How Many Integers Satisfy This Inequality?

[anemone]

How many integers satisfy the following relation?

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

 

[kaliprasad]

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]

...

so 330 solutions

BUT...I got 332.

 

[kaliprasad]

BUT...I got 332.

I would like to have a look at the correct solution

 

[anemone]

My solution:

Spoiler

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
\(\displaystyle |x-9-98| \le 82\) \(\displaystyle |x-107| \le 82\) \(\displaystyle -82 \le x-107 \le 82\) \(\displaystyle 25 \le x \le 189\)	\(\displaystyle |-(x-9)-98| \le 82\) \(\displaystyle |-x-89| \le 82\) \(\displaystyle -82 \le -x-89 \le 82\) \(\displaystyle -171\le x \le -7\)
View attachment 1395 The number of integers that satisfy the aforementioned relation is thus $165$.	View attachment 1396 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.