Join the IDNLearn.com community and start getting the answers you need today. Find the information you need quickly and easily with our comprehensive and accurate Q&A platform.
Sagot :
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].
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.
