What is the length of the segment of the number line consisting of the points that satisfy (x-4)^2 \le 9?

I thought it was 7 but that was wrong...
Can't x be 1, 2, 3, 4, 5, 6, and 7?

*This question is from AoPS (AoPS info below).
Subject: Algebra
Focus: Quadratic Inequalities

1 Answer
Jun 7, 2018

6

Explanation:

OHHHH OKAY SO I'M DUMB. I got it wrong because it's asking for the length, and even though there are 7 numbers, the distance is 6.

On to the Real Explanation

First, take the square root of both sides. Then you get:
x-4\le3
Add 4 to both sides.
x\le7
However, if you think about it (and look at what the question is asking), x cannot possibly equal all of the values less than 7.
Checking different values, you can see that 0 doesn't work.
And so,
x can be anywhere from 1 to 7.
Not a very good solution, I know, but...
oh! here's

AoPS' Solution:

Since the square of x-4 is at most 9, the value of x-4 must be between -3 and 3 (or equal to either). So, we have -3 \le x-4 \le 3. Thus, 1 \le x \le 7. Therefore, our answer is 6.

OR -

If (x-4)^2 \le 9, then x can be no more than 3 away from 4. Therefore, the values of x from 1 to 7 satisfy the inequality, and our answer in 6.