Solve for x: (x - 1)/2 = 2 - (x + 2)/5.

Step-by-step solution

1. 1

   Multiply every term by the LCD. The denominators are $2$ and $5$, so the least common denominator is $10$. Multiplying each term — including the standalone $2$ — by $10$:

   $10\cdot\frac{x-1}{2}=10\cdot 2-10\cdot\frac{x+2}{5}\;\Longrightarrow\;5(x-1)=20-2(x+2).$

   Skipping the $10\times2=20$ is the error that most often derails this problem.

2. 2

   Expand both sides. On the left, $5(x-1)=5x-5$. On the right the bracket is subtracted, so both of its terms flip sign:

   $20-2(x+2)=20-2x-4=16-2x.$

   Writing $20-2x-4$ before combining makes the sign of the $-4$ explicit; going straight to $16-2x$ in one jump is where a $+4$ tends to appear.

3. 3

   Collect $x$ terms on one side. The equation is now $5x-5=16-2x$. Adding $2x$ to both sides and adding $5$:

   $5x+2x=16+5\;\Longrightarrow\;7x=21.$

   Moving the $-2x$ to the left keeps the coefficient positive and avoids a division by a negative number.

4. 4

   Divide and state the answer. Dividing both sides by $7$:

   $x=\frac{21}{7}=3.$

5. 5

   Check in the original equation. Left side: $\frac{3-1}{2}=1$. Right side: $2-\frac{3+2}{5}=2-1=1$ ✓. Both sides equal $1$, so $x=3$ is correct — and checking in the original fractional form, not the cleared version, confirms the LCD step as well.