If |x + 3| = 4x - 3 and |y + 1| = 3, and x y, what is the product of

Question

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

Bunuel · Answer

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

ccarson · Answer

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

Andrew89 · Answer

Hi Bunuel,

What I did is, calculating the 4 possibilities:

1- X+3=4X-3 ,then  X=2 
2- X+3= -(4X-3) = -4X+3, then  X=0 
3- Y+1=3, then  Y=2 
4- Y+1=-3, then  Y=-4

When I tried to plug in all the outcomes, then all give a correct answers including 0 (0+3=-4(0)+3) so 0+3=0+3)

So, what's wrong here?

Bunuel · Answer

x = 0 doe not satisfy |x + 3| = 4x - 3. As said in the solution you quote, x must be more than or equal to 3/4.

Suyash1331 · Answer

very important to note that x is not equal to y. else the answer will be wrong

Kinshook · Answer

If |x + 3| = 4x - 3 and |y + 1| = 3, and x ≠ y, what is the product of the unique solutions for x and y?

Inflection points:
x = -3; y = -1

Case 1: x < -3; y< -1
-3-x = 4x - 3; x = 0; But x<-3; Not feasible

Case 2: x < -3; y>=-1
-3-x = 4x - 3; x = 0; But x<-3; Not feasible

Case 3: x >= -3; y<-1
x+3 = 4x -3; x = 2
-y-1 = 3; y=-4;
xy = 2*(-4) = -8; Feasible

Case 4: x >= -3; y>=-1
x+3 = 4x -3; x = 2
y+1=3; y =2;
xy = 2*2 = 4; Not unique values of x & y; Not feasible

IMO B

If |x + 3| = 4x - 3 and |y + 1| = 3, and x y, what is the product of

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GMAT Problem Solving (PS) Questions

If |x + 3| = 4x - 3 and |y + 1| = 3, and x y, what is the product of

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

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From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards