Solve. \(\frac{x+3}{4} - \frac{x-2}{3} = 1\)
Question 1 Report
Solve.
\(\frac{x+3}{4} - \frac{x-2}{3} = 1\)
Answer Details
When an equation contains fractions, clear them first by multiplying every term by the lowest common multiple of the denominators. Here the denominators are 4 and 3, so the LCM is 12.
\[ 12 \times \frac{x+3}{4} - 12 \times \frac{x-2}{3} = 12 \times 1 \] \[ 3(x+3) - 4(x-2) = 12 \] [M1]Now expand both brackets, taking care with the subtraction: the \(-4\) multiplies both terms inside the second bracket, so \(-4 \times (-2) = +8\).
\[ 3x + 9 - 4x + 8 = 12 \] \[ -x + 17 = 12 \] [M1]Solve the resulting linear equation:
\[ -x = 12 - 17 = -5 \;\Rightarrow\; x = 5. \] [A1]Check in the original equation: \(\frac{5+3}{4} - \frac{5-2}{3} = \frac{8}{4} - \frac{3}{3} = 2 - 1 = 1\), which is correct.
Two errors dominate here. The first is forgetting to multiply the right-hand side by 12, leaving \(3(x+3) - 4(x-2) = 1\) and giving \(x = 16\). The second is the sign inside the second bracket: writing \(-4x - 8\) instead of \(-4x + 8\) leads to \(x = -1\). Both are caught immediately by substituting the answer back into the original fractions.
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