To solve the inequality $|2x + 4| \leq 3$, we must consider the definition of absolute value. The inequality $|A| \leq B$ can be split into two inequalities:

$-B \leq A \leq B$

For the given inequality $|2x + 4| \leq 3$, we split it as follows:

$-3 \leq 2x + 4 \leq 3$

Step 1: Solve the left inequality

$-3 \leq 2x + 4$

Subtract 4 from both sides:

$-7 \leq 2x$

Now, divide both sides by 2:

$-\frac{7}{2} \leq x$

Or:

$x \geq -\frac{7}{2}$

Step 2: Solve the right inequality

$2x + 4 \leq 3$

Subtract 4 from both sides:

$2x \leq -1$

Now, divide both sides by 2:

$x \leq -\frac{1}{2}$

Final Solution

Combining the results from both inequalities, we have:

$-\frac{7}{2} \leq x \leq -\frac{1}{2}$

In interval notation, the solution is:

$\left[-\frac{7}{2}, -\frac{1}{2}\right]$

This means $x$ can be any value between $-\frac{7}{2}$ and $-\frac{1}{2}$, inclusive.

Would you like further details or have any questions?

Related Questions:

1. How do you solve inequalities involving absolute values when the inequality sign is $\geq$?
2. What is the difference between solving $|A| \leq B$ and $|A| \geq B$?
3. How do you graph the solution set of an absolute value inequality on a number line?
4. How do you solve compound inequalities?
5. Can absolute value inequalities have no solution?

Tip: Always remember to split absolute value inequalities into two separate inequalities when solving.