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

Question

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

Bunuel · Accepted Answer

|x - 5| ≤ 2.5;

Get rid of modulus: -2.5 ≤ x - 5 ≤ 2.5;

Add 5 to all three parts: 2.5 ≤ x ≤ 7.5

Integer values of x that satisfy the inequality are 3, 4, 5, 6, and 7. So, 5 values.

Answer: C.

CrackverbalGMAT · Answer

Hi saswata4s,

We can solve this question using number line too, because modulus (or absolute value) is nothing but the length,

Here the question says,

|x-5| ≤ 2.5

That is distance between x and 5 has to be less than or equal to 2.5,

So, let’s draw this in the number line, we can see that from the below diagram, that

x has to be between 2.5 and 7.5,

Since question ask about integer values,

The values are 3,4,5,6 and 7.

So there are 5 values,

Answer has to be C.

Hope this helps.

KarishmaB · Answer

Check out this video: https://youtu.be/oqVfKQBcnrs

sarugiri · Answer

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

Diwakar003 · Answer

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!

saswata4s · Answer

Hi,

Could you please explain this in little more details.

Cheers.

Bunuel · Answer

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?

JeffTargetTestPrep · Answer

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

EMPOWERgmatRichC · Answer

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:  C

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

guptarahul · Answer

(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

dave13 · Answer

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 ?

EMPOWERgmatRichC · Answer

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

dave13 · Answer

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!

Bunuel · Answer

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 For more check Ultimate GMAT Quantitative Megathread Hope it helps.

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

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

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