Certainly! Let's match each absolute value inequality with its corresponding solution set:
1. Inequality: \(3|x+5|-1 \leq 8\)
To find the solution set, let's isolate the absolute value expression. First, we'll add 1 to both sides to get:
[tex]\[
3|x+5| \leq 9
\][/tex]
Then, we'll divide both sides by 3:
[tex]\[
|x+5| \leq 3
\][/tex]
The solution to \( |x+5| \leq 3 \) is:
[tex]\[
-3 \leq x+5 \leq 3
\][/tex]
Subtracting 5 from all parts of the inequality, we get:
[tex]\[
-8 \leq x \leq -2
\][/tex]
So, the matching solution is:
[tex]\[
-8 \leq x \leq -2
\][/tex]
2. Inequality: \(|2x-6|-4 \geq 6\)
First, add 4 to both sides:
[tex]\[
|2x-6| \geq 10
\][/tex]
This gives us two cases to consider:
[tex]\[
2x - 6 \geq 10 \quad \text{or} \quad 2x - 6 \leq -10
\][/tex]
Solving for \(x\), we get:
[tex]\[
2x \geq 16 \implies x \geq 8 \quad \text{or} \quad 2x \leq -4 \implies x \leq -2
\][/tex]
Therefore, the solution set is:
[tex]\[
x \leq -2 \text{ or } x \geq 8
\][/tex]
3. Inequality: \(4|x-3|+1 \leq 9\)
First, subtract 1 from both sides:
[tex]\[
4|x-3| \leq 8
\][/tex]
Then, divide both sides by 4:
[tex]\[
|x-3| \leq 2
\][/tex]
Solving the absolute value inequality, we get:
[tex]\[
-2 \leq x-3 \leq 2
\][/tex]
Adding 3 to all parts of the inequality, we get:
[tex]\[
1 \leq x \leq 5
\][/tex]
So, the solution is:
[tex]\[
1 \leq x \leq 5
\][/tex]
4. Inequality: \(3|x-9|-8 \geq 4\)
First, add 8 to both sides:
[tex]\[
3|x-9| \geq 12
\][/tex]
Then, divide both sides by 3:
[tex]\[
|x-9| \geq 4
\][/tex]
This gives us two cases to consider:
[tex]\[
x-9 \geq 4 \quad \text{or} \quad x-9 \leq -4
\][/tex]
Solving for \(x\), we get:
[tex]\[
x \geq 13 \quad \text{or} \quad x \leq 5
\][/tex]
Therefore, the solution set is:
[tex]\[
x \leq 5 \text{ or } x \geq 13
\][/tex]
So, the matches are:
[tex]\[
\begin{array}{lcl}
3|x+5|-1 \leq 8 & \rightarrow & -8 \leq x \leq -2 \\
|2x-6|-4 \geq 6 & \rightarrow & x \leq-2 \text { or } x \geq 8 \\
4|x-3|+1 \leq 9 & \rightarrow & 1 \leq x \leq 5 \\
3|x-9|-8 \geq 4 & \rightarrow & x \leq 5 \text { or } x \geq 13 \\
\end{array}
\][/tex]
