If |x - 5| + |x - 7|

Question

If |x - 5| + |x - 7| < 4, what is the range of values of x ?

(A) 4 < x < 8
(B) -4 < x < 8
(C) 3 < x < 5
(D) -8 < x < -4
(E) -4 < x < 4

BrentGMATPrepNow · Accepted Answer

For these kinds of questions, I typically find that the easiest/fastest approach is to simply test numbers.

For example, is x = 0 a solution to |x - 5| + |x - 7| < 4?
Plug x = 0 into the inequality to get: |0 - 5| + |0 - 7| < 4
Simplify: 5 + 7 < 4. Doesn't work!
So x = 0 is NOT a solution, which means we can eliminate B and E, since they say that x = 0 IS a solution.

Now let's test x = 4 
Plug x = 4 into the inequality to get: |4 - 5| + |4 - 7| < 4
Simplify: 1 + 3 < 4. Doesn't work!
Since x = 4 is NOT a solution, we can eliminate C, since it says x = 4 IS a solution. 
 
We're down to A and D. 
We'll test x = 6 to get: |6 - 5| + |6 - 7| < 4
Simplify: 1 + 1 < 4. WORKS!
Since x = 6 is a solution, we can eliminate D, since it says x = 6 is NOT a solution.

Answer: A

akshat26 · Answer

Create a number line,
plot 5 and 7 then, solve the equation for taking x as less than 5 , 5 , between 5 and 7 , at 7 and greater than 7.
You will get the domain as 4<x<8.

Hence,
option A.

dave13 · Answer

BrentGMATPrepNow what would be algebraic method to solve it ? like finding critical points ... etc

BrentGMATPrepNow · Answer

It's sometimes useful to think about |x - k| as the distance (on the number line) from point x to point k.

So, for this question, |x - 5| represents the distance from x to 5 on the number line. 
Similarly, |x - 7| represents the distance from x to 7 on the number line.

So, |x - 5| + |x - 7| represents (the distance from x to 5 on the number line) + (the distance from x to 7 on the number line)
First recognize that, if x is any value between 5 and 7, then (the distance from x to 5 on the number line) + (the distance from x to 7 on the number line) will always equal 2 (try it!!)

The question tells us that |x - 5| + |x - 7| < 4. 
In other words, the sum of the two distances must be  less than 4 . 
Let's find values of x where are  the sum of the two distances is exactly 4

For example, if x = 4, then the distance from x to 5 is 1, and the distance from x to 7 is 3. This gives us a total sum of 4. 
Likewise, if x = 8, then the distance from x to 5 is 3, and the distance from x to 7 is 1. This gives us a total sum of 4.

Since the sum of the two distances must be  less than 4 , we can conclude that 4 < x < 8

Answer: A

Kinshook · Answer

Asked: If |x - 5| + |x - 7| < 4, what is the range of values of x ?

|x - 5| + |x - 7| < 4

Distance of x from 5 & 7 combined is less than 4.

4 < x < 8

IMO A

ktzsikka · Answer

Hi
Can you please explain the penultimate step
"Distance of x from 5 & 7 combined is less than 4."

And will this method work even when we have 4 mods instead of 2?

Thanks

Posted from my mobile device

ManifestDreamMBA · Answer

An approach I thought of for this question.
Given the equation, distance of the point x from 5 and 7 have t be less then 4.
Any point x will be combined at least 2 units away from 5 and 7 (=7-5), doesn't matter where it is, in b/w 5 and 7, a 5 or at 7. (Think of this as fixed base)
Now it can only add less than 2 more units to meet the constraint. (This is my variable)
For any unit x it moves away from either point, it will add x, x+2 (say at 4, x = 1 (from 5), x+2 = 3(from 7))
so 2x+2<4
x<1
That means it cannot move more than 1 unit beyond the points 5 and 7
So, it has to be between 4 and 8

egmat · Answer

Great question! Let me walk you through this step by step.

|x - 5| + |x - 7| < 4 has two critical points: 5 and 7. We break the number line into three intervals and remove the absolute values in each.

Case 1: x < 5
Both expressions inside are negative, so we flip them:
(5 - x) + (7 - x) < 4
12 - 2x < 4
-2x < -8
x > 4
Combined with x < 5, this gives us 4 < x < 5.

Case 2: 5 ≤ x ≤ 7
Here (x - 5) is positive but (x - 7) is negative:
(x - 5) + (7 - x) < 4
2 < 4
This is ALWAYS true! So every x between 5 and 7 works.

Case 3: x > 7
Both expressions are positive:
(x - 5) + (x - 7) < 4
2x - 12 < 4
2x < 16
x < 8
Combined with x > 7, this gives us 7 < x < 8.

Now combine all three cases:
4 < x < 5, plus 5 ≤ x ≤ 7, plus 7 < x < 8
This merges into one clean interval: 4 < x < 8.

Answer: A

Quick intuition check: |x - 5| + |x - 7| represents the total distance from x to the points 5 and 7 on a number line. The minimum total distance is 2 (when x is between 5 and 7). We need that total distance to be less than 4, meaning we have 2 units of 'slack' to distribute equally on either side — giving us 4 on the left and 8 on the right.

If |x - 5| + |x - 7| < 4, what is the range of values of x ?

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If |x - 5| + |x - 7| < 4, what is the range of values of x ?

Prep Toolkit

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