Q.Solve the combined inequation $-3<2x+1\le 5$ for $x\in\mathbb{R}$ and write the solution in interval form.

Operate on all three parts simultaneously.

Subtract $1$: $-4<2x\le4$; divide by $2$: $-2<x\le2$.

A double inequation is two conditions at once; apply each operation to all three parts.

Subtract $1$ throughout:

Divide throughout by the positive $2$ (signs unchanged):

The left end is strict ($<$, open) and the right end is weak ($\le$, closed), so the solution set is $(-2,2]$.

Check (independent verification): the two boundaries must behave correctly. At $x=-2$: $2(-2)+1=-3$, and $-3<-3$ is false ✓ excluded. At $x=2$: $2(2)+1=5$, and $5\le5$ is true ✓ included. A middle value $x=0$: $-3<1\le5$ ✓.

Splitting into two inequations but forgetting to apply an operation to the middle part, or mismatching the bracket types at the two ends — the left stays open $(-2$ and the right closed $2]$.