Answer :
To determine which one of the given compound inequalities has no solution, we'll solve each set of inequalities step by step.
### Compound Inequality 1:
[tex]\[ -2x + 9 > 4x + 3 \quad \text{and} \quad -8x - 9 < -7x - 3 \][/tex]
#### Solving the first inequality:
[tex]\[ -2x + 9 > 4x + 3 \][/tex]
Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[ -2x - 4x + 9 > 3 \][/tex]
Combine like terms:
[tex]\[ -6x + 9 > 3 \][/tex]
Subtract 9 from both sides:
[tex]\[ -6x > -6 \][/tex]
Divide by [tex]\(-6\)[/tex] and reverse the inequality sign:
[tex]\[ x < 1 \][/tex]
#### Solving the second inequality:
[tex]\[ -8x - 9 < -7x - 3 \][/tex]
Add [tex]\(7x\)[/tex] to both sides:
[tex]\[ -8x + 7x - 9 < -3 \][/tex]
Combine like terms:
[tex]\[ -x - 9 < -3 \][/tex]
Add 9 to both sides:
[tex]\[ -x < 6 \][/tex]
Divide by [tex]\(-1\)[/tex] and reverse the inequality sign:
[tex]\[ x > -6 \][/tex]
#### Combined solution:
-6 < x < 1
This compound inequality has a solution.
### Compound Inequality 2:
[tex]\[ 3x + 1 < 5x + 7 \quad \text{and} \quad -2x + 5 \leq -5x - 10 \][/tex]
#### Solving the first inequality:
[tex]\[ 3x + 1 < 5x + 7 \][/tex]
Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[ 3x - 5x + 1 < 7 \][/tex]
Combine like terms:
[tex]\[ -2x + 1 < 7 \][/tex]
Subtract 1 from both sides:
[tex]\[ -2x < 6 \][/tex]
Divide by [tex]\(-2\)[/tex] and reverse the inequality sign:
[tex]\[ x > -3 \][/tex]
#### Solving the second inequality:
[tex]\[ -2x + 5 \leq -5x - 10 \][/tex]
Add [tex]\(5x\)[/tex] to both sides:
[tex]\[ -2x + 5x + 5 \leq -10 \][/tex]
Combine like terms:
[tex]\[ 3x + 5 \leq -10 \][/tex]
Subtract 5 from both sides:
[tex]\[ 3x \leq -15 \][/tex]
Divide by 3:
[tex]\[ x \leq -5 \][/tex]
#### Combined solution:
[tex]\[ x > -3 \quad \text{and} \quad x \leq -5 \][/tex]
Since there are no values of [tex]\(x\)[/tex] that can satisfy both conditions simultaneously, this compound inequality has no solution.
### Compound Inequality 3:
[tex]\[ -2(x-9) < 3(x+4) \quad \text{and} \quad -4(x-1) > -5(x-2) \][/tex]
#### Solving the first inequality:
[tex]\[ -2(x-9) < 3(x+4) \][/tex]
Distribute:
[tex]\[ -2x + 18 < 3x + 12 \][/tex]
Add [tex]\(2x\)[/tex] to both sides:
[tex]\[ 18 < 5x + 12 \][/tex]
Subtract 12 from both sides:
[tex]\[ 6 < 5x \][/tex]
Divide by 5:
[tex]\[ \frac{6}{5} < x \quad \text{or} \quad x > \frac{6}{5} \][/tex]
#### Solving the second inequality:
[tex]\[ -4(x-1) > -5(x-2) \][/tex]
Distribute:
[tex]\[ -4x + 4 > -5x + 10 \][/tex]
Add [tex]\(5x\)[/tex] to both sides:
[tex]\[ x + 4 > 10 \][/tex]
Subtract 4 from both sides:
[tex]\[ x > 6 \][/tex]
#### Combined solution:
[tex]\[ x > \frac{6}{5} \quad \text{and} \quad x > 6 \][/tex]
Since the [tex]\(x > 6\)[/tex] is the more restrictive condition, the combined solution is [tex]\(x > 6\)[/tex]. This compound inequality has a solution.
### Compound Inequality 4:
[tex]\[ 3x - 7 \leq 5x + 5 \quad \text{and} \quad -4x + 8 < 5x - 1 \][/tex]
#### Solving the first inequality:
[tex]\[ 3x - 7 \leq 5x + 5 \][/tex]
Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[ 3x - 5x - 7 \leq 5 \][/tex]
Combine like terms:
[tex]\[ -2x - 7 \leq 5 \][/tex]
Add 7 to both sides:
[tex]\[ -2x \leq 12 \][/tex]
Divide by [tex]\(-2\)[/tex] and reverse the inequality sign:
[tex]\[ x \geq -6 \][/tex]
#### Solving the second inequality:
[tex]\[ -4x + 8 < 5x - 1 \][/tex]
Add [tex]\(4x\)[/tex] to both sides:
[tex]\[ 8 < 9x - 1 \][/tex]
Add 1 to both sides:
[tex]\[ 9 < 9x \][/tex]
Divide by 9:
[tex]\[ 1 < x \quad \text{or} \quad x > 1 \][/tex]
#### Combined solution:
[tex]\[ x \geq -6 \quad \text{and} \quad x > 1 \][/tex]
Since the [tex]\(x > 1\)[/tex] is the more restrictive condition, the combined solution is [tex]\(x > 1\)[/tex]. This compound inequality has a solution.
### Conclusion:
Compound Inequality 2, [tex]\(3x + 1 < 5x + 7\)[/tex] and [tex]\(-2x + 5 \leq -5x - 10\)[/tex], has no solution.
### Compound Inequality 1:
[tex]\[ -2x + 9 > 4x + 3 \quad \text{and} \quad -8x - 9 < -7x - 3 \][/tex]
#### Solving the first inequality:
[tex]\[ -2x + 9 > 4x + 3 \][/tex]
Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[ -2x - 4x + 9 > 3 \][/tex]
Combine like terms:
[tex]\[ -6x + 9 > 3 \][/tex]
Subtract 9 from both sides:
[tex]\[ -6x > -6 \][/tex]
Divide by [tex]\(-6\)[/tex] and reverse the inequality sign:
[tex]\[ x < 1 \][/tex]
#### Solving the second inequality:
[tex]\[ -8x - 9 < -7x - 3 \][/tex]
Add [tex]\(7x\)[/tex] to both sides:
[tex]\[ -8x + 7x - 9 < -3 \][/tex]
Combine like terms:
[tex]\[ -x - 9 < -3 \][/tex]
Add 9 to both sides:
[tex]\[ -x < 6 \][/tex]
Divide by [tex]\(-1\)[/tex] and reverse the inequality sign:
[tex]\[ x > -6 \][/tex]
#### Combined solution:
-6 < x < 1
This compound inequality has a solution.
### Compound Inequality 2:
[tex]\[ 3x + 1 < 5x + 7 \quad \text{and} \quad -2x + 5 \leq -5x - 10 \][/tex]
#### Solving the first inequality:
[tex]\[ 3x + 1 < 5x + 7 \][/tex]
Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[ 3x - 5x + 1 < 7 \][/tex]
Combine like terms:
[tex]\[ -2x + 1 < 7 \][/tex]
Subtract 1 from both sides:
[tex]\[ -2x < 6 \][/tex]
Divide by [tex]\(-2\)[/tex] and reverse the inequality sign:
[tex]\[ x > -3 \][/tex]
#### Solving the second inequality:
[tex]\[ -2x + 5 \leq -5x - 10 \][/tex]
Add [tex]\(5x\)[/tex] to both sides:
[tex]\[ -2x + 5x + 5 \leq -10 \][/tex]
Combine like terms:
[tex]\[ 3x + 5 \leq -10 \][/tex]
Subtract 5 from both sides:
[tex]\[ 3x \leq -15 \][/tex]
Divide by 3:
[tex]\[ x \leq -5 \][/tex]
#### Combined solution:
[tex]\[ x > -3 \quad \text{and} \quad x \leq -5 \][/tex]
Since there are no values of [tex]\(x\)[/tex] that can satisfy both conditions simultaneously, this compound inequality has no solution.
### Compound Inequality 3:
[tex]\[ -2(x-9) < 3(x+4) \quad \text{and} \quad -4(x-1) > -5(x-2) \][/tex]
#### Solving the first inequality:
[tex]\[ -2(x-9) < 3(x+4) \][/tex]
Distribute:
[tex]\[ -2x + 18 < 3x + 12 \][/tex]
Add [tex]\(2x\)[/tex] to both sides:
[tex]\[ 18 < 5x + 12 \][/tex]
Subtract 12 from both sides:
[tex]\[ 6 < 5x \][/tex]
Divide by 5:
[tex]\[ \frac{6}{5} < x \quad \text{or} \quad x > \frac{6}{5} \][/tex]
#### Solving the second inequality:
[tex]\[ -4(x-1) > -5(x-2) \][/tex]
Distribute:
[tex]\[ -4x + 4 > -5x + 10 \][/tex]
Add [tex]\(5x\)[/tex] to both sides:
[tex]\[ x + 4 > 10 \][/tex]
Subtract 4 from both sides:
[tex]\[ x > 6 \][/tex]
#### Combined solution:
[tex]\[ x > \frac{6}{5} \quad \text{and} \quad x > 6 \][/tex]
Since the [tex]\(x > 6\)[/tex] is the more restrictive condition, the combined solution is [tex]\(x > 6\)[/tex]. This compound inequality has a solution.
### Compound Inequality 4:
[tex]\[ 3x - 7 \leq 5x + 5 \quad \text{and} \quad -4x + 8 < 5x - 1 \][/tex]
#### Solving the first inequality:
[tex]\[ 3x - 7 \leq 5x + 5 \][/tex]
Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[ 3x - 5x - 7 \leq 5 \][/tex]
Combine like terms:
[tex]\[ -2x - 7 \leq 5 \][/tex]
Add 7 to both sides:
[tex]\[ -2x \leq 12 \][/tex]
Divide by [tex]\(-2\)[/tex] and reverse the inequality sign:
[tex]\[ x \geq -6 \][/tex]
#### Solving the second inequality:
[tex]\[ -4x + 8 < 5x - 1 \][/tex]
Add [tex]\(4x\)[/tex] to both sides:
[tex]\[ 8 < 9x - 1 \][/tex]
Add 1 to both sides:
[tex]\[ 9 < 9x \][/tex]
Divide by 9:
[tex]\[ 1 < x \quad \text{or} \quad x > 1 \][/tex]
#### Combined solution:
[tex]\[ x \geq -6 \quad \text{and} \quad x > 1 \][/tex]
Since the [tex]\(x > 1\)[/tex] is the more restrictive condition, the combined solution is [tex]\(x > 1\)[/tex]. This compound inequality has a solution.
### Conclusion:
Compound Inequality 2, [tex]\(3x + 1 < 5x + 7\)[/tex] and [tex]\(-2x + 5 \leq -5x - 10\)[/tex], has no solution.
