Given k is a nonzero integer, is k 0 ?

Question

Given k is a nonzero integer, is  k > 0  ?

(1)  |k-4| =| k| +4

(2)  k>k^3

KarishmaB · Accepted Answer

k is a non zero integer so the values it can take are ... -2, -1, 1, 2, 3, ... Is k > 0? 1) |k-4| =| k| +4 According to the definition of absolute values, |k - 4| = k - 4 if (k - 4) >= 0 |k - 4| = -(k - 4) = -k + 4 if (k - 4) < 0 Note that the the right hand side is |k| + 4. This can be equal to the second case only and that too when k < 0. So we know that k must be negative. We can answer the question with 'No'. Sufficient. 2) k>k^3 On the number line, where is x greater than x^3? When either 0 < x< 1 or x < -1. Here, since k must be an integer, it will not lie between 0 and 1. So k must be less than -1 i.e. k must be negative. We can answer the question with 'No'. Sufficient. Answer (D) P.S. - You must know the relation between x, x^2 and x^3 on the number line.

hellosanthosh2k2 · Answer

Statement 1: |k-4| = |k| + 4
 squaring both sides, (as both LHS and RHS is positive, safe to square)
  k^2 - 8k + 16 = k^2 + 8|k| + 16 
 => -8k = 8|k|
 => -k = |k|
 => k <= 0 sufficient to answer our question

Statement 2:  k > k^3 
 => since k is an integer, k has to be negative to hold the equality true => sufficient

Answer (D)

BrushMyQuant · Answer

Learn More about Absolute Values Here

STAT1: |k-4| = |k| + 4

|k| + 4 will be a positive number and the absolute values of k and 4 will be added up.
|k-4| = |k + (-4)| this will give us the sum of absolute values of k and 4 only when k and -4 add up and this is possible only when k is also negative.
=> k < 0
=> SUFFICIENT

STAT2: k > k^3
=> possible values of k which will satisfy this statement are k<0 and k being a positive fraction. But k is a non-zero integer so k<0 is theonly solution
=> SUFFICIENT

So, answer will be D
Hope it helps!

rahul16singh28 · Answer

Statemnt 1: As  |K|  is always positive, so RHS will be always greater than 4. So, on LHS K has to be negative.

Statement 2:  k(1-k)(1+k)   > 0 . This is possible only when K is negative.

Hence, D.

spetznaz · Answer

We are required to find if k>0 or not. 
St 1 : Square both sides. We get k=0 ; Sufficient
St 2 : Solve for inequality , we get k<-1. Keep in mind that k is an integer ; Sufficient

Option D it is !!

KarishmaB · Answer

Responding to a pm: Detailing stmnt 1 As per definition of absolute values, |x| = x if x >= 0 |x| = -x is x < 0 Look at the LHS first: Using the definition, |k - 4| = k - 4 ......... if (k - 4) >= 0 i.e. if k >= 4 |k - 4| = -(k - 4) = -k + 4 .................if (k - 4) < 0 i.e. if k < 4 So |k - 4| can take two values: k - 4 OR -k + 4 depending on some constraints Now look at the RHS: |k| + 4 |k| = k if k >= 0 |k| = -k if k < 0 So |k| + 4 can take two values: k + 4 OR -k + 4 depending on whether k is positive or negative The only common value between LHS and RHS is -k + 4 which happens when k < 0. So for LHS to be equal to RHS, k must be negative.

KrishnakumarKA1 · Answer

Solution:

We can solve this question either by applying the properties of inequalities or by performing the squaring operation on both sides.
Given: “k” is a non zero integer. That means “k” can be -1,-2, 1, 2…….. It can take negative and positive integers except ZERO.
To find: Is k > 0?
Note: Properties of inequalities:
i) |k| = k if k>= 0

ii) |k| = -k is k < 0

Analysis of statement 1: |k-4 |= |k|+ 4

As there is modulus function both sides of the inequality we have to consider with respect to LHS and RHS.
Using the properties we can have these two cases for LHS.
Case 1: \(|k-4 |=k-4 if (k-4)≥0\) i.e.\(k≥4\)
Case 2: \(|k-4 |= -k+4\) if \((k-4)<0\) i.e.\(k<4\)
So, \(|k-4 |\) can be \(k-4 or -k+4.\)
Let’s check for RHS.
Case 1: \(|k|+ 4 = k + 4\) if \(k≥0\)
Case 2: \(|k|+ 4= -k+4\) if \(k<0\)
So, \(|k|+ 4\) can be \(k + 4 or -k+4\) depending upon the “k” is positive or negative.
If we take the common between LHS and RHS, we get; \(-k+4\) this is true when \(“k"\) is negative i.e. \(k<0\). We can definitely answer the question as “NO”.
Hence statement 1 is sufficient to answer. We can eliminate options B, C and E.

Analysis of statement 2: \(k>k^3\)
Think over a bit, you get the answer

When can the value of \(k>k^3\) be true? Its only possible when k is negative or \(k<0\).
We can definitely answer the question as “NO”.
Hence statement 2 is sufficient to answer. We can eliminate option A.

The correct answer option is “D”.

When can the value of k>k^3 be true? Its only possible when k is negative or k<0 . We can definitely answer the question as “NO”. Hence statement 2 is sufficient to answer. We can eliminate option A. The correct answer option is “D”.

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Given k is a nonzero integer, is k > 0 ?

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