Math Snap

STEP 1

1. The equation given is $2|3r - 9| - 1 = 23$.
2. The problem involves solving an absolute value equation.
3. We will solve for $r$ and check which of the provided options match the solution.

STEP 2

1. Isolate the absolute value expression.
2. Solve the equation inside the absolute value for both positive and negative scenarios.
3. Verify the solutions against the given options.

STEP 3

First, add 1 to both sides of the equation to isolate the absolute value term:
$$2|3r - 9| - 1 = 23$$ $$2|3r - 9| = 23 + 1$$ $$2|3r - 9| = 24$$

STEP 4

Divide both sides by 2 to further isolate the absolute value expression:
$$|3r - 9| = \frac{24}{2}$$ $$|3r - 9| = 12$$

STEP 5

The absolute value equation $|3r - 9| = 12$ implies two scenarios:
1. $3r - 9 = 12$
2. $3r - 9 = -12$
Solve each scenario separately.

STEP 6

Solve the first scenario $3r - 9 = 12$:
Add 9 to both sides:
$$3r - 9 + 9 = 12 + 9$$ $$3r = 21$$ Divide by 3:
$$r = \frac{21}{3}$$ $$r = 7$$

STEP 7

Solve the second scenario $3r - 9 = -12$:
Add 9 to both sides:
$$3r - 9 + 9 = -12 + 9$$ $$3r = -3$$ Divide by 3:
$$r = \frac{-3}{3}$$ $$r = -1$$

The solutions for $r$ are $r = 7$ and $r = -1$. Check which option matches these solutions:
- Option D: $r = \{-1, 7\}$
The correct answer is:
$$\boxed{\{-1, 7\}}$$