Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Let's solve each inequality step-by-step and match each solution on the left side to the correct answer on the right side.
### Inequality 1: [tex]\(-9(x-1) \geq -3(x+5)\)[/tex]
First, we distribute the terms inside the parentheses:
[tex]\[ -9(x-1) \geq -3(x+5) \][/tex]
[tex]\[ -9x + 9 \geq -3x - 15 \][/tex]
Next, combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ -9x + 9 + 3x \geq -3x - 15 + 3x \][/tex]
[tex]\[ -6x + 9 \geq -15 \][/tex]
Subtract 9 from both sides:
[tex]\[ -6x + 9 - 9 \geq -15 - 9 \][/tex]
[tex]\[ -6x \geq -24 \][/tex]
Divide by -6, and remember to reverse the inequality sign:
[tex]\[ x \leq 4 \][/tex]
### Inequality 2: [tex]\(\frac{2x+7}{5} \leq x-1\)[/tex]
Start by eliminating the fraction. Multiply both sides by 5:
[tex]\[ 2x + 7 \leq 5(x-1) \][/tex]
[tex]\[ 2x + 7 \leq 5x - 5 \][/tex]
Next, combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ 2x + 7 - 2x \leq 5x - 5 - 2x \][/tex]
[tex]\[ 7 \leq 3x - 5 \][/tex]
Add 5 to both sides:
[tex]\[ 7 + 5 \leq 3x - 5 + 5 \][/tex]
[tex]\[ 12 \leq 3x \][/tex]
Divide by 3:
[tex]\[ 4 \leq x \][/tex]
Or:
[tex]\[ x \geq 4 \][/tex]
### Inequality 3: [tex]\(2(2x+7) \leq 6(x+2)\)[/tex]
Distribute the constants:
[tex]\[ 2(2x+7) \leq 6(x+2) \][/tex]
[tex]\[ 4x + 14 \leq 6x + 12 \][/tex]
Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ 4x + 14 - 4x \leq 6x + 12 - 4x \][/tex]
[tex]\[ 14 \leq 2x + 12 \][/tex]
Subtract 12 from both sides:
[tex]\[ 14 - 12 \leq 2x + 12 - 12 \][/tex]
[tex]\[ 2 \leq 2x \][/tex]
Divide by 2:
[tex]\[ 1 \leq x \][/tex]
Or:
[tex]\[ x \geq 1 \][/tex]
### Inequality 4: [tex]\(-3(2x-7) \geq -5(x-4)\)[/tex]
Distribute the constants:
[tex]\[ -3(2x-7) \geq -5(x-4) \][/tex]
[tex]\[ -6x + 21 \geq -5x + 20 \][/tex]
Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ -6x + 21 + 5x \geq -5x + 20 + 5x \][/tex]
[tex]\[ -x + 21 \geq 20 \][/tex]
Subtract 21 from both sides:
[tex]\[ -x + 21 - 21 \geq 20 - 21 \][/tex]
[tex]\[ -x \geq -1 \][/tex]
Divide by -1, and remember to reverse the inequality sign:
[tex]\[ x \leq 1 \][/tex]
### Matching the answers:
1. [tex]\(-9(x-1) \geq -3(x+5)\)[/tex] corresponds to [tex]\(x \leq 4\)[/tex].
2. [tex]\(\frac{2x+7}{5} \leq x-1\)[/tex] corresponds to [tex]\(x \geq 4\)[/tex].
3. [tex]\(2(2x+7) \leq 6(x+2)\)[/tex] corresponds to [tex]\(x \geq 1\)[/tex].
4. [tex]\(-3(2x-7) \geq -5(x-4)\)[/tex] corresponds to [tex]\(x \leq 1\)[/tex].
So, the final answers will be:
[tex]\[ \begin{array}{l} -9(x-1) \geq -3(x+5) \implies x \leq 4 \\ \frac{2x+7}{5} \leq x-1 \implies x \geq 4 \\ 2(2x+7) \leq 6(x+2) \implies x \geq 1 \\ -3(2x-7) \geq -5(x-4) \implies x \leq 1 \end{array} \][/tex]
### Inequality 1: [tex]\(-9(x-1) \geq -3(x+5)\)[/tex]
First, we distribute the terms inside the parentheses:
[tex]\[ -9(x-1) \geq -3(x+5) \][/tex]
[tex]\[ -9x + 9 \geq -3x - 15 \][/tex]
Next, combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ -9x + 9 + 3x \geq -3x - 15 + 3x \][/tex]
[tex]\[ -6x + 9 \geq -15 \][/tex]
Subtract 9 from both sides:
[tex]\[ -6x + 9 - 9 \geq -15 - 9 \][/tex]
[tex]\[ -6x \geq -24 \][/tex]
Divide by -6, and remember to reverse the inequality sign:
[tex]\[ x \leq 4 \][/tex]
### Inequality 2: [tex]\(\frac{2x+7}{5} \leq x-1\)[/tex]
Start by eliminating the fraction. Multiply both sides by 5:
[tex]\[ 2x + 7 \leq 5(x-1) \][/tex]
[tex]\[ 2x + 7 \leq 5x - 5 \][/tex]
Next, combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ 2x + 7 - 2x \leq 5x - 5 - 2x \][/tex]
[tex]\[ 7 \leq 3x - 5 \][/tex]
Add 5 to both sides:
[tex]\[ 7 + 5 \leq 3x - 5 + 5 \][/tex]
[tex]\[ 12 \leq 3x \][/tex]
Divide by 3:
[tex]\[ 4 \leq x \][/tex]
Or:
[tex]\[ x \geq 4 \][/tex]
### Inequality 3: [tex]\(2(2x+7) \leq 6(x+2)\)[/tex]
Distribute the constants:
[tex]\[ 2(2x+7) \leq 6(x+2) \][/tex]
[tex]\[ 4x + 14 \leq 6x + 12 \][/tex]
Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ 4x + 14 - 4x \leq 6x + 12 - 4x \][/tex]
[tex]\[ 14 \leq 2x + 12 \][/tex]
Subtract 12 from both sides:
[tex]\[ 14 - 12 \leq 2x + 12 - 12 \][/tex]
[tex]\[ 2 \leq 2x \][/tex]
Divide by 2:
[tex]\[ 1 \leq x \][/tex]
Or:
[tex]\[ x \geq 1 \][/tex]
### Inequality 4: [tex]\(-3(2x-7) \geq -5(x-4)\)[/tex]
Distribute the constants:
[tex]\[ -3(2x-7) \geq -5(x-4) \][/tex]
[tex]\[ -6x + 21 \geq -5x + 20 \][/tex]
Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ -6x + 21 + 5x \geq -5x + 20 + 5x \][/tex]
[tex]\[ -x + 21 \geq 20 \][/tex]
Subtract 21 from both sides:
[tex]\[ -x + 21 - 21 \geq 20 - 21 \][/tex]
[tex]\[ -x \geq -1 \][/tex]
Divide by -1, and remember to reverse the inequality sign:
[tex]\[ x \leq 1 \][/tex]
### Matching the answers:
1. [tex]\(-9(x-1) \geq -3(x+5)\)[/tex] corresponds to [tex]\(x \leq 4\)[/tex].
2. [tex]\(\frac{2x+7}{5} \leq x-1\)[/tex] corresponds to [tex]\(x \geq 4\)[/tex].
3. [tex]\(2(2x+7) \leq 6(x+2)\)[/tex] corresponds to [tex]\(x \geq 1\)[/tex].
4. [tex]\(-3(2x-7) \geq -5(x-4)\)[/tex] corresponds to [tex]\(x \leq 1\)[/tex].
So, the final answers will be:
[tex]\[ \begin{array}{l} -9(x-1) \geq -3(x+5) \implies x \leq 4 \\ \frac{2x+7}{5} \leq x-1 \implies x \geq 4 \\ 2(2x+7) \leq 6(x+2) \implies x \geq 1 \\ -3(2x-7) \geq -5(x-4) \implies x \leq 1 \end{array} \][/tex]
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.