Solve the following inequality algebraically. $$ 5|x-2|-7 \leq 43 $$

Question

UpStudy AI · Accepted Answer

To solve the inequality algebraically:

$$
5|x - 2| - 7 \leq 43
$$

**Step 1: Isolate the absolute value term.**

Add 7 to both sides of the inequality:

$$
5|x - 2| \leq 43 + 7
$$

$$
5|x - 2| \leq 50
$$

**Step 2: Divide both sides by 5 to further isolate the absolute value.**

$$
|x - 2| \leq \frac{50}{5}
$$

$$
|x - 2| \leq 10
$$

**Step 3: Remove the absolute value by considering both possible cases.**

The inequality $$ |x - 2| \leq 10 $$ means that the expression inside the absolute value is between -10 and 10:

$$
-10 \leq x - 2 \leq 10
$$

**Step 4: Solve for $$ x $$ by adding 2 to all parts of the inequality.**

$$
-10 + 2 \leq x \leq 10 + 2
$$

$$
-8 \leq x \leq 12
$$

**Final Solution:**

$$
\boxed{-8\le x \le12}
$$
The solution to the inequality is $$ -8 \leq x \leq 12 $$.

Solve the following inequality algebraically. \[ 5|x-2|-7 \leq 43 \]