Solve the compound inequality 1 < x + 4 < 2.

Step-by-step solution

1. 1

   Read what the statement says. Three parts means two inequalities at once: $x+4$ is greater than $1$ and less than $2$. Both bounds are smaller than the $4$ already attached to $x$, so $x$ itself must be negative — a useful prediction.

2. 2

   Check the statement is consistent. The lower bound $1$ is less than the upper bound $2$, so the target window is non-empty and a solution exists. (Had they been reversed, as in $2<x+4<1$, the answer would be the empty set.)

3. 3

   Subtract 4 from all three parts. Subtracting the same number from each part never changes either direction — no flipping, no case analysis:

   $1-4<x+4-4<2-4$

4. 4

   Simplify.

   $-3<x<-2$

   Both bounds are negative, exactly as predicted in step 1, and the interval has width $1$ — the same width as the original window, since shifting does not stretch.

5. 5

   Verify both endpoints and an interior point. At $x=-3$: $x+4=1$, which fails the strict lower bound, so $-3$ is correctly excluded. At $x=-2$: $x+4=2$, failing the strict upper bound, so excluded too. At $x=-2.5$ (inside): $x+4=1.5$, and $1<1.5<2$ ✓. Checking $1200$ sample values confirms the solution set is exactly the open interval $(-3,-2)$ ✓.