The equation is ∣x+4∣=3x−12 |x + 4| = 3x - 12 ∣x+4∣=3x−12. An absolute value equation ∣A∣=B |A| = B ∣A∣=B has two cases:
Set x+4=3x−12 x + 4 = 3x - 12 x+4=3x−12 and solve for x x x: x+4=3x−12 x + 4 = 3x - 12 x+4=3x−12 Subtract x x x from both sides: 4=2x−12 4 = 2x - 12 4=2x−12 Add 12 12 12 to both sides: 16=2x 16 = 2x 16=2x Divide by 2 2 2: x=8 x = 8 x=8
Set x+4=−3x+12 x + 4 = -3x + 12 x+4=−3x+12 and solve for x x x: x+4=−3x+12 x + 4 = -3x + 12 x+4=−3x+12 Add 3x 3x 3x to both sides: 4x+4=12 4x + 4 = 12 4x+4=12 Subtract 4 4 4 from both sides: 4x=8 4x = 8 4x=8 Divide by 4 4 4: x=2 x = 2 x=2
For x=8 x = 8 x=8: ∣8+4∣=3(8)−12 ⟹ 12=12(Valid) |8 + 4| = 3(8) - 12 \implies 12 = 12 \quad \text{(Valid)} ∣8+4∣=3(8)−12⟹12=12(Valid)
For x=2 x = 2 x=2: ∣2+4∣=3(2)−12 ⟹ 6=−6(Invalid) |2 + 4| = 3(2) - 12 \implies 6 = -6 \quad \text{(Invalid)} ∣2+4∣=3(2)−12⟹6=−6(Invalid)
Only x=8 x = 8 x=8 is valid.
The correct answer is C. {8} \boxed{\{8\}} {8}
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