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]
