Solve the inequality 1 - (x + 6)/2 < (2x + 1)/3.

Step-by-step solution

1. 1

   Find the least common denominator. The denominators are $2$ and $3$, so the LCD is $6$. Because $6>0$, multiplying both sides by it preserves the direction of the inequality; this is the step that removes all fractions at once instead of fraction by fraction.

2. 2

   Multiply every term by 6 and simplify the coefficients.

   $6\cdot 1-6\cdot\frac{x+6}{2}<6\cdot\frac{2x+1}{3}\ \Longrightarrow\ 6-3(x+6)<2(2x+1)$

   Here $6/2=3$ and $6/3=2$, and the standalone $1$ becomes $6$ — a term with no denominator still gets multiplied.

3. 3

   Expand both sides. The $-3$ multiplies both parts of $(x+6)$:

   $6-3x-18<4x+2\ \Longrightarrow\ -3x-12<4x+2$

   Writing $6-3x+18$ instead would be the classic sign slip.

4. 4

   Move the $x$ terms to the right so the coefficient stays positive. Add $3x$ to both sides:

   $-12<7x+2$

   Then subtract $2$:

   $-14<7x$

5. 5

   Divide by 7 and state the solution set. Since $7>0$ the sign does not flip:

   $-2<x\ \Longrightarrow\ x>-2,\qquad\text{i.e. }(-2,\infty)$

6. 6

   Check the boundary and one interior point. At $x=-2$: left $=1-2=-1$ and right $=\tfrac{-3}{3}=-1$, so the two sides are equal — the endpoint is excluded, exactly as a strict inequality requires. At $x=0$: left $=1-3=-2$, right $=\tfrac13$, and $-2<\tfrac13$ ✓.