The equation |2x - 3| = x - 5 has how many solutions for x?

Question

The equation |2x - 3| = x - 5 has how many solutions for x? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 https://gmatclub.com/forum/download/file.php?id=81458 GMAT-Club-Forum-3yplsche.png

Bunuel · Accepted Answer

(A) 0
(B) 1
(C) 2 
(D) 3
(E) 4

The left hand side, we have an absolute value, |2x - 3|, which we know is always non-negative (so, 0 or positive), thus the right-hand side must also be non-negative, which implies x - 5 ≥ 0, resulting in x ≥ 5.

If x ≥ 5, then the expression within the absolute value, 2x - 3, is positive, leading us to drop the absolute value to get |2x - 3| = 2x - 3 (since for any a ≥ 0, |a| = a).

This simplifies our equation to 2x - 3 = x - 5. Solving for x gives us x = -2. However, since x must be greater than or equal to 5, this solution is not valid. Therefore, the equation |2x - 3| = x - 5 has no solution.

Answer: A.

egmat · Answer

Watch this video to see solution of a relatively straightfoward absolute value equation in action.  Be sure to not fall in any of the traps that such equations set for you.

https://www.youtube.com/watch?v=IVZRHQLn0SU

LoneWarrior23 · Answer

The answer is 0, but I thought it was 2 (-2 and 8/3). Can someone please help me figure it out?

Bunuel

oiploiloi · Answer

0 Modulus means that (2x - 3) can be positive or negative. If (2x - 3) positive: 2x - 3 = x - 5 x = -2. But -2 is not positive so this answer is invalid If (2x - 3) is negative: -(2x - 3) = x - 5 x = 8/3. But 8/3 is not negative so this answer is also invalid Both answers are invalid, therefore there are 0 correct answers. LoneWarrior23 , you're missing the final step of checking if your answer is valid i.e. fulfilling the negative/positive condition. If you're confused by the conditions, take the next hour and go through this thread: https://gmatclub.com/forum/math-absolut ... 86462.html I was confused by this exact question just a couple of days ago and that thread cleared it right up. You'll be able to solve pretty much every inequality question after reading through.

LoneWarrior23 · Answer

Thank you for this great explanation!

Posted from my mobile device

unraveled · Answer

The equation |2x - 3| = x - 5 has how many solutions for x?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

x - 5 has to be greater than 0 since absolute value is always positive. 
Also, x - 5 can be equal to 0.

|2x - 3| can be either positive or negative. 
Either value that we get from this is not satisfying the condition that x - 5 >= 0.

Answer A.

Oppenheimer1945 · Answer

The Lines are parallel with no points of intersection . Hence 0 solution

javierjohnson · Answer

I had the same problem.

But reviweing i realized that i forgot to test the solutions:

1. With x = -2 you get 7 = -7
2. With x = 8/3 you get 7/3 = -7/3

So there isnt a solutions that satisfies the equation

quetzalferry · Answer

Definitely a more dumb way, but I just jumped directly to brute forcing it

|2x - 5| = x - 5
(2x - 5)^2 = (x - 5)^2 => squaring both sides, absolute operation removed
4x^2 - 12x + 9 = x^2 - 10x + 25
3x^2 - 2x - 16 = 0
(3x - 8)(x + 2) = 0
x = -2 or x = 8/3
since |2x- 5| = x - 5 and |2x - 5| > 0 then x - 5 > 0 => x > 5
none of the roots > 5, so zero solution

kumar884 · Answer

To solve the equation ∣2x−3∣=x−5|2x - 3| = x - 5∣2x−3∣=x−5, we need to consider two cases due to the absolute value.

In this case, ∣2x−3∣=2x−3. So the equation becomes: 
2x−3=x−5 
 Solving for x:  
 2x−x=−5+3  
 x=−2  
 We must check if this solution satisfies the condition 2x−3≥0:  
 2(−2)−3=−4−3=−7≥0  
 Since −7≥0, x=−2x = -2x=−2 is not a valid solution in this case.

spy2206 · Answer

this is a right way only to do no?? you said dumb, so confused

5monthGMAT700a · Answer

I think he said it's "dumb way" because it takes some more time than using a graph technic and we don't have to get the correct x value.

Of course that's a right way but we can choose one of several way to solve the problem.

As for me, I approached this problem using graph which took only 43 seconds.

ramjas1234 · Answer

Hi, why did you use only 2x-3 = x-5 i.e taking only positive value of inside modulus content and not the second possible case also i.e -2x+3 =x-5? Although answer remains same even after testing both the cases but just wanted to understand if taking only +ve value of inside modulus is sufficient or we should check for both +ve and -ve cases of inside modulus part.

Thanks

Thanks

Bunuel · Answer

The solution you quote already explains why we are doing this. Here it is again, more clearly. The left-hand side, an absolute value |2x - 3|, is by definition always non-negative, meaning 0 or positive. Thus, the right-hand side must also be non-negative, which implies that x - 5 ≥ 0, giving x ≥ 5. If x ≥ 5, then the expression inside the absolute value, 2x - 3, is positive, so we drop the absolute value and write |2x - 3| = 2x - 3.

pulkitt · Answer

| 2x -3 | = x - 5 Testing when |2x - 3| is 0 or positive. So, 2x - 3 = x - 5. Solving we will get x = -2. Now you have two options, either put -2 back in the original equation i.e 2x - 3 = x - 5 and see, if both the side matches ( they don't) OR I use a different method Alternative. Instead of testing, we know we assumed |2x - 3| to be either 0 or positive. So, let's see what happens if the value of x is the least, that is when |2x - 3| is >= 0 Solving, 2x - 3 >= 0 we get, 2x >= 3 or x>= 3/2 that means x>= 1.5, is it more than or equal to 1.5? Clearly not Similarly test for the negative values, | 2x -3 | = x - 5 Now, assuming value inside modulus is negative. Hence - 2x + 3 = x -5 Solving this we will get x = 8/3 Now either put these values and test, or try this alternative method 2x-3 <= 0 So, 2x <= 3 Now x <= 1.5 That means x has to less than or equal to 1.5 IS it? No. Hence, there are 0 solutions for this equation. Hope it helped ((:

clubmate · Answer

I think there is a simple and straight forward way here.

See if this makes sense.

|2x-3| = x-5

To open the mod we must first open it with the rule, |x| = x, when x>=0
This means,
2x-3 = x-5, when 2x-3>=0
(Solve right side to see the condition clearly)
2x-3 = x-5, when 2x>=3
2x-3 = x-5, when x>=3/2
(Solve left side to see the possible values of x clearly)
x=-2, when x>=3/2
(This is not compatible as x = -2 is not equal or greater than 3/2.)
Hence not possible.

Now the negative
|2x-3| = x-5
2x-3 = -(x-5) {|x| = -x, when x<0; this means 2x-3<0 OR x<3/2 OR x<1.5}
2x-3 = -(x-5)
2x-3 = -x+5
3x = 8
x = 8/3 = 2.67 (BUT this is greater than 1.5)
Hence not possible

Since neither 2x-3>=0 or 2x-3<=0 gives any solution, answer is 0.

The equation |2x - 3| = x - 5 has how many solutions for x?

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The equation |2x - 3| = x - 5 has how many solutions for x?

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