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Re: If x is any integer from -6 to 2 inclusive, how many
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25 Oct 2016, 08:37

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GMATPrepNow wrote:

If x is any integer from -6 to 2 inclusive, how many values of x satisfy the following inequality: 2 - |x³ + 10x² - 24x| < 2 ?

A) 5 B) 6 C) 7 D) 8 E) 9

My solution:

First note that there are 9 integers from -6 to 2 inclusive.

Given: 2 - |x³ + 10x² - 24x| < 2 Add |x³ + 10x² - 24x| to both sides: 2 < 2 + |x³ + 10x² - 24x| Subtract 2 from both sides to get: 0 < |x³ + 10x² - 24x| Now let's apply some number sense. We know that the absolute value of something will always be greater than or equal to zero. So the ONLY values that DON'T satisfy the inequality 0 < |x³ + 10x² - 24x| will be those values where |x³ + 10x² - 24x| = 0 In other words, x³ + 10x² - 24x = 0 Factor to get: x(x² + 10x - 24) = 0 Factor more: x(x + 12)(x - 2) = 0 Solve to get: x = 0, or x = -12 or x = 2 So, x = 0, or x = -12 or x = 2 are the ONLY values of x that DON'T satisfy the inequality 0 < |x³ + 10x² - 24x| Since x = -12 is not within the given range of values (-6 to 2 inclusive), we need only consider the 2 values (x = 0 and x = 2) So, the number of integer values that DO satisfy the inequality = 9 - 2 = 7

If x is any integer from -6 to 2 inclusive, how many
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Updated on: 29 Nov 2016, 10:48

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1

GMATPrepNow wrote:

If x is any integer from -6 to 2 inclusive, how many values of x satisfy the following inequality: 2 - |x³ + 10x² - 24x| < 2 ?

A) 5 B) 6 C) 7 D) 8 E) 9

*Kudos for all correct solutions

2 - |x³ + 10x² - 24x| < 2 =|x³ + 10x² - 24x| > 0 (cancelling 2 both sides and multiplying by -1 both sides ,flipping the sign) factor out:- x³ + 10x² - 24x = x(x-2)(x+12) After removing mod sign,Two cases:- Case 1:- if x(x-2)(x+12) > 0 then -12<x<0 & x>2 but as x is integer between -6 & 2 x satisfies for -6,-5,-4,-3,-2,-1 (5 nos.)

Case 2:- if x(x-2)(x+12) < 0 then x<-12 & 0<x<2 but as x is integer between -6 & 2 x satisfies for 1 (1 nos.)

x cannot be 0 & 2 or -12(not in range) ..because then inequality does not hold and equation =0

Total satisfying values =-6,-5,-4,-3,-2,-1(case 1) && 1(case 2)=7nos.

Ans C

Originally posted by rohit8865 on 24 Oct 2016, 20:39.
Last edited by rohit8865 on 29 Nov 2016, 10:48, edited 1 time in total.

Re: If x is any integer from -6 to 2 inclusive, how many
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12 May 2018, 20:52

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