Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Find reliable answers to your questions from a wide community of knowledgeable experts on our user-friendly Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To determine which inequality has no solution, we will analyze each inequality step-by-step.
### Inequality 1: [tex]\(6(x + 2) > x - 3\)[/tex]
1. Distribute the 6 on the left side:
[tex]\[ 6x + 12 > x - 3 \][/tex]
2. Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[ 5x + 12 > -3 \][/tex]
3. Subtract 12 from both sides:
[tex]\[ 5x > -15 \][/tex]
4. Divide by 5:
[tex]\[ x > -3 \][/tex]
This inequality has a solution [tex]\(x > -3\)[/tex].
### Inequality 2: [tex]\(3 + 4x \leq 2(1 + 2x)\)[/tex]
1. Distribute the 2 on the right side:
[tex]\[ 3 + 4x \leq 2 + 4x \][/tex]
2. Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[ 3 \leq 2 \][/tex]
This is a contradiction because 3 is never less than or equal to 2. Therefore, this inequality has no solution.
### Inequality 3: [tex]\(-2(x + 6) < x - 20\)[/tex]
1. Distribute the [tex]\(-2\)[/tex] on the left side:
[tex]\[ -2x - 12 < x - 20 \][/tex]
2. Add [tex]\(2x\)[/tex] to both sides:
[tex]\[ -12 < 3x - 20 \][/tex]
3. Add 20 to both sides:
[tex]\[ 8 < 3x \][/tex]
4. Divide by 3:
[tex]\[ \frac{8}{3} < x \][/tex]
or
[tex]\[ x > \frac{8}{3} \][/tex]
This inequality has a solution [tex]\(x > \frac{8}{3}\)[/tex].
### Inequality 4: [tex]\(x - 9 < 3(x - 3)\)[/tex]
1. Distribute the 3 on the right side:
[tex]\[ x - 9 < 3x - 9 \][/tex]
2. Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[ -9 < 2x - 9 \][/tex]
3. Add 9 to both sides:
[tex]\[ 0 < 2x \][/tex]
4. Divide by 2:
[tex]\[ 0 < x \][/tex]
or
[tex]\[ x > 0 \][/tex]
This inequality has a solution [tex]\(x > 0\)[/tex].
### Conclusion
The inequality that has no solution is:
[tex]\[ 3 + 4x \leq 2(1 + 2x) \][/tex]
### Inequality 1: [tex]\(6(x + 2) > x - 3\)[/tex]
1. Distribute the 6 on the left side:
[tex]\[ 6x + 12 > x - 3 \][/tex]
2. Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[ 5x + 12 > -3 \][/tex]
3. Subtract 12 from both sides:
[tex]\[ 5x > -15 \][/tex]
4. Divide by 5:
[tex]\[ x > -3 \][/tex]
This inequality has a solution [tex]\(x > -3\)[/tex].
### Inequality 2: [tex]\(3 + 4x \leq 2(1 + 2x)\)[/tex]
1. Distribute the 2 on the right side:
[tex]\[ 3 + 4x \leq 2 + 4x \][/tex]
2. Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[ 3 \leq 2 \][/tex]
This is a contradiction because 3 is never less than or equal to 2. Therefore, this inequality has no solution.
### Inequality 3: [tex]\(-2(x + 6) < x - 20\)[/tex]
1. Distribute the [tex]\(-2\)[/tex] on the left side:
[tex]\[ -2x - 12 < x - 20 \][/tex]
2. Add [tex]\(2x\)[/tex] to both sides:
[tex]\[ -12 < 3x - 20 \][/tex]
3. Add 20 to both sides:
[tex]\[ 8 < 3x \][/tex]
4. Divide by 3:
[tex]\[ \frac{8}{3} < x \][/tex]
or
[tex]\[ x > \frac{8}{3} \][/tex]
This inequality has a solution [tex]\(x > \frac{8}{3}\)[/tex].
### Inequality 4: [tex]\(x - 9 < 3(x - 3)\)[/tex]
1. Distribute the 3 on the right side:
[tex]\[ x - 9 < 3x - 9 \][/tex]
2. Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[ -9 < 2x - 9 \][/tex]
3. Add 9 to both sides:
[tex]\[ 0 < 2x \][/tex]
4. Divide by 2:
[tex]\[ 0 < x \][/tex]
or
[tex]\[ x > 0 \][/tex]
This inequality has a solution [tex]\(x > 0\)[/tex].
### Conclusion
The inequality that has no solution is:
[tex]\[ 3 + 4x \leq 2(1 + 2x) \][/tex]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.