Solve the compound inequality -2 is less than 8 - 2x, which is less than or equal to -1.

Step-by-step solution

1. 1

   Treat it as one object with three parts. A compound inequality can be solved in a single chain, as long as every operation is applied to all three parts. The goal is to end with $x$ alone in the middle.

2. 2

   Subtract 8 from all three parts. Addition and subtraction never change the direction of an inequality:

   $-2-8<8-2x-8\le -1-8\quad\Rightarrow\quad -10<-2x\le -9$

3. 3

   Divide all three parts by $-2$ and reverse both signs. Dividing by a negative flips each comparison:

   $\frac{-10}{-2}>x\ge\frac{-9}{-2}\quad\Rightarrow\quad 5>x\ge\frac92$

   Both the $<$ and the $\le$ turn around; flipping only one of them is the classic mistake.

4. 4

   Rewrite in increasing order. Reading the chain from left to right with the smaller number first:

   $\frac92\le x<5$

   Note how the strict and non-strict ends have swapped position, following the numbers they belong to.

5. 5

   Verify the endpoints and an interior point. At $x=\frac92$: $8-9=-1$, and $-1\le -1$ holds, so $\frac92$ is included. At $x=5$: $8-10=-2$, and $-2<-2$ is false, so $5$ is excluded. At $x=4.75$: $8-9.5=-1.5$, and $-2<-1.5\le -1$ ✓. The solution set is $\left[\frac92,5\right)$.