Answer :
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]
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]
