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

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

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

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

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

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards