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Math  /  Algebra

QuestionFind the real solutions of the equation. 9−∣3x∣=69-|3 x|=6
Select the correct choice below and fill in any answer boxes within your choice. A. The solution set is □\square \}. (Simplify your answer. Use a comma to separate answers as needed.) B. There is no real solution.

Studdy Solution

STEP 1

1. The equation 9−∣3x∣=6 9 - |3x| = 6 involves an absolute value.
2. We need to consider the definition of absolute value, which can split the equation into two cases.

STEP 2

1. Isolate the absolute value term.
2. Solve the equation for the two cases of the absolute value.
3. Verify the solutions and determine the solution set.

STEP 3

First, isolate the absolute value term by subtracting 9 from both sides of the equation:
9−∣3x∣=6 9 - |3x| = 6 −∣3x∣=6−9 -|3x| = 6 - 9 −∣3x∣=−3 -|3x| = -3
Divide both sides by -1 to get:
∣3x∣=3 |3x| = 3

STEP 4

The equation ∣3x∣=3 |3x| = 3 implies two cases:
Case 1: 3x=3 3x = 3
Case 2: 3x=−3 3x = -3
Solve each case separately.

STEP 5

For Case 1, solve 3x=3 3x = 3 :
3x=3 3x = 3 x=33 x = \frac{3}{3} x=1 x = 1

STEP 6

For Case 2, solve 3x=−3 3x = -3 :
3x=−3 3x = -3 x=−33 x = \frac{-3}{3} x=−1 x = -1

STEP 7

Verify the solutions x=1 x = 1 and x=−1 x = -1 by substituting back into the original equation:
For x=1 x = 1 : 9−∣3(1)∣=9−3=6 9 - |3(1)| = 9 - 3 = 6 (True)
For x=−1 x = -1 : 9−∣3(−1)∣=9−3=6 9 - |3(-1)| = 9 - 3 = 6 (True)
Both solutions are valid.
The solution set is: {1,−1} \boxed{\{1, -1\}}

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