How many possible values of m satisfy the inequality |m + 1| – |m – 3|

Question

How many possible values of m satisfy the inequality |m + 1| – |m – 3| > 4, if m is an integer?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

QuantMadeEasy · Accepted Answer

Step 1: Understanding the question 
|m + 1| is the distance of m from -1
|m – 3| is the distance of m from 3
m lies in the region a, b or c (refer to the attachment) 
Difference between (distance of m from -1) and (distance of m from 3) is required

Step 2: Calculation 
As the distance between -1 and 3 is 4
Hence |m + 1| – |m – 3| can be maximum 4, but cannot be greater than 4

A is correct

TestPrepUnlimited · Answer

We may try plugging in the important values to consider. m = 3 to m = -1 are the boundaries as  m > 3  would make both  m + 1 ,  m - 3  positive while on the other hand  m < -1  makes both negative.

m = 0 gives 1 - 3 > 4, not accepted.
m = 1 gives 2 - 1 > 4, not accepted.
Finally, m = 2 gives 3 - 0 > 4, not accepted.

Then no values satisfy. Also note when  m \leq{-1}  the left side is always -4, when  m \geq{3}  the left side is always 4, hence why we were able to narrow down to this range.

Ans: A

Kinshook · Answer

Asked: How many possible values of m satisfy the inequality |m + 1| – |m – 3| > 4, if m is an integer?

|m + 1| – |m – 3| > 4
|m+1| > 4+ |m-3|

Case 1: m<-1
-m-1 - (-m+3) > 4
-4 > 4
NOT FEASIBLE

Case 2: -1<=m<3
m+1 - (-m+3) >4
2m -2 > 4
m > 3
NOT FEASIBLE

Case 3: m>=3
m+1 - (m-3) > 4
4>4
NOT FEASIBLE

IMO A

gmatcafe · Answer

|m + 1| – |m – 3| > 4
We have three cases:

1st case: m>=3. Then m+1-m+3=4. In this case we do not have any solution.

2nd case: -1=<m<3. Then m+1-3+m=2m-2>4 ----> m-1>2 ----> m>3. In this case also we don't have any solution.

3rd case: m<-1. Then -m-1-3+m=-4. In this case also we don't have any solution.

So the inequality does not have any solution. A)

OjhaShishir · Answer

Three areas are on the number line:

1) m < -1

Here, the inequality is:

-(m+1) + (m-3) > 4

-4 > 4

Not possible.

2) m between 1 and 3

Here, the inequality is:

(m+1) + (m-3) > 4

2m - 2 > 4

2m > 6

m > 3

Not possible, since m is between 1 and 3

3) m > 3

Here, the inequality is:

(m+1) - (m-3) > 4

4 > 4

Not possible.

So, no solution.

chetan2u · Answer

For those, who want to understand the concept of MOD.
The question asks us whether the distance of m from -1 is greater than the distance of m from 3 by a quantity of 4.

Let us draw a number line, and mark -1 with brown and 3 with blue.
m can be at three locations
1) On the left side of -1, shown as m1...
Clearly BROWN line is closer to m than BLUE line, so answer will always be negative and -4....... Is -4>4?...NO, so no value will fit in.
2) In between 3 and -1, shown as m2...
Now, it is the SUM that is 4, so the difference has to be less than 4. It will be max at 3, that is 4, and minimum at -1, that is -4....... Is (-4 to 4) >4?...NO, so no value will fit in.
3) On the right side of 3, shown as m...
Clearly BLUE line is closer to m than BROWN line, so answer will always be positive, but there is OVERLAP after 3, so Brown-Blue=4 ....... Is 4>4?...NO, so no value will fit in.

Hence 0 values will fit in.'

A

nisthagupta28 · Answer

Can we solve this by squaring as well?

(M+1+M-3) (M+1-M+3) >16
(2M-2)(4)>16 (dividing the equation by 2)
(M-1)>4
M>5

checking by replugging any value which is M>5.
the answer will always be 4>4 which can't be possible. Hence, 0.

Posted from my mobile device

PocusFocus · Answer

great solution! thanks, it's so elegant.

How many possible values of m satisfy the inequality |m + 1| – |m – 3|

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How many possible values of m satisfy the inequality |m + 1| – |m – 3|

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