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Match each inequality on the left with its correct solution on the right. Some answer choices on the right will be used more than once.

[tex]\[
\begin{array}{l}
4x + 2 \ \textgreater \ 10 \text{ and } -3x - 1 \ \textgreater \ 5 \\
|3x| + 4 \ \textless \ 10 \\
|x + 2| + 4 \ \textless \ 3 \\
|2x + 4| + 2 \ \textgreater \ 4
\end{array}
\][/tex]

[tex]\[
\begin{array}{l}
-2 \ \textless \ x \ \textless \ 2 \\
x \ \textgreater \ -1 \text{ or } x \ \textless \ -3
\end{array}
\][/tex]

No solution

Sagot :

GinnyAnsweridn
Certainly! Let's match each inequality with its correct solution based on our logical steps and final results.

### Given Inequalities:

1. [tex]\(4x + 2 > 10 \text{ and } -3x - 1 > 5\)[/tex]
2. [tex]\(|3x| + 4 < 10\)[/tex]
3. [tex]\(|x + 2| + 4 < 3\)[/tex]
4. [tex]\(|2x + 4| + 2 > 4\)[/tex]

### Potential Solutions:

a. [tex]\(-2 < x < 2\)[/tex]

b. [tex]\(x > -1 \text{ or } x < -3\)[/tex]

c. No solution

### Inequality Analysis:

1. Inequality: [tex]\(4x + 2 > 10 \text{ and } -3x - 1 > 5\)[/tex]
- Solution: This system of inequalities has no solution because there is no single value of [tex]\(x\)[/tex] that can satisfy both inequalities simultaneously.
- Answer: No solution

2. Inequality: [tex]\(|3x| + 4 < 10\)[/tex]
- Solution: Simplify to find:
[tex]\[ |3x| < 6 \implies -6 < 3x < 6 \implies -2 < x < 2 \][/tex]
- Answer: [tex]\(-2 < x < 2\)[/tex]

3. Inequality: [tex]\(|x + 2| + 4 < 3\)[/tex]
- Solution: Simplify to find:
[tex]\[ |x + 2| < -1 \quad (\text{which is impossible since the absolute value is always non-negative}) \][/tex]
- Answer: No solution

4. Inequality: [tex]\(|2x + 4| + 2 > 4\)[/tex]
- Solution: Simplify to find:
[tex]\[ |2x + 4| > 2 \implies 2x + 4 > 2 \text{ or } 2x + 4 < -2 \implies x > -1 \text{ or } x < -3 \][/tex]
- Answer: [tex]\(x > -1 \text{ or } x < -3\)[/tex]

### Matching Each Inequality with Solutions:

1. [tex]\(4x + 2 > 10 \text{ and } -3x - 1 > 5\)[/tex]
Matches with: No solution

2. [tex]\(|3x| + 4 < 10\)[/tex]
Matches with: [tex]\(-2 < x < 2\)[/tex]

3. [tex]\(|x + 2| + 4 < 3\)[/tex]
Matches with: No solution

4. [tex]\(|2x + 4| + 2 > 4\)[/tex]
Matches with: [tex]\(x > -1 \text{ or } x < -3\)[/tex]

Therefore, the matched solutions are:

- [tex]\(4x + 2 > 10 \text{ and } -3x - 1 > 5\)[/tex] -> No solution
- [tex]\(|3x| + 4 < 10\)[/tex] -> [tex]\(-2 < x < 2\)[/tex]
- [tex]\(|x + 2| + 4 < 3\)[/tex] -> No solution
- [tex]\(|2x + 4| + 2 > 4\)[/tex] -> [tex]\(x > -1 \text{ or } x < -3\)[/tex]
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