Find the number of integers

anemone

MHB POTW Director
Staff member
How many integers satisfy the following relation?

$$\displaystyle |||x+9|-18|-98| \le 82$$

Last edited:

Well-known member
How many integers satisfy the following relation?

$$\displaystyle |||x+9|-18|-98| \le 82$$
if we put x + 9 >= 0 we get

- 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

anemone

Staff member

Well-known member
BUT...I got 332.
I would like to have a look at the correct solution

anemone

MHB POTW Director
Staff member
My solution:

We have two cases to consider here, one is when $x+9\ge 0$ and the other is when $x+9< 0$.

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

$$\displaystyle ||x+9-18|-98| \le 82$$

$$\displaystyle ||x-9|-98| \le 82$$

 i. ii. Now, let $x-9\ge 0$ (i.e. $x \ge 9$), we have Now, let $x-9< 0$ (i.e. $x \ge 9$), we have $$\displaystyle |x-9-98| \le 82$$ $$\displaystyle |x-107| \le 82$$ $$\displaystyle -82 \le x-107 \le 82$$ $$\displaystyle 25 \le x \le 189$$ $$\displaystyle |-(x-9)-98| \le 82$$ $$\displaystyle |-x-89| \le 82$$ $$\displaystyle -82 \le -x-89 \le 82$$ $$\displaystyle -171\le x \le -7$$ 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 $2$.

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 $165+3+165+2=335$.

Hey kaliprasad, I'm sorry because according to my previous reply, I told you the answer that I've gotten was 332, which isn't the correct answer.