M41-25

Question

How many different values of  x  are there such that  |x+10|=2x+8 ?

A. None
B. 1
C. 2
D. 3
E. Infinitely many

Josvit · Accepted Answer

Liked this. Thanks for the friendly remainder that if we complete a question in less than 50 seconds, it is valuable to double check for some seconds the answer.

The thing here is that we have two restrictions,   the equation and that 2x + 8 needs to be non-negative.

|x+10|= 2x + 8;

x+10 = 2x + 8 OR x+10= -2x -8

Then we have x= 2 OR x= -6; but -6 doesn´t meet that 2x+8 has to be non-negative. So, only x=2 works.

Bunuel · Answer

Official Solution:

How many different values of  x  are there such that  |x+10|=2x+8 ?

A. None
B. 1
C. 2
D. 3
E. Infinitely many

The left-hand side of the equation is an absolute value, which is always non-negative. Thus, the right-hand side must also be non-negative:  2x+8 \ge 0 , which gives us  x \ge -4 . For this range,  x+10  is positive, so  |x+10|=x+10 . Therefore,  |x+10|=2x+8  can be rewritten as  x+10=2x+8 . Solving for  x  gives  x=2 . Therefore, only one value of  x  satisfies  |x+10|=2x+8 .

Answer: B

Vikramaditya00 · Answer

I think this is a  high-quality  question and the explanation isn't clear enough, please elaborate. For this range, x+10 is positive, how do we know this?

Bunuel · Answer

x \ge -4  implies  x +4 \ge 0 , thus  x+10 > 0 .

sssdd1234 · Answer

I think this is a  high-quality  question and I agree with explanation.

rishabh2310 · Answer

kindly explain why 2x+8 has to be non negative ??? thanks in advance

Bunuel · Answer

The left-hand side of the equation is an absolute value ,  which is always non-negative .  Thus, the right-hand side must also be non-negative:  2x+8 \ge 0  , which gives us  x \ge -4 . For this range,  x+10  is positive, so  |x+10|=x+10 . Therefore,  |x+10|=2x+8  can be rewritten as  x+10=2x+8 . Solving for  x  gives  x=2 . Therefore, only one value of  x  satisfies  |x+10|=2x+8 .

bondi123 · Answer

I like the solution - it’s helpful.

Leesam03 · Answer

I did not quite understand the solution. What do we mean by an absolute value? Since it's a modulus and an equation, it has to be positive but the sign would be ambiguous if it were an inequality?

Bunuel · Answer

The absolute value |x + 10| is always non-negative. So for the equation |x + 10| = 2x + 8 to be valid, the right side (2x + 8) must also be non-negative. That’s why we start with 2x + 8 ≥ 0. Please check the rest above.

Leesam03 · Answer

Thank you, I get it now.

SwethaReddyL · Answer

quick doubt here, if it is mentioned as an inequality rather than equality, will we then consider the case -2x-8?

I made a mistake by taking both +ve and -ve scenario and came to solution as 2
In absolute values we do consider both +ve and -ve scenarios right?

Bunuel · Answer

This has nothing to do with it being an equation or inequality. The solution just uses a shortcut to drop the absolute value. But we could have done it the conventional way by splitting into two cases. The approach used is just a faster way to handle the modulus, not the only valid method.

aashnaag · Answer

Hey, I have the same question - I also input both positive and negative values of x and arrived at answer 2. I understand that the modulus is supposed to return only positive values, but if we remove the modulus, can someone explain why it's not -x-10 and x+10= both? In most such Qs, we use both values and arrive at solutions, in this case - 2.

Bunuel · Answer

You can also solve the question in the conventional way:

|x + 10| = 2x + 8

Case 1: x < -10

In this case, x + 10 is negative, so |x + 10| = -(x + 10). So, we get -(x + 10) = 2x + 8, which gives x = -6. But x = -6 is not in the range x < -10, so we discard it.

Case 2: x ≥ -10

In this case, x + 10 is non-negative, so |x + 10| = x + 10. So, we get x + 10 = 2x + 8, which gives x = 2. x = 2 is in the range and satisfies the equation, so it is valid.

Therefore, the equation has only one solution: x = 2.

Again, the method used in the solution is simply a shortcut to skip the full case analysis and solve the question faster.

M41-25

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