Let's solve each inequality step by step and determine which one requires reversing the inequality symbol. Reversing the inequality symbol typically occurs when we multiply or divide both sides of the inequality by a negative number.
### Inequality A: [tex]\(4(5 - y) \geq 80\)[/tex]
1. Distribute the 4:
[tex]\[
4 \cdot 5 - 4 \cdot y \geq 80
\][/tex]
[tex]\[
20 - 4y \geq 80
\][/tex]
2. Isolate the y term by subtracting 20 from both sides:
[tex]\[
-4y \geq 60
\][/tex]
3. To solve for y, divide both sides by -4. Remember, dividing by a negative number reverses the inequality:
[tex]\[
y \leq -\frac{60}{4}
\][/tex]
[tex]\[
y \leq -15
\][/tex]
### Inequality B: [tex]\(2(y - 3) \geq 8\)[/tex]
1. Distribute the 2:
[tex]\[
2 \cdot y - 2 \cdot 3 \geq 8
\][/tex]
[tex]\[
2y - 6 \geq 8
\][/tex]
2. Isolate the y term by adding 6 to both sides:
[tex]\[
2y \geq 14
\][/tex]
3. To solve for y, divide both sides by 2:
[tex]\[
y \geq 7
\][/tex]
### Inequality C: [tex]\(3y - 4 < 11\)[/tex]
1. Isolate the y term by adding 4 to both sides:
[tex]\[
3y < 15
\][/tex]
2. To solve for y, divide both sides by 3:
[tex]\[
y < 5
\][/tex]
### Inequality D: [tex]\(-\frac{3}{4} + 6y > \frac{1}{4}\)[/tex]
1. Isolate the y term by adding [tex]\(\frac{3}{4}\)[/tex] to both sides:
[tex]\[
6y > \frac{1}{4} + \frac{3}{4}
\][/tex]
[tex]\[
6y > 1
\][/tex]
2. To solve for y, divide both sides by 6:
[tex]\[
y > \frac{1}{6}
\][/tex]
From the steps above, we can see that Inequality A requires reversing the inequality symbol when dividing by -4. None of the other inequalities involved multiplication or division by a negative number.
Therefore, the inequality that requires reversing the symbol to solve is:
[tex]\[
\textbf{A } \quad 4(5 - y) \geq 80
\][/tex]
