Let's carefully analyze Jerome's steps in solving the equation [tex]\(\frac{1}{3} x + \frac{5}{6} = 1\)[/tex].
1. Subtraction property:
Jerome applied the subtraction property correctly:
[tex]\[
\frac{1}{3} x + \frac{5}{6} - \frac{5}{6} = 1 - \frac{5}{6}
\][/tex]
Simplifying the right side:
[tex]\[
\frac{1}{3} x = 1 - \frac{5}{6}
\][/tex]
Since [tex]\(1\)[/tex] can be written as [tex]\(\frac{6}{6}\)[/tex]:
[tex]\[
1 - \frac{5}{6} = \frac{6}{6} - \frac{5}{6}
\][/tex]
This simplifies correctly to:
[tex]\[
\frac{1}{6}
\][/tex]
2. LCD:
Jerome then correctly expressed the equation:
[tex]\[
\frac{1}{3} x = \frac{1}{6}
\][/tex]
3. Multiply by the reciprocal:
Jerome attempted to isolate [tex]\(x\)[/tex] by multiplying both sides by the reciprocal of [tex]\(\frac{1}{3}\)[/tex], which is [tex]\(3\)[/tex]:
[tex]\[
\left(\frac{3}{1}\right) \cdot \frac{1}{3} x = \frac{1}{6} \cdot \left(\frac{3}{1}\right)
\][/tex]
On the left side:
[tex]\[
\left(\frac{3}{1}\right) \cdot \frac{1}{3} x = x
\][/tex]
On the right side:
[tex]\[
\frac{1}{6} \cdot 3 = \frac{3}{6}
\][/tex]
4. Solve and simplify:
Jerome then simplified [tex]\(\frac{3}{6}\)[/tex]:
[tex]\[
x = \frac{3}{6}
\][/tex]
However, Jerome incorrectly simplified this fraction. Instead of simplifying [tex]\(\frac{3}{6}\)[/tex] to [tex]\(\frac{1}{2}\)[/tex], he wrote it as [tex]\(x = -2\)[/tex].
Therefore, Jerome's error occurred in step 4 where he failed to correctly simplify the fraction [tex]\(\frac{3}{6}\)[/tex] to [tex]\(\frac{1}{2}\)[/tex].
The correct solution is:
[tex]\[
x = \frac{1}{2}
\][/tex]
