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Match the inequality on the left to the appropriate solution on the right.

[tex]\[
\begin{array}{l}
1. \ 3x + 2 \ \textless \ 11 \\
2. \ \frac{1}{2}x - 2 \ \textgreater \ 2 \\
3. \ 4 \ \textless \ 2x - 2 \ \textless \ 14 \\
4. \ 3x + 4 \ \textless \ 13 \ \text{or} \ x - 7 \ \textgreater \ 1 \\
5. \ -3x + 2 \ \textless \ -7 \ \text{and} \ x - 4 \ \textgreater \ 4
\end{array}
\][/tex]

[tex]\[
\begin{array}{l}
A. \ x \ \textless \ 3 \\
B. \ x \ \textgreater \ 8 \\
C. \ 3 \ \textless \ x \ \textless \ 8 \\
D. \ x \ \textless \ 3 \ \text{or} \ x \ \textgreater \ 8 \\
E. \ x \ \textless \ -\frac{5}{3} \ \text{and} \ x \ \textgreater \ 4
\end{array}
\][/tex]

Match each inequality with the correct solution letter (A-E).

Sagot :

GinnyAnsweridn
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]
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