4\)[/tex] is:
[tex]\[
x \in (-\infty, -1) \cup \left( \frac{5}{3}, \infty \right)
\][/tex]
### Combining the Solutions
We need to find the values of [tex]\(x\)[/tex] that satisfy both inequalities:
1. The first inequality gives: [tex]\(x \in [9, 17]\)[/tex]
2. The second inequality gives: [tex]\(x \in (-\infty, -1) \cup \left( \frac{5}{3}, \infty \right)\)[/tex]
We look for the intersection of these sets.
- [tex]\([-9, 17] \cap (-\infty, -1)\)[/tex] results in an empty set because [tex]\([-9, 17]\)[/tex] doesn't overlap with [tex]\((- \infty, -1)\)[/tex].
- [tex]\([-9, 17] \cap \left(\frac{5}{3}, \infty\right)\)[/tex] results in [tex]\([9, 17]\)[/tex] where [tex]\((1.\overline{6}, \infty)\)[/tex] overlaps from 9 to 17.
Therefore, the combined solution for [tex]\(x\)[/tex] is:
[tex]\[
x \in [9, 17]
\][/tex]
