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31 May 2025, 14:14
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B
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Difficulty:
85%
(hard)
Question Stats:
43% (01:49) correct
57%
(01:48)
wrong
based on 251
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If |x + 3| = 4x - 3 and |y + 1| = 3, and x ≠ y, what is the product of the unique solutions for x and y?
A. -16
B. -8
C. -4
D. 0
E. 4
A. -16
B. -8
C. -4
D. 0
E. 4
31 May 2025, 14:23
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ccarson
If |x + 3| = 4x - 3 and |y + 1| = 3, and x ≠ y, what is the product of the unique solutions for x and y?
A. -16
B. -8
C. -4
D. 0
E. 4
A. -16
B. -8
C. -4
D. 0
E. 4
We’re given |x + 3| = 4x - 3
Since the left-hand side is an absolute value, it must be non-negative. So the right-hand side must also be non-negative:
4x - 3 ≥ 0, which gives x ≥ 3/4
If x ≥ 3/4, then x + 3 > 0, so |x + 3| = x + 3
Now the equation becomes x + 3 = 4x - 3, which gives x = 2.
Next, |y + 1| = 3. That gives two cases: y + 1 = 3 or y + 1 = -3, so y = 2 or y = -4
Since x ≠ y and x = 2, we must take y = -4
Then the product is x * y = 2 * (-4) = -8
Answer: B
02 Jun 2025, 18:02
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What I find more simple to remember, if you want another way to look at it is as follows:
|x+3| = 4x-3
square both sides
x^2 + 6x + 9 = 16x^2 - 24x + 9
simplify both sides
15x^2 - 30x = 0
divide by 15
x^2 - 2x = 0
factor
x(x-2) = 0
Then it's straightforward that x = 2 or x = 0. If I get 0 as an answer when I do this method, I'll plug it back into the equation and see what happens:
|x+3| = 4x - 3 -> |0+3| = 0 - 3
Since this doesn't work because 3 doesn't equal negative 3, you use 2 as your answer to x.
Bunuel answered how to find y very well and I used the same method but I hope this different method for finding x helps too.
P.S. It took me ~1 minute to solve this so this method isn't slow at all, it's what works for me
|x+3| = 4x-3
square both sides
x^2 + 6x + 9 = 16x^2 - 24x + 9
simplify both sides
15x^2 - 30x = 0
divide by 15
x^2 - 2x = 0
factor
x(x-2) = 0
Then it's straightforward that x = 2 or x = 0. If I get 0 as an answer when I do this method, I'll plug it back into the equation and see what happens:
|x+3| = 4x - 3 -> |0+3| = 0 - 3
Since this doesn't work because 3 doesn't equal negative 3, you use 2 as your answer to x.
Bunuel answered how to find y very well and I used the same method but I hope this different method for finding x helps too.
P.S. It took me ~1 minute to solve this so this method isn't slow at all, it's what works for me
