Sure! Let's match each inequality with its correct solution step by step.
1. Inequality: [tex]\(4x + 1 > 9\)[/tex] and [tex]\(-6x - 2 > 10\)[/tex]
- To solve [tex]\(4x + 1 > 9\)[/tex]:
[tex]\[ 4x + 1 > 9 \][/tex]
[tex]\[ 4x > 8 \][/tex]
[tex]\[ x > 2 \][/tex]
- To solve [tex]\(-6x - 2 > 10\)[/tex]:
[tex]\[ -6x - 2 > 10 \][/tex]
[tex]\[ -6x > 12 \][/tex]
[tex]\[ x < -2 \][/tex]
- Since [tex]\( x > 2 \)[/tex] and [tex]\( x < -2 \)[/tex] cannot be true simultaneously, there is no solution.
Solution: no solution
2. Inequality: [tex]\(|-3x| < 6\)[/tex]
- Since [tex]\(|-3x|\)[/tex] is the absolute value of [tex]\(-3x\)[/tex], it means:
[tex]\[ |-3x| < 6 \][/tex]
- This can be split into two parts:
[tex]\[ -3x < 6 \][/tex]
[tex]\[ x > -2 \][/tex]
[tex]\[ 3x < 6 \][/tex]
[tex]\[ x < 2 \][/tex]
- Combining these two parts, we get:
[tex]\[ -2 < x < 2 \][/tex]
Solution: [tex]\(-2 < x < 2\)[/tex]
3. Inequality: [tex]\(|3x| + 5 < 4\)[/tex]
- To solve [tex]\(|3x| + 5 < 4\)[/tex], we first isolate the absolute value:
[tex]\[ |3x| + 5 < 4 \][/tex]
[tex]\[ |3x| < -1 \][/tex]
- Since an absolute value cannot be negative, there is no solution.
Solution: no solution
4. Inequality: [tex]\(|x + 2| > 1\)[/tex]
- Since [tex]\(|x+2|\)[/tex] is the absolute value, it gives us two cases:
[tex]\[ x+2 > 1 \text{ or } x+2 < -1 \][/tex]
- For [tex]\( x + 2 > 1 \)[/tex]:
[tex]\[ x > -1 \][/tex]
- For [tex]\( x + 2 < -1 \)[/tex]:
[tex]\[ x < -3 \][/tex]
- Combining these two parts, we get:
[tex]\[ x > -1 \text{ or } x < -3 \][/tex]
Solution: [tex]\(x > -1 \text{ or } x < -3\)[/tex]
Therefore, the matches are:
- [tex]\(4x + 1 > 9\)[/tex] and [tex]\(-6x - 2 > 10\)[/tex] → no solution
- [tex]\(|-3x| < 6\)[/tex] → [tex]\(-2 < x < 2\)[/tex]
- [tex]\(|3x| + 5 < 4\)[/tex] → no solution
- [tex]\(|x + 2| > 1\)[/tex] → [tex]\(x > -1 \text{ or } x < -3\)[/tex]
