A. Solve for [tex]a[/tex]: [tex]4a + 5 \ \textgreater \ 9a + 15[/tex]

B. Solve for [tex]y[/tex]: [tex]2y - 3 \ \textless \ 9 + y[/tex]

C. Solve for [tex]y[/tex]: [tex]3y - 10 \ \textgreater \ 11[/tex]

D. Solve for [tex]x[/tex]: [tex]7 - 2x \ \textgreater \ 3y[/tex]



Answer :

Sure, let's solve each of these inequalities step-by-step:

### A. Solve for [tex]\( a \)[/tex]:
Given: [tex]\( 4a + 5 > 9a + 15 \)[/tex]

1. Subtract [tex]\( 4a \)[/tex] from both sides to isolate the variable on one side:
[tex]\[ 5 > 9a - 4a + 15 \][/tex]
This simplifies to:
[tex]\[ 5 > 5a + 15 \][/tex]

2. Subtract 15 from both sides to further isolate the term with the variable:
[tex]\[ 5 - 15 > 5a \][/tex]
[tex]\[ -10 > 5a \][/tex]

3. Divide both sides by 5 to solve for [tex]\( a \)[/tex]:
[tex]\[ \frac{-10}{5} > a \][/tex]
[tex]\[ -2 > a \][/tex]

So the solution for [tex]\( a \)[/tex] is:
[tex]\[ a < -2 \][/tex]

### B. Solve for [tex]\( y \)[/tex]:
Given: [tex]\( 2y - 3 < 9 + y \)[/tex]

1. Subtract [tex]\( y \)[/tex] from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ 2y - y - 3 < 9 \][/tex]
This simplifies to:
[tex]\[ y - 3 < 9 \][/tex]

2. Add 3 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y - 3 + 3 < 9 + 3 \][/tex]
[tex]\[ y < 12 \][/tex]

So the solution for [tex]\( y \)[/tex] is:
[tex]\[ y < 12 \][/tex]

### C. Solve for [tex]\( y \)[/tex]:
Given: [tex]\( 3y - 10 > 11 \)[/tex]

1. Add 10 to both sides to isolate the term with the variable:
[tex]\[ 3y - 10 + 10 > 11 + 10 \][/tex]
[tex]\[ 3y > 21 \][/tex]

2. Divide both sides by 3 to solve for [tex]\( y \)[/tex]:
[tex]\[ \frac{3y}{3} > \frac{21}{3} \][/tex]
[tex]\[ y > 7 \][/tex]

So the solution for [tex]\( y \)[/tex] is:
[tex]\[ y > 7 \][/tex]

### D. Solve for [tex]\( x \)[/tex]:
Given: [tex]\( 7 - 2x > 3y \)[/tex]

1. Subtract 7 from both sides to move the constant term:
[tex]\[ 7 - 7 - 2x > 3y - 7 \][/tex]
[tex]\[ -2x > 3y - 7 \][/tex]

2. Divide both sides by -2 and remember to reverse the inequality:
[tex]\[ \frac{-2x}{-2} < \frac{3y - 7}{-2} \][/tex]
[tex]\[ x < \frac{7 - 3y}{2} \][/tex]

So the solution for [tex]\( x \)[/tex] is:
[tex]\[ x < \frac{7 - 3y}{2} \][/tex]