To find the least and greatest amounts the two friends could spend on food, we need to solve the equation given by:
[tex]\[ \frac{1}{2} | x - 110 | = 6 \][/tex]
Let's solve this step-by-step:
1. Eliminate the fraction: To remove the fraction, multiply both sides of the equation by 2.
[tex]\[ | x - 110 | = 12 \][/tex]
2. Interpret the absolute value: Absolute value equations can be split into two separate linear equations because the absolute value of a number is always positive. So, we have:
[tex]\[ x - 110 = 12 \][/tex]
[tex]\[ x - 110 = -12 \][/tex]
3. Solve each linear equation:
- For the first equation [tex]\( x - 110 = 12 \)[/tex]:
[tex]\[ x = 110 + 12 \][/tex]
[tex]\[ x = 122 \][/tex]
- For the second equation [tex]\( x - 110 = -12 \)[/tex]:
[tex]\[ x = 110 - 12 \][/tex]
[tex]\[ x = 98 \][/tex]
Therefore, the least amount the two friends could spend on food is [tex]$\$[/tex]98[tex]$, and the greatest amount they could spend is $[/tex]\[tex]$122$[/tex].
Comparing this with the multiple-choice options provided:
- [tex]$ \[ \ \ \ \ \ \$[/tex] 104, \[tex]$ 116 \]
- $[/tex] [tex]\[ \ \ \ \ \ \$ 107, \$ 113 \][/tex]
- [tex]$ \[ \ \ \ \ \ \$[/tex] 110, \[tex]$ 122 \]
None of these options correctly list both \$[/tex]98 and \[tex]$122. However, based on the calculations, the correct answer should include these two values.
Since the correct calculations for the food budget yield \( \$[/tex]98 \) and [tex]\( \$122 \)[/tex], and none of the provided options matches exactly, the problem might have a miswritten or mislabeled solution set.
