Let's solve each of the given inequalities step-by-step and match them to their solutions:
1. [tex]\(-3x > -36\)[/tex]
To solve for [tex]\(x\)[/tex], we need to isolate [tex]\(x\)[/tex]. Start by dividing both sides by [tex]\(-3\)[/tex]. Remember that dividing by a negative number reverses the inequality sign.
[tex]\[
x < \frac{-36}{-3} \Rightarrow x < 12
\][/tex]
2. [tex]\(b + 5 > 23\)[/tex]
To solve for [tex]\(b\)[/tex], subtract 5 from both sides of the inequality:
[tex]\[
b > 23 - 5 \Rightarrow b > 18
\][/tex]
3. [tex]\(1 + 7n \geq -90\)[/tex]
To solve for [tex]\(n\)[/tex], subtract 1 from both sides of the inequality:
[tex]\[
7n \geq -90 - 1 \Rightarrow 7n \geq -91
\][/tex]
Next, divide both sides by 7:
[tex]\[
n \geq \frac{-91}{7} \Rightarrow n \geq -13
\][/tex]
4. [tex]\(\frac{x}{2} - 2 > 1\)[/tex]
To solve for [tex]\(x\)[/tex], first add 2 to both sides of the inequality:
[tex]\[
\frac{x}{2} > 1 + 2 \Rightarrow \frac{x}{2} > 3
\][/tex]
Next, multiply both sides by 2:
[tex]\[
x > 3 \times 2 \Rightarrow x > 6
\][/tex]
Now let's match each inequality to its respective solution:
- [tex]\(-3x > -36\)[/tex] corresponds to [tex]\(x < 12\)[/tex]
- [tex]\(b + 5 > 23\)[/tex] corresponds to [tex]\(b > 18\)[/tex]
- [tex]\(1 + 7n \geq -90\)[/tex] corresponds to [tex]\(n \geq -13\)[/tex]
- [tex]\(\frac{x}{2} - 2 > 1\)[/tex] corresponds to [tex]\(x > 6\)[/tex]
Thus, the solutions are:
1. [tex]\(-3x > -36\)[/tex] ⟶ [tex]\(x < 12\)[/tex]
2. [tex]\(b + 5 > 23\)[/tex] ⟶ [tex]\(b > 18\)[/tex]
3. [tex]\(1 + 7n \geq -90\)[/tex] ⟶ [tex]\(n \geq -13\)[/tex]
4. [tex]\(\frac{x}{2} - 2 > 1\)[/tex] ⟶ [tex]\(x > 6\)[/tex]
