Solve for x: (5x - 7)/2 - (3x + 4)/3 = (x + 5)/6 + 1.

Step-by-step solution

1. 1

   Find the least common denominator of every term. The denominators are $2$, $3$ and $6$ (the $1$ on the right counts as $\tfrac11$), so

   $\text{LCD}=6.$

2. 2

   Multiply every term on both sides by 6. Multiply the whole term, brackets and all — this is where signs get lost otherwise:

   $3(5x-7)-2(3x+4)=(x+5)+6.$

   Each fraction has vanished because $6/2=3$, $6/3=2$ and $6/6=1$.

3. 3

   Expand both sides, distributing the minus sign carefully.

   $15x-21-6x-8=x+11.$

   The $-2$ multiplies both $3x$ and $+4$, giving $-6x-8$.

4. 4

   Collect like terms.

   $9x-29=x+11\ \Longrightarrow\ 8x=40\ \Longrightarrow\ x=5.$

5. 5

   Substitute back into the original equation. Left side:

   $\frac{25-7}{2}-\frac{15+4}{3}=\frac{18}{2}-\frac{19}{3}=9-\frac{19}{3}=\frac{8}{3}.$

   Right side:

   $\frac{5+5}{6}+1=\frac{10}{6}+1=\frac{5}{3}+1=\frac{8}{3}.$

   Both sides equal $\tfrac83$, so $x=5$ is correct.