Jen solved the equation [tex]$100 - \frac{1}{2}x = 92$[/tex] to determine the number of questions, [tex]$x$[/tex], she answered incorrectly on a test. She began by subtracting 100 from both sides, then multiplying both sides by -2. Which best describes the accuracy of Jen's work?

A. Incorrect. She needed to add 100 to both sides instead of subtracting 100.
B. Incorrect. She needed to multiply both sides by -2 first, then subtract 100.
C. Correct. She could have also subtracted 100 from both sides, then divided by [tex]$-\frac{1}{2}$[/tex].
D. Correct. She could have also multiplied both sides by -2 first, then subtracted 100.



Answer :

Let's solve the equation [tex]\( 100 - \frac{1}{2} x = 92 \)[/tex] step by step to determine the number of questions [tex]\( x \)[/tex] that Jen answered incorrectly.

### Step-by-Step Solution:

1. Start with the given equation:
[tex]\[ 100 - \frac{1}{2} x = 92 \][/tex]

2. Subtract 100 from both sides:
[tex]\[ 100 - \frac{1}{2} x - 100 = 92 - 100 \][/tex]
Simplifying this, we get:
[tex]\[ -\frac{1}{2} x = -8 \][/tex]

3. Multiply both sides by -2 to isolate [tex]\( x \)[/tex]:
[tex]\[ -2 \left( -\frac{1}{2} x \right) = -2 \left( -8 \right) \][/tex]
Simplifying this, we get:
[tex]\[ x = 16 \][/tex]

So, the value of [tex]\( x \)[/tex] is 16.

### Analysis of Jen's Steps:

Jen's steps were as follows:
1. She subtracted 100 from both sides, which is the correct first step:
[tex]\[ 100 - \frac{1}{2} x = 92 \][/tex]
[tex]\[ -\frac{1}{2} x = -8 \][/tex]

2. She then multiplied both sides by -2, which is also correct:
[tex]\[ x = 16 \][/tex]

Based on the provided options, the correct one should match the above steps. The accurate option is:

- Correct. She could have also subtracted 100 from both sides, then divided by [tex]\(-\frac{1}{2}\)[/tex].

Indeed, after subtracting 100 from both sides to get:
[tex]\[ -\frac{1}{2} x = -8 \][/tex]
she could have alternatively divided both sides by [tex]\(-\frac{1}{2}\)[/tex] instead of multiplying by -2. Dividing both sides by [tex]\(-\frac{1}{2}\)[/tex] would also isolate [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-8}{-\frac{1}{2}} = 16 \][/tex]

Thus, the most accurate description of Jen's work is:

Correct. She could have also subtracted 100 from both sides, then divided by [tex]\( -\frac{1}{2} \)[/tex].