Answer :
Let's analyze each of the given compound inequalities step-by-step to find out which one has no solution.
1. [tex]\(3(x+1) \geq 2(x+5)\)[/tex] and [tex]\(-2(x+1) \leq 3(x+1)\)[/tex]
First inequality:
[tex]\[ 3(x + 1) \geq 2(x + 5) \][/tex]
[tex]\[ 3x + 3 \geq 2x + 10 \][/tex]
[tex]\[ 3x - 2x \geq 10 - 3 \][/tex]
[tex]\[ x \geq 7 \][/tex]
Second inequality:
[tex]\[ -2(x + 1) \leq 3(x + 1) \][/tex]
[tex]\[ -2x - 2 \leq 3x + 3 \][/tex]
[tex]\[ -2 - 3 \leq 3x + 2x \][/tex]
[tex]\[ -5 \leq 5x \][/tex]
[tex]\[ -1 \leq x \][/tex]
Combining the two:
[tex]\[ -1 \leq x \quad \text{and} \quad x \geq 7 \][/tex]
This simplifies to:
[tex]\[ x \geq 7 \][/tex]
So, this set of inequalities has a solution where [tex]\( x \geq 7 \)[/tex].
2. [tex]\(2x - 1 < x - 4\)[/tex] and [tex]\(-4x + 3 > 2x - 9\)[/tex]
First inequality:
[tex]\[ 2x - 1 < x - 4 \][/tex]
[tex]\[ 2x - x < -4 + 1 \][/tex]
[tex]\[ x < -3 \][/tex]
Second inequality:
[tex]\[ -4x + 3 > 2x - 9 \][/tex]
[tex]\[ -4x - 2x > -9 - 3 \][/tex]
[tex]\[ -6x > -12 \][/tex]
[tex]\[ x < 2 \][/tex]
Combining the two:
[tex]\[ x < -3 \quad \text{and} \quad x < 2 \][/tex]
This simplifies to:
[tex]\[ x < -3 \][/tex]
So, this set of inequalities has a solution where [tex]\( x < -3 \)[/tex].
3. [tex]\(-3x - 1 \leq 2x + 9\)[/tex] and [tex]\(-4x + 3 > -2x + 11\)[/tex]
First inequality:
[tex]\[ -3x - 1 \leq 2x + 9 \][/tex]
[tex]\[ -3x - 2x \leq 9 + 1 \][/tex]
[tex]\[ -5x \leq 10 \][/tex]
[tex]\[ x \geq -2 \][/tex]
Second inequality:
[tex]\[ -4x + 3 > -2x + 11 \][/tex]
[tex]\[ -4x + 2x > 11 - 3 \][/tex]
[tex]\[ -2x > 8 \][/tex]
[tex]\[ x < -4 \][/tex]
Combining the two:
[tex]\[ x \geq -2 \quad \text{and} \quad x < -4 \][/tex]
There is no [tex]\( x \)[/tex] that satisfies both conditions simultaneously. Therefore, this set of inequalities has no solution.
4. [tex]\(2x - 5 \leq 3x + 5\)[/tex] and [tex]\(-4x - 1 > 2x + 3\)[/tex]
First inequality:
[tex]\[ 2x - 5 \leq 3x + 5 \][/tex]
[tex]\[ 2x - 3x \leq 5 + 5 \][/tex]
[tex]\[ -x \leq 10 \][/tex]
[tex]\[ x \geq -10 \][/tex]
Second inequality:
[tex]\[ -4x - 1 > 2x + 3 \][/tex]
[tex]\[ -4x - 2x > 3 + 1 \][/tex]
[tex]\[ -6x > 4 \][/tex]
[tex]\[ x < -\frac{2}{3} \][/tex]
Combining the two:
[tex]\[ x \geq -10 \quad \text{and} \quad x < -\frac{2}{3} \][/tex]
This simplifies to:
[tex]\[ -10 \leq x < -\frac{2}{3} \][/tex]
So, this set of inequalities has a solution in the range [tex]\([-10, -\frac{2}{3}]\)[/tex].
After analyzing all the inequalities, we find that the compound inequality:
[tex]\[ -3x - 1 \leq 2x + 9 \quad \text{and} \quad -4x + 3 > -2x + 11 \][/tex]
has no solution.
Therefore, the answer is [tex]\( \boxed{3} \)[/tex].
1. [tex]\(3(x+1) \geq 2(x+5)\)[/tex] and [tex]\(-2(x+1) \leq 3(x+1)\)[/tex]
First inequality:
[tex]\[ 3(x + 1) \geq 2(x + 5) \][/tex]
[tex]\[ 3x + 3 \geq 2x + 10 \][/tex]
[tex]\[ 3x - 2x \geq 10 - 3 \][/tex]
[tex]\[ x \geq 7 \][/tex]
Second inequality:
[tex]\[ -2(x + 1) \leq 3(x + 1) \][/tex]
[tex]\[ -2x - 2 \leq 3x + 3 \][/tex]
[tex]\[ -2 - 3 \leq 3x + 2x \][/tex]
[tex]\[ -5 \leq 5x \][/tex]
[tex]\[ -1 \leq x \][/tex]
Combining the two:
[tex]\[ -1 \leq x \quad \text{and} \quad x \geq 7 \][/tex]
This simplifies to:
[tex]\[ x \geq 7 \][/tex]
So, this set of inequalities has a solution where [tex]\( x \geq 7 \)[/tex].
2. [tex]\(2x - 1 < x - 4\)[/tex] and [tex]\(-4x + 3 > 2x - 9\)[/tex]
First inequality:
[tex]\[ 2x - 1 < x - 4 \][/tex]
[tex]\[ 2x - x < -4 + 1 \][/tex]
[tex]\[ x < -3 \][/tex]
Second inequality:
[tex]\[ -4x + 3 > 2x - 9 \][/tex]
[tex]\[ -4x - 2x > -9 - 3 \][/tex]
[tex]\[ -6x > -12 \][/tex]
[tex]\[ x < 2 \][/tex]
Combining the two:
[tex]\[ x < -3 \quad \text{and} \quad x < 2 \][/tex]
This simplifies to:
[tex]\[ x < -3 \][/tex]
So, this set of inequalities has a solution where [tex]\( x < -3 \)[/tex].
3. [tex]\(-3x - 1 \leq 2x + 9\)[/tex] and [tex]\(-4x + 3 > -2x + 11\)[/tex]
First inequality:
[tex]\[ -3x - 1 \leq 2x + 9 \][/tex]
[tex]\[ -3x - 2x \leq 9 + 1 \][/tex]
[tex]\[ -5x \leq 10 \][/tex]
[tex]\[ x \geq -2 \][/tex]
Second inequality:
[tex]\[ -4x + 3 > -2x + 11 \][/tex]
[tex]\[ -4x + 2x > 11 - 3 \][/tex]
[tex]\[ -2x > 8 \][/tex]
[tex]\[ x < -4 \][/tex]
Combining the two:
[tex]\[ x \geq -2 \quad \text{and} \quad x < -4 \][/tex]
There is no [tex]\( x \)[/tex] that satisfies both conditions simultaneously. Therefore, this set of inequalities has no solution.
4. [tex]\(2x - 5 \leq 3x + 5\)[/tex] and [tex]\(-4x - 1 > 2x + 3\)[/tex]
First inequality:
[tex]\[ 2x - 5 \leq 3x + 5 \][/tex]
[tex]\[ 2x - 3x \leq 5 + 5 \][/tex]
[tex]\[ -x \leq 10 \][/tex]
[tex]\[ x \geq -10 \][/tex]
Second inequality:
[tex]\[ -4x - 1 > 2x + 3 \][/tex]
[tex]\[ -4x - 2x > 3 + 1 \][/tex]
[tex]\[ -6x > 4 \][/tex]
[tex]\[ x < -\frac{2}{3} \][/tex]
Combining the two:
[tex]\[ x \geq -10 \quad \text{and} \quad x < -\frac{2}{3} \][/tex]
This simplifies to:
[tex]\[ -10 \leq x < -\frac{2}{3} \][/tex]
So, this set of inequalities has a solution in the range [tex]\([-10, -\frac{2}{3}]\)[/tex].
After analyzing all the inequalities, we find that the compound inequality:
[tex]\[ -3x - 1 \leq 2x + 9 \quad \text{and} \quad -4x + 3 > -2x + 11 \][/tex]
has no solution.
Therefore, the answer is [tex]\( \boxed{3} \)[/tex].
