Answer :
To solve the problem of finding an equivalent absolute value inequality for the situation [tex]\(69 \leq t \leq 91\)[/tex], let's follow these steps:
1. Identify the Midpoint:
- The midpoint of the given temperature range can be found by averaging the high and low temperatures.
- Calculating the midpoint: [tex]\( \frac{69 + 91}{2} = \frac{160}{2} = 80 \)[/tex].
2. Determine the Range:
- The range is the distance from the midpoint to either the high or low temperature.
- The distance from 80 to 91 (or from 80 to 69) is [tex]\(91 - 80 = 11\)[/tex] (or [tex]\(80 - 69 = 11\)[/tex]).
3. Formulate the Absolute Value Inequality:
- An absolute value inequality representing the interval [tex]\(69 \leq t \leq 91\)[/tex] can be expressed as [tex]\( |t - 80| \leq 11 \)[/tex].
- This is because the absolute value inequality [tex]\( |t - 80| \)[/tex] represents all values [tex]\( t \)[/tex] that are within 11 units of 80, thus capturing the original range from 69 to 91.
Now let's review the provided options:
- F. [tex]\(69 \leq |t| \leq 91\)[/tex]: This combines the temperatures into an incorrect form and is not equivalent to the interval [tex]\(69 \leq t \leq 91\)[/tex].
- G. [tex]\( |t-80| \leq 11 \)[/tex]: This correctly captures our calculated absolute value inequality.
- H. [tex]\( |t-69| \leq 91 \)[/tex]: This does not accurately describe the interval [tex]\(69 \leq t \leq 91\)[/tex].
- J. [tex]\( |t-11| \leq 80 \)[/tex]: This also does not correctly represent the interval [tex]\(69 \leq t \leq 91\)[/tex].
Therefore, the correct answer is:
G. [tex]\( |t-80| \leq 11 \)[/tex].
1. Identify the Midpoint:
- The midpoint of the given temperature range can be found by averaging the high and low temperatures.
- Calculating the midpoint: [tex]\( \frac{69 + 91}{2} = \frac{160}{2} = 80 \)[/tex].
2. Determine the Range:
- The range is the distance from the midpoint to either the high or low temperature.
- The distance from 80 to 91 (or from 80 to 69) is [tex]\(91 - 80 = 11\)[/tex] (or [tex]\(80 - 69 = 11\)[/tex]).
3. Formulate the Absolute Value Inequality:
- An absolute value inequality representing the interval [tex]\(69 \leq t \leq 91\)[/tex] can be expressed as [tex]\( |t - 80| \leq 11 \)[/tex].
- This is because the absolute value inequality [tex]\( |t - 80| \)[/tex] represents all values [tex]\( t \)[/tex] that are within 11 units of 80, thus capturing the original range from 69 to 91.
Now let's review the provided options:
- F. [tex]\(69 \leq |t| \leq 91\)[/tex]: This combines the temperatures into an incorrect form and is not equivalent to the interval [tex]\(69 \leq t \leq 91\)[/tex].
- G. [tex]\( |t-80| \leq 11 \)[/tex]: This correctly captures our calculated absolute value inequality.
- H. [tex]\( |t-69| \leq 91 \)[/tex]: This does not accurately describe the interval [tex]\(69 \leq t \leq 91\)[/tex].
- J. [tex]\( |t-11| \leq 80 \)[/tex]: This also does not correctly represent the interval [tex]\(69 \leq t \leq 91\)[/tex].
Therefore, the correct answer is:
G. [tex]\( |t-80| \leq 11 \)[/tex].
