A chain of inequalities

Statement

Find the values of $x$ that satisfy the chain $4 - x < 2x + 1 \le x + 7$.

Solution

The chain is equivalent to two inequalities that must hold at the same time.

First: $4 - x < 2x + 1 \implies 4 - 1 < 2x + x \implies 3 < 3x \implies x > 1$.

Second: $2x + 1 \le x + 7 \implies 2x - x \le 7 - 1 \implies x \le 6$.

The solution set is the intersection: $1 < x \le 6$. $\boxed{1 < x \le 6}$