To solve the inequality $4x + 5 \leq -x - 2$, follow these steps:

Step 1: Move all $x$-terms to one side of the inequality

Add $x$ to both sides: $4x + x + 5 \leq -2$ This simplifies to: $5x + 5 \leq -2$

Step 2: Move the constant terms to the other side

Subtract 5 from both sides: $5x \leq -7$

Step 3: Isolate $x$ by dividing by 5

$x \leq -\frac{7}{5}$

Solution in Interval Notation

The solution set for $x$ is all values less than or equal to $-\frac{7}{5}$. In interval notation, this is: $(-\infty, -\frac{7}{5}]$

Would you like more details or have any questions about this solution?

---

Here are some related questions to expand your understanding:

1. How would the solution change if the inequality were $4x + 5 < -x - 2$?
2. What is the interval notation if the inequality were $4x + 5 \geq -x - 2$?
3. How would the steps differ if solving $4x + 5 = -x - 2$ (an equation instead of an inequality)?
4. What would the solution look like if $4x + 5$ was greater than or equal to $-x - 2$?
5. How can you graph the solution of $x \leq -\frac{7}{5}$ on a number line?

Tip: Always double-check interval notation by considering whether the inequality includes or excludes the boundary point (i.e., $\leq$ or $<$).