Solve |x+1| ≤ 7-x: Absolute Value Inequality Analysis

To solve $\left|x + 1\right| \leq 7 - x$, we split it into two cases due to the absolute value:

1. $x + 1 \leq 7 - x$

2. $-x - 1 \leq 7 - x$

Solving case 1:

$x + 1 \leq 7 - x \Rightarrow x + x \leq 7 - 1 \Rightarrow 2x \leq 6 \Rightarrow x \leq 3$

Solving case 2:

$-x - 1 \leq 7 - x \Rightarrow -1 \leq 7$ (always true)

Both parts confirm that $x \leq 3$ is the only true inequality.