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Answer :
To find the correct inequality for the isosceles triangle that gives a perimeter of at least 137 feet, we need to evaluate each of the given inequality options and see which one meets this condition.
Let's analyze each option:
1. [tex]\(8x + 17 \geq 137\)[/tex]
- To solve for [tex]\(x\)[/tex], subtract 17 from both sides:
[tex]\[
8x \geq 137 - 17
\][/tex]
[tex]\[
8x \geq 120
\][/tex]
- Now, divide by 8:
[tex]\[
x \geq 15
\][/tex]
2. [tex]\(8x + 21 \geq 137\)[/tex]
- Subtract 21 from both sides:
[tex]\[
8x \geq 137 - 21
\][/tex]
[tex]\[
8x \geq 116
\][/tex]
- Divide by 8:
[tex]\[
x \geq 14.5
\][/tex]
3. [tex]\(6x + 10 \geq 137\)[/tex]
- Subtract 10 from both sides:
[tex]\[
6x \geq 137 - 10
\][/tex]
[tex]\[
6x \geq 127
\][/tex]
- Divide by 6:
[tex]\[
x \geq 21.1667
\][/tex]
4. [tex]\(6x + 10 \geq 127\)[/tex]
- Subtract 10 from both sides:
[tex]\[
6x \geq 127 - 10
\][/tex]
[tex]\[
6x \geq 117
\][/tex]
- Divide by 6:
[tex]\[
x \geq 19.5
\][/tex]
Now, let's determine the inequality that satisfies the condition of having a perimeter of at least 137 feet.
- Inequality 1: [tex]\(x \geq 15\)[/tex]
- Inequality 2: [tex]\(x \geq 14.5\)[/tex]
- Inequality 3: [tex]\(x \geq 21.1667\)[/tex]
- Inequality 4: [tex]\(x \geq 19.5\)[/tex]
The inequality that provides a sufficient perimeter of at least 137 feet, with the smallest valid [tex]\(x\)[/tex], is option 2: [tex]\(8x + 21 \geq 137\)[/tex]. This means the solution to our problem is option [tex]\(8x + 21 \geq 137\)[/tex].
Let's analyze each option:
1. [tex]\(8x + 17 \geq 137\)[/tex]
- To solve for [tex]\(x\)[/tex], subtract 17 from both sides:
[tex]\[
8x \geq 137 - 17
\][/tex]
[tex]\[
8x \geq 120
\][/tex]
- Now, divide by 8:
[tex]\[
x \geq 15
\][/tex]
2. [tex]\(8x + 21 \geq 137\)[/tex]
- Subtract 21 from both sides:
[tex]\[
8x \geq 137 - 21
\][/tex]
[tex]\[
8x \geq 116
\][/tex]
- Divide by 8:
[tex]\[
x \geq 14.5
\][/tex]
3. [tex]\(6x + 10 \geq 137\)[/tex]
- Subtract 10 from both sides:
[tex]\[
6x \geq 137 - 10
\][/tex]
[tex]\[
6x \geq 127
\][/tex]
- Divide by 6:
[tex]\[
x \geq 21.1667
\][/tex]
4. [tex]\(6x + 10 \geq 127\)[/tex]
- Subtract 10 from both sides:
[tex]\[
6x \geq 127 - 10
\][/tex]
[tex]\[
6x \geq 117
\][/tex]
- Divide by 6:
[tex]\[
x \geq 19.5
\][/tex]
Now, let's determine the inequality that satisfies the condition of having a perimeter of at least 137 feet.
- Inequality 1: [tex]\(x \geq 15\)[/tex]
- Inequality 2: [tex]\(x \geq 14.5\)[/tex]
- Inequality 3: [tex]\(x \geq 21.1667\)[/tex]
- Inequality 4: [tex]\(x \geq 19.5\)[/tex]
The inequality that provides a sufficient perimeter of at least 137 feet, with the smallest valid [tex]\(x\)[/tex], is option 2: [tex]\(8x + 21 \geq 137\)[/tex]. This means the solution to our problem is option [tex]\(8x + 21 \geq 137\)[/tex].
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