Question

solve the following inequality algebraically. 5|x - 5| + 8 ≤ 48

Explanation:

Step 1: Isolate the absolute value expression

Subtract 8 from both sides of the inequality:
$$5|x - 5| + 8 - 8 \leq 48 - 8$$
Simplify to get:
$$5|x - 5| \leq 40$$

Step 2: Divide both sides by 5

Divide each side of the inequality by 5:
$$\frac{5|x - 5|}{5} \leq \frac{40}{5}$$
Simplify to obtain:
$$|x - 5| \leq 8$$

Step 3: Solve the compound inequality

The absolute value inequality \(|x - 5| \leq 8\) is equivalent to the compound inequality:
$$-8 \leq x - 5 \leq 8$$

Step 4: Solve for \(x\)

Add 5 to all parts of the compound inequality:
$$-8 + 5 \leq x - 5 + 5 \leq 8 + 5$$
Simplify each part:
$$-3 \leq x \leq 13$$

Answer:

The solution to the inequality is \(-3 \leq x \leq 13\)