Let's carefully examine Rohan's solution step-by-step to identify where he made a mistake.
Given equation:
[tex]\[ 2x - 3 = \frac{x}{2} - 5 \][/tex]
Step 1: Eliminating the fraction
To eliminate the fraction, multiply both sides of the equation by 2:
[tex]\[ 2 \cdot (2x - 3) = 2 \cdot \left(\frac{x}{2} - 5 \right) \][/tex]
Step 2: Simplifying both sides
Carrying out the multiplication:
[tex]\[ 4x - 6 = x - 10 \][/tex]
At this stage, Step 2 is already different from Rohan’s:
Rohan’s Step 2:
[tex]\[ 4x - 6 = x - 5 \][/tex]
Therefore, Rohan made an error here. Correctly simplifying both sides should give:
[tex]\[ 4x - 6 = x - 10 \][/tex]
Step 3: Rearranging terms to isolate [tex]\(x\)[/tex]
Moving [tex]\(x\)[/tex] to one side and constants to the other:
[tex]\[ 4x - x = -10 + 6 \][/tex]
[tex]\[ 3x = -4 \][/tex]
Notice in Rohan’s work:
Rohan’s Step 3:
[tex]\[ 4x - x = 6 - 5 \][/tex]
[tex]\[ 3x = 1 \][/tex]
Rohan’s Step 3 is incorrect. He mistakenly added 6 to the right side instead of subtracting it correctly.
Correct Step 4: Solving for [tex]\(x\)[/tex]
Now, divide both sides by 3 to isolate [tex]\(x\)[/tex]:
[tex]\[ x = \frac{-4}{3} \][/tex]
Therefore, Rohan made a mistake in Step 3 when he incorrectly handled the constants. His final solution [tex]\(x = \frac{1}{3}\)[/tex] is incorrect due to this mistake. The correct solution is [tex]\( x = \frac{-4}{3} \)[/tex].
