Let's solve each of the given inequalities step-by-step and match them to their solutions.
### Inequality 1: [tex]\(-3x > -36\)[/tex]
1. Step 1: Isolate [tex]\(x\)[/tex] by dividing both sides by [tex]\(-3\)[/tex]. Remember that dividing by a negative number reverses the inequality sign.
[tex]\[
x < 12
\][/tex]
### Inequality 2: [tex]\(b + 5 > 23\)[/tex]
1. Step 1: Isolate [tex]\(b\)[/tex] by subtracting 5 from both sides.
[tex]\[
b > 18
\][/tex]
### Inequality 3: [tex]\(1 + 7n \geq -90\)[/tex]
1. Step 1: Subtract 1 from both sides to isolate the term with [tex]\(n\)[/tex].
[tex]\[
7n \geq -91
\][/tex]
2. Step 2: Divide both sides by 7.
[tex]\[
n \geq -13
\][/tex]
### Inequality 4: [tex]\(\frac{x}{2} - 2 > 1\)[/tex]
1. Step 1: Add 2 to both sides to isolate the term with [tex]\(x\)[/tex].
[tex]\[
\frac{x}{2} > 3
\][/tex]
2. Step 2: Multiply both sides by 2.
[tex]\[
x > 6
\][/tex]
### Matching the Solutions:
- 1. [tex]\(-3x > -36\)[/tex] corresponds to [tex]\(\boxed{x < 12}\)[/tex].
- 2. [tex]\(b + 5 > 23\)[/tex] corresponds to [tex]\(\boxed{b > 18}\)[/tex].
- 3. [tex]\(1 + 7n \geq -90\)[/tex] corresponds to [tex]\(\boxed{n \geq -13}\)[/tex].
- 4. [tex]\(\frac{x}{2} - 2 > 1\)[/tex] corresponds to [tex]\(\boxed{x > 6}\)[/tex].
So, the matched inequalities and their solutions are:
a. [tex]\(x < 12\)[/tex] [tex]\(\rightarrow\)[/tex] 1. [tex]\(-3x > -36\)[/tex]
b. [tex]\(b > 18\)[/tex] [tex]\(\rightarrow\)[/tex] 2. [tex]\(b + 5 > 23\)[/tex]
c. [tex]\(n \geq -13\)[/tex] [tex]\(\rightarrow\)[/tex] 3. [tex]\(1 + 7n \geq -90\)[/tex]
d. [tex]\(x > 6\)[/tex] [tex]\(\rightarrow\)[/tex] 4. [tex]\(\frac{x}{2} - 2 > 1\)[/tex]
I hope this helps you understand how to solve and match these inequalities with their respective solutions!
