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What is the sum of all integers x satisfying |x^2 - 5x| < 6 ?

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Bunuel
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What is the sum of all integers x satisfying |x^2 - 5x| < 6 ? [#permalink] New post   05 Mar 2020, 03:06
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What is the sum of all integers x satisfying \(|x^2 - 5x| < 6\) ?

A. 0
B. 5
C. 10
D. 15
E. 20

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C
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What is the sum of all integers x satisfying |x^2 - 5x| < 6 ? [#permalink] New post   05 Mar 2020, 03:17
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\(| x^{2} —5x | < 6 \)
—\(6 < x^{2} —5x < 6\)

\(x^{2}—5x +6 > 0\)
—>\(( x—2)(x—3) > 0\)
\(x< 2\) and \(x >3\)

\(x^{2} —5x —6 < 0\)
—> \((x+ 1)(x—6) < 0\)
—\(1 < x < 6 \)

Taken together,
—\(1 < x < 2\) and \(3 < x < 6\)
\(0+ 1 + 4+ 5 = 10\)

The answer is C

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Re: What is the sum of all integers x satisfying |x^2 - 5x| < 6 ? [#permalink] New post   05 Mar 2020, 03:27
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Bunuel wrote:
What is the sum of all integers x satisfying \(|x^2 - 5x| < 6\) ?

A. 0
B. 5
C. 10
D. 15
E. 20



\(|x^2 - 5x| < 6\)

i.e. \(-6 < x^2 - 5x < 6\)

i.e. \(-6 < x(x - 5) < 6\)

i.e. product of two integers separated by 5 must fall between -6 and +6

-1*(-6) does NOT lie in the range
0*(-5)
1*(-4)
2*(-3) does NOT lie in the range
3*(-2) does NOT lie in the range
4*(-1)
5*(0)

i.e. may be only {0, 1, 4 and 5}

Sum = 0+1+4+5 = 10

Answer: Option C
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What is the sum of all integers x satisfying |x^2 - 5x| < 6 ? [#permalink] New post   05 Mar 2020, 05:58
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Bunuel wrote:
What is the sum of all integers x satisfying \(|x^2 - 5x| < 6\) ?

A. 0
B. 5
C. 10
D. 15
E. 20

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Solution:


    • \(|x^2 -5x| < 6\)
      o \(-6 < x^2 -5x < 6\)
    • Let’s break it down and solve
      o \(x^2 -5x <6\)
      o \(x^2 -5x -6 <0\)
      o \(x^2 -6x + x -6 <0\)
      o \((x-6)(x+1)<0\)
         \(-1< x < 6\)

    • Now, \(x^2 -5x > -6\)
      o \(x^2 -5x +6 >0\)
      o \(x^2-3x-2x+6>0\)
      o \((x-3)(x-2)>0\)
      o \(x >3\) and \(x < 2\)

    Let's combining the both result.

    • Integer which satisfy both the equations are 0, 1, 4, 5
      o Their sum = 0 +1 +4 +5 =10
Hence, the correct answer is Option C.
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Re: What is the sum of all integers x satisfying |x^2 - 5x| < 6 ? [#permalink] New post   05 Mar 2020, 07:36
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Bunuel wrote:
What is the sum of all integers x satisfying \(|x^2 - 5x| < 6\) ?

A. 0
B. 5
C. 10
D. 15
E. 20



\(|x^2 - 5x| < 6\)

Case i)
\(x^2-5x-6<0 \) when x^2-5x > 0 i.e. when x(x-5) > 0 i.e. x>5 or x<0
\(x^2-5x-6<0 \) = (x-6)(x+1) < 0 => -1 < x < 6
There will be no integer values within this range

Case ii)
\(x^2-5x>-6 \) when \(x^2-5x \leq 0\) i.e. when \(x(x-5) \leq 0\) i.e. \(0 \leq x \leq 5\)
\(x^2-5x+6>0 \) = (x-3)(x-2) > 0 => x>3 or x<2 => Range is \(0 \leq x < 2\) and \(3 < x \leq 5\)

Integer values = 0,1,4 and 5

Hence Sum = 10

Answer - C
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Re: What is the sum of all integers x satisfying |x^2 - 5x| < 6 ? [#permalink] New post   05 Mar 2020, 09:32
shameekv1989 wrote:
Bunuel wrote:
What is the sum of all integers x satisfying \(|x^2 - 5x| < 6\) ?

A. 0
B. 5
C. 10
D. 15
E. 20



\(|x^2 - 5x| < 6\)

Case i)
\(x^2-5x-6<0 \) when x^2-5x > 0 i.e. when x(x-5) > 0 i.e. x>5 or x<0
\(x^2-5x-6<0 \) = (x-6)(x+1) < 0 => -1 < x < 6
There will be no integer values within this range

Case ii)
\(x^2-5x>-6 \) when \(x^2-5x \leq 0\) i.e. when \(x(x-5) \leq 0\) i.e. \(0 \leq x \leq 5\)
\(x^2-5x+6>0 \) = (x-3)(x-2) > 0 => x>3 or x<2 => Range is \(0 \leq x < 2\) and \(3 < x \leq 5\)

Integer values = 0,1,4 and 5

Hence Sum = 10

Answer - C


Hi Can anyone confirm whether I am right in opening up the modulus?

|x^2-5x| =>

x(x−5)≤0 or x(x-5)> 0(Is it \(\leq\) and \(\geq\) for both cases? or only \(\leq\) while considering \(\leq 0\) or is it only \(\geq\) while taking \(\geq 0\))
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Re: What is the sum of all integers x satisfying |x^2 - 5x| < 6 ? [#permalink] New post   11 Mar 2020, 09:45
lacktutor wrote:
\(| x^{2} —5x | < 6 \)
—\(6 < x^{2} —5x < 6\)

\(x^{2}—5x +6 > 0\)
—>\(( x—2)(x—3) > 0\)
\(x< 2\) and \(x >3\)

\(x^{2} —5x —6 < 0\)
—> \((x+ 1)(x—6) < 0\)
—\(1 < x < 6 \)

Taken together,
—\(1 < x < 2\) and \(3 < x < 6\)
\(0+ 1 + 4+ 5 = 10\)

The answer is C

Posted from my mobile device


lacktutor hi there :) \(( x—2)(x—3) > 0\) from here i get X=2 and x = 3 how did you x< 2 ?
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Re: What is the sum of all integers x satisfying |x^2 - 5x| < 6 ? [#permalink] New post   11 Mar 2020, 10:29
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dave13 wrote:
lacktutor wrote:
\(| x^{2} —5x | < 6 \)
—\(6 < x^{2} —5x < 6\)

\(x^{2}—5x +6 > 0\)
—>\(( x—2)(x—3) > 0\)
\(x< 2\) and \(x >3\)

\(x^{2} —5x —6 < 0\)
—> \((x+ 1)(x—6) < 0\)
—\(1 < x < 6 \)

Taken together,
—\(1 < x < 2\) and \(3 < x < 6\)
\(0+ 1 + 4+ 5 = 10\)

The answer is C

Posted from my mobile device


lacktutor hi there :) \(( x—2)(x—3) > 0\) from here i get X=2 and x = 3 how did you x< 2 ?


hi, dave13
It’s an inequality question, not the equation.
Well, in order (x—2)(x—3) to be greater than zero, x should be less than 2(x<2) and greater than 3(x>3).

Hope it helps
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Re: What is the sum of all integers x satisfying |x^2 - 5x| < 6 ? [#permalink] New post   19 Jun 2023, 09:58
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Re: What is the sum of all integers x satisfying |x^2 - 5x| < 6 ? [#permalink]
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