How many integer values of x satisfy the inequality |x - 5| ≤ 2.5?

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Please correct me if i'm wrong.
For the inequality |x-5| <= 2.5 to be +ve, x>5
For the inequality |x-5| <= 2.5 to be -ve, x<5
By removing the modulus,
(x-5) = 2.5; x = 7.5
-(x-5) = 2.5; x = 2.5
Hence the inequality lies between 2.5 < (x-5) < 7.5
Since the question asks for integers within this range, we have 3,4,5,6 and 7. Hence answer is C.

Is there any possibility this question be asked for non-integer values.? Just trying to understand the concept better.


TIA

Hello there,

For the given set of options, the question cannot ask for non-integer values as there are infinite non integral values between any two integers. However, had the options were limits, then there's a possibility to ask for non-integeral values.

Cheers!

Hi,

Could you please explain this in little more details.

Cheers.

The point is that there are infinitely many numbers in any interval. For example, if we don't limit the values of x to integers only, then infinitely many values of x satisfy 2.5 ≤ x ≤ 7.5.

Does this make sense?

Let’s solve for when (x - 5) is positive and when (x - 5) is negative.

When (x - 5) is positive:

x - 5 ≤ 2.5

x ≤ 7.5

When (x - 5) is negative:

-(x - 5) ≤ 2.5

-x + 5 ≤ 2.5

-x ≤ -2.5

x ≥ 2.5

Thus, 2.5 ≤ x ≤ 7.5.

We see that there are 5 integer values that satisfy the inequality: 3, 4, 5, 6, and 7.

Answer: C

Hi saswata4s,

A certain number of questions in the Quant section of the GMAT can be solved rather easily with 'brute force' arithmetic. You don't need to know/complete any complex math - you just have to put the pen on the pad and work your way through the possibilities. The answer choices to this question show that there are at least 3, but no more than 7, integer values that 'fit' this inequality. I bet that you can find them all in under a minute if you try.

|X - 5| < 2.5

Let's start with an obvious value: X = 5

|5 - 5| = 0, which is clearly less than 2.5

X = 6 ---> |1| is less than 2.5
X = 7 ---> |7| is less than 2.5
X = 8 ---> |3| is NOT less than 2.5.... Increasing the value of X will just increase the value of the inequality, so there's no reason to check any higher integers. Let's stop here and try going in the other direction....

X = 4 ---> |-1| = 1 is less than 2.5
X = 3 ---> |-2| = 2 is less than 2.5
X = 2 ---> |-3| = 3 is NOT less than 2.5....Decreasing the value of X will just increase the value of the inequality, so there's no reason to check any lower integers. Thus, we're done - and there are 5 integer values that fit the inequality.

Final Answer:
GMAT assassins aren't born, they're made,
Rich

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

2.5<=x<=7.5

As x is integer so x can be 3,4,5,6,7

Hence 5 values can x take

Hello Bunuel, Zeus of all real numbers and unreal numbers please show your kindness to mortal GMAT student by explaining what i did wrong in my solution below:-)

I approached this question by opening the modulus - modulus always is associated with opening modulus no ?

|x - 5| ≤ 2.5

Case 1: x is positive x - 5≤ 2.5 --> x = 7.5

Case 2: x is negative x - 5≤ -2.5 --> x = 2.5 (Not valid, our initial condition was that x is negative but we got positive.

So whats wrong with my approach ?

Hi dave13,

Your approach to solving this problem isn't "wrong", but it appears "incomplete."

You've determined the "upper limit" and lower limit" for what X could be (meaning that 2.5 <= X <= 7.5), but you did not factor in ALL of the given information. We're told that X MUST be an INTEGER, so given the range that you've determined... how many INTEGER values could X be?

X could be 3, 4, 5, 6 or 7... meaning that there are 5 potential INTEGER solutions to this question.

GMAT assassins aren't born, they're made,
Rich

Hello EMPOWERgmatRichC,

Many thanks for your reply :) you know what confused? :? CASE 2

Case 2: x is negative x - 5≤ -2.5 --> x = 2.5 (Not valid, our initial condition was that x is negative but we got positive

case two was not valid so that's why I thought I again did something wrong, case two was invalid...

could you explain the logic behind case TWO? what does "INVALID" mean ?

thanks!

When opening the modulus you should consider the EXPRESSION being positive or negative, not x. The simplest approach is given in my post but if we solve the way you are going then we'd have the following.

|x - 5| ≤ 2.5

When x - 5 < 0 (NOT x), so when x < 5, we'd have -(x - 5) ≤ 2.5 --> \(x \geq 2.5\). Since we consider the range when x < 5, then for this range we'd have \(2.5 \leq x < 5\)

When x - 5 >= 0 (NOT x), so when x >= 5, we'd have x - 5 ≤ 2.5 --> \(x \leq 7.5\). Since we consider the range when x >= 5, then for this range we'd have \(5 \leq x \leq 7.5\)

Combining both: \(2.5 \leq x \leq 7.5\)


You should study articles below carefully:

10. Absolute Value

• Theory
  Math: Absolute value (Modulus)
  Absolute Value: Tips and hints
  Properties of Absolute Values on the GMAT
  Absolute modulus : A better understanding
  GMAT Question Patterns: Abolute Value
  The Reason Behind Absolute Value Questions on the GMAT
  
  • Questions
    Inequality and absolute value questions from my collection
    The E-GMAT Question Series on ABSOLUTE VALUE
    DS Questions
    PS Questions

[/list]

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.

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