MHB How Many Integers Satisfy This Inequality?

[anemone]

How many integers satisfy the following relation?

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

 

[kaliprasad]

How many integers satisfy the following relation?

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

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

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

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

[TABLE="class: grid, width: 500"]
[TR]
[TD]i.[/TD]
[TD]ii.[/TD]
[/TR]
[TR]
[TD]Now, let $x-9\ge 0$ (i.e. $x \ge 9$), we have[/TD]
[TD]Now, let $x-9< 0$ (i.e. $x \ge 9$), we have[/TD]
[/TR]
[TR]
[TD]$$|x-9-98| \le 82$$

$$|x-107| \le 82$$

$$-82 \le x-107 \le 82$$

$$25 \le x \le 189$$[/TD]
[TD]$$|-(x-9)-98| \le 82$$

$$|-x-89| \le 82$$

$$-82 \le -x-89 \le 82$$

$$-171\le x \le -7$$[/TD]
[/TR]
[TR]
[TD]View attachment 1395

The number of integers that satisfy the aforementioned relation is thus $165$.[/TD]
[TD]View attachment 1396

The number of integers that satisfy the aforementioned relation in this particular case is thus $2$.[/TD]
[/TR]
[/TABLE]

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

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

$$||-x-27|-98| \le 82$$

[TABLE="class: grid, width: 500"]
[TR]
[TD]i.[/TD]
[TD]ii.[/TD]
[/TR]
[TR]
[TD]Now, let $-x-27\ge 0$, we have[/TD]
[TD]Now, let $-x-27< 0$, we have[/TD]
[/TR]
[TR]
[TD]$$|-x-27-98| \le 82$$

$$|-x-125| \le 82$$

$$-207 \le x \le -43$$[/TD]
[TD]$$|-(-x-27)-98| \le 82$$

$$|x-71| \le 82$$

$$-11\le x \le 153$$[/TD]
[/TR]
[TR]
[TD]The number of integers that satisfy the aforementioned relation is thus $165$.[/TD]
[TD]The number of integers that satisfy the aforementioned relation in this particular case is thus $2$.[/TD]
[/TR]
[/TABLE]

Therefore, the total number of integers satisfy the relation $$|||x+9|-18|-98| \le 82$$ 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.