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- #1

- Feb 14, 2012

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How many integers satisfy the following relation?

\(\displaystyle |||x+9|-18|-98| \le 82\)

\(\displaystyle |||x+9|-18|-98| \le 82\)

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- Thread starter anemone
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- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,683

How many integers satisfy the following relation?

\(\displaystyle |||x+9|-18|-98| \le 82\)

\(\displaystyle |||x+9|-18|-98| \le 82\)

Last edited:

- Mar 31, 2013

- 1,309

if we put x + 9 >= 0 we getHow many integers satisfy the following relation?

\(\displaystyle |||x+9|-18|-98| \le 82\)

- 82 <= | x- 9 | - 98 <= 82

so 165 solutions as |x -9| - 98 can be atleast -98

similarly if we put | x+ 9| <= 0 so 165 solutions

so 330 solutions

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- #3

- Feb 14, 2012

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BUT...I got 332.......

so 330 solutions

- Mar 31, 2013

- 1,309

I would like to have a look at the correct solutionBUT...I got 332.

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- Feb 14, 2012

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If $x+9\ge 0$ (i.e. $x \ge -9$), then the inequality becomes

\(\displaystyle ||x+9-18|-98| \le 82\)

\(\displaystyle ||x-9|-98| \le 82\)

But if $x+9< 0$ (i.e. $x<-9$), then the inequality becomes

\(\displaystyle ||-x-9-18|-98| \le 82\)

\(\displaystyle ||-x-27|-98| \le 82\)

i. | ii. |

Now, let $-x-27\ge 0$, we have | Now, let $-x-27< 0$, we have |

\(\displaystyle |-x-27-98| \le 82\) \(\displaystyle |-x-125| \le 82\) \(\displaystyle -207 \le x \le -43\) | \(\displaystyle |-(-x-27)-98| \le 82\) \(\displaystyle |x-71| \le 82\) \(\displaystyle -11\le x \le 153\) |

The number of integers that satisfy the aforementioned relation is thus $165$. | The number of integers that satisfy the aforementioned relation in this particular case is thus $2$. |

Therefore, the total number of integers satisfy the relation \(\displaystyle |||x+9|-18|-98| \le 82\) is $165+3+165+2=335$.

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