Let’s review George's work and identify if there are any mistakes step-by-step.
[tex]\(\frac{8+6}{2}=\frac{11}{2}\)[/tex] is just the midpoint between 8 and 6:
1. George calculated the midpoint correctly:
[tex]\[
\frac{8 + 6}{2} = \frac{14}{2} = 7
\][/tex]
However, it seems there is a typo in the calculations above, as [tex]\( \frac{11}{2} \)[/tex] is not equal to [tex]\(\frac{8 + 6}{2} = 7\)[/tex].
2. In Step 4, George evaluates the rewritten equation at [tex]\(x = \frac{11}{2}\)[/tex] (which should have been at [tex]\(x = 7\)[/tex], but we'll overlook this typo for the current explanation).
3. Step 5 mentions using this value of [tex]\(x\)[/tex] to replace the previous upper bound that yielded a negative difference and sets [tex]\(x = 6\)[/tex] accordingly.
To identify the correct step where George made his first mistake, let's consider the options:
A. In step 3, George should have averaged the two differences instead of the two bounds.
- No, since the goal is to find the midpoint between bounds, not differences.
B. George made no mistakes, and his work is correct.
- This cannot be correct due to the calculation error and the replacement strategy.
C. In step 5, George should have used [tex]\(x = \frac{11}{2}\)[/tex] as the new lower bound.
- George's mistake here lies in misinterpreting the new bound. Since he evaluated at [tex]\(x = \frac{11}{2}\)[/tex] (let’s correct it as [tex]\(7\)[/tex]) and found a negative difference, he should use it as the new lower bound instead of the upper bound.
D. In step 1, George should have subtracted in the opposite order to set the equation equal to 0.
- Step 1 doesn't relate to subtraction order mistakes based on the current context.
So, George made his first mistake in Step 5 by misapplying the new bound. Therefore, the correct answer is:
C. In step 5, George should have used [tex]\(x = \frac{11}{2}\)[/tex] as the new lower bound.
