Solve the inequality (x + 2)/4 - x/5 > 1 and give the solution in interval notation.

Step-by-step solution

1. 1

   Find the LCD and multiply through. The denominators $4$ and $5$ are coprime, so the least common denominator is $20$. Multiplying every term — including the lone $1$ on the right — by the positive number $20$:

   $20\cdot\frac{x+2}{4}-20\cdot\frac{x}{5}>20\cdot 1\;\Longrightarrow\;5(x+2)-4x>20.$

   Forgetting to multiply the right-hand side is the classic error; it would leave $5(x+2)-4x>1$ and shift the answer by $19$.

2. 2

   Expand the bracket. Distributing the $5$:

   $5x+10-4x>20.$

   Only the first term carries a bracket, so this is a single distribution — but the $+10$ from $5\times 2$ is the part that ends up controlling the answer.

3. 3

   Combine like terms. The two $x$ terms almost cancel:

   $5x-4x=x,\qquad\text{so}\qquad x+10>20.$

   A coefficient of exactly $1$ is a good sign that the arithmetic is on track — it comes from $\tfrac14-\tfrac15=\tfrac{1}{20}$ scaled by $20$.

4. 4

   Isolate $x$. Subtracting $10$ from both sides (an operation that never changes the inequality direction):

   $x>10.$

   In interval notation the solution set is $(10,\infty)$, with $10$ excluded because the inequality is strict.

5. 5

   Test around the boundary. At $x=10.4$: $\frac{12.4}{4}-\frac{10.4}{5}=3.1-2.08=1.02>1$ ✓. At $x=9.6$: $\frac{11.6}{4}-\frac{9.6}{5}=2.9-1.92=0.98$, which is not $>1$ ✓. At $x=10$ the left side is exactly $3-2=1$, confirming the boundary.