Let's match each inequality to the appropriate solution:
1. Inequality: [tex]\(3x + 2 < 11\)[/tex]
First, solve the inequality:
[tex]\[ 3x + 2 < 11 \][/tex]
Subtract 2 from both sides:
[tex]\[ 3x < 9 \][/tex]
Divide by 3:
[tex]\[ x < 3 \][/tex]
The solution is [tex]\( (-\infty < x < 3) \)[/tex].
2. Inequality: [tex]\(\frac{1}{2}x - 2 > 2\)[/tex]
First, solve the inequality:
[tex]\[ \frac{1}{2}x - 2 > 2 \][/tex]
Add 2 to both sides:
[tex]\[ \frac{1}{2}x > 4 \][/tex]
Multiply both sides by 2:
[tex]\[ x > 8 \][/tex]
The solution is [tex]\( (8 < x < \infty) \)[/tex].
3. Inequality: [tex]\(4 < 2x - 2 < 14\)[/tex]
This is a compound inequality, so we solve it in two parts:
[tex]\[ 4 < 2x - 2 \][/tex]
Add 2 to both sides:
[tex]\[ 6 < 2x \][/tex]
Divide by 2:
[tex]\[ 3 < x \][/tex]
The second part:
[tex]\[ 2x - 2 < 14 \][/tex]
Add 2 to both sides:
[tex]\[ 2x < 16 \][/tex]
Divide by 2:
[tex]\[ x < 8 \][/tex]
Combining both solutions, we get [tex]\( (3 < x < 8) \)[/tex].
4. Inequality: [tex]\(3x + 4 < 13 \text{ and } x - 7 > 1\)[/tex]
First part:
[tex]\[ 3x + 4 < 13 \][/tex]
Subtract 4 from both sides:
[tex]\[ 3x < 9 \][/tex]
Divide by 3:
[tex]\[ x < 3 \][/tex]
Second part:
[tex]\[ x - 7 > 1 \][/tex]
Add 7 to both sides:
[tex]\[ x > 8 \][/tex]
However, there is no value of [tex]\( x \)[/tex] that satisfies both inequalities simultaneously. So, the solution is [tex]\( \text{False} \)[/tex].
5. Inequality: [tex]\(-3x + 2 < -7 \text{ and } x - 4 > 4\)[/tex]
First part:
[tex]\[ -3x + 2 < -7 \][/tex]
Subtract 2 from both sides:
[tex]\[ -3x < -9 \][/tex]
Divide by -3 (and flip the inequality):
[tex]\[ x > 3 \][/tex]
Second part:
[tex]\[ x - 4 > 4 \][/tex]
Add 4 to both sides:
[tex]\[ x > 8 \][/tex]
Combining both solutions, we take the more restrictive condition:
[tex]\[ x > 8 \][/tex]
The complete solution is [tex]\( (8 < x < \infty) \)[/tex].
Now, match each inequality to its corresponding solution:
1. [tex]\( 3x + 2 < 11 \)[/tex] matches [tex]\( (-\infty < x < 3) \)[/tex]
2. [tex]\( \frac{1}{2} x - 2 > 2 \)[/tex] matches [tex]\( (8 < x < \infty) \)[/tex]
3. [tex]\( 4 < 2x - 2 < 14 \)[/tex] matches [tex]\( (3 < x < 8) \)[/tex]
4. [tex]\( 3x + 4 < 13 \)[/tex] and [tex]\( x - 7 > 1 \)[/tex] matches [tex]\( \text{False} \)[/tex]
5. [tex]\( -3x + 2 < -7 \)[/tex] and [tex]\( x - 4 > 4 \)[/tex] matches [tex]\( (8 < x < \infty) \)[/tex]
