Solve the compound inequality 1 < x + 4 <= 2 and write the answer in interval notation.

Step-by-step solution

1. 1

   Read the double inequality as two statements at once. $1<x+4\le 2$ is shorthand for $1<x+4$ and $x+4\le 2$. Because the variable appears only in the middle part, both can be handled in one pass instead of separately.

2. 2

   Subtract $4$ from every part. Whatever you do to the middle must be done to both outer parts to keep the chain equivalent:

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

   $-3<x\le -2$

3. 3

   Confirm no sign flipped. Adding or subtracting a constant never reverses an inequality; only multiplying or dividing by a negative number does. Since we only subtracted, the strict $<$ stays strict and the $\le$ stays inclusive.

4. 4

   Translate to interval notation. The left endpoint is excluded (round bracket) and the right endpoint is included (square bracket):

   $(-3,\,-2]$

5. 5

   Test one point inside and both endpoints. At $x=-2.5$: $x+4=1.5$, and $1<1.5\le 2$ $\checkmark$. At $x=-2$: $x+4=2$, and $2\le 2$ holds while $1<2$ holds $\checkmark$. At $x=-3$: $x+4=1$, but $1<1$ is false $\checkmark$ — correctly excluded.