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Sagot :
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]
### 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]
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