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Match each absolute value inequality on the left side with its solution set on the right.

[tex]\[
\begin{array}{ll}
1. & 3|x+5|-1 \leq 8 \\
2. & |2x-6|-4 \geq 6 \\
3. & 4|x-3|+1 \leq 9 \\
4. & 3|x-9|-8 \geq 4 \\
\end{array}
\][/tex]

[tex]\[
\begin{array}{ll}
A. & x \leq -2 \text{ or } x \geq 8 \\
B. & -8 \leq x \leq -2 \\
C. & x \leq 5 \text{ or } x \geq 13 \\
D. & 1 \leq x \leq 5 \\
\end{array}
\][/tex]

Match the inequalities with their solution sets:
1. \(3|x+5|-1 \leq 8\) →
2. \(|2x-6|-4 \geq 6\) →
3. \(4|x-3|+1 \leq 9\) →
4. [tex]\(3|x-9|-8 \geq 4\)[/tex] →

Sagot :

GinnyAnswer
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]
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