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Math  /  Algebra

QuestionIve the equation beior 2∣3r−9∣−1=23r={−7,7}r={1,7} t) r={−7,1} D) r={−1,7} posting on on Twitter  D)  singing waffles karaoke \begin{array}{c} 2|3 r-9|-1=23 \\ \begin{array}{c} r=\{-7,7\} \\ r=\{1,7\} \\ \text { t) } r=\{-7,1\} \\ \text { D) } r=\{-1,7\} \\ \text { posting on on Twitter } \\ \text { D) } \quad \text { singing waffles karaoke } \end{array} \end{array}

Studdy Solution

STEP 1

1. The equation given is 2∣3r−9∣−1=23 2|3r - 9| - 1 = 23 .
2. The problem involves solving an absolute value equation.
3. We will solve for r 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| - 1 = 23 2∣3r−9∣=23+1 2|3r - 9| = 23 + 1 2∣3r−9∣=24 2|3r - 9| = 24

STEP 4

Divide both sides by 2 to further isolate the absolute value expression:
∣3r−9∣=242 |3r - 9| = \frac{24}{2} ∣3r−9∣=12 |3r - 9| = 12

STEP 5

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

STEP 6

Solve the first scenario 3r−9=12 3r - 9 = 12 :
Add 9 to both sides:
3r−9+9=12+9 3r - 9 + 9 = 12 + 9 3r=21 3r = 21
Divide by 3:
r=213 r = \frac{21}{3} r=7 r = 7

STEP 7

Solve the second scenario 3r−9=−12 3r - 9 = -12 :
Add 9 to both sides:
3r−9+9=−12+9 3r - 9 + 9 = -12 + 9 3r=−3 3r = -3
Divide by 3:
r=−33 r = \frac{-3}{3} r=−1 r = -1

STEP 8

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

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