Verified answer

The way its normally done:

Note that a continuous function such as this one can only change sign by passing through a zero first.

Now, x² - x - 20 = (x + 4)(x - 5)

So the only time that x² - x - 20 can change sign is at

x = -4 and x = 5

Now in the interval

(-∞, -4), we have that the function is positive

(We only need to check on value in the interval)

In the interval

(-4, 5), the function is negative

In the interval

(5, ∞), the function is positive again.

So the function is only negative (< 0) in the interval

(-4, 5)

Also, it is equal to 0 at -4 and 5, so the the interval that satisfies the inequality is

[-4, 5]

======================

Another way to do it:

Add 20.25 to each side and you get

x² - x + .25 ≤ 20.25

So that

(x - ½)² ≤ 20.25 = 81/4

This gives us

|x - ½| ≤ 9/2

So,

x - ½ ≤ 9/2 and

x - ½ ≥ -9/2

Giving us

-4 ≤ x ≤ 5

Giving us exactly the same interval as before

You can ignore the "less than or equal sign" which i will represent by "=<" and treat it as an equal sign for most of the problem. Until you have separate x terms by themselves, only then does it have some meaning.

So we have x^2-x-20=<0

(x-5)(x+4)=<0

So x-5=<0 and x+4=<0

Therefore x=<5 and x=<-4

The solution is the one that both solutions "agree on".

If we say x is less than or equal to 5, it is wrong because it could also be greater than -4. That's a contradiction.

If we say x is less than or equal to -4 it's correct because it is less than -4 and any number at or below -4 is also below 5. So x=<-4

x² - x - 20 <= 0

(x - 5)(x + 4) = 0

Equate each equation to 0.

x - 5 = 0

x = 5

x <= 5

***

x + 4 = 0

x = -4

x >= -4

***

-4 <= x <= 5

x² -5x+4x -20 < or = 0

(x-5)(x+4) < or = 0

x = 5 or -4

x≥-20

x² -x -20 < or = 0

x² -5x+4x -20 < or = 0

x(x-5)+4(x-5) < or = 0

(x-5)(x+4) < or = 0

x = 5 or -4

so x = -4 which is less than 0

x² -x -20 < or =0

x² -5x+4x -20 < or =0

x² -5x+4x -20 < or = 0

(x-5)(x+4) < or = 0

x = 5 or -4

x = -4

I was thinking it was along these lines:

x² -x -20 ≤ 0

x² -x ≤ 0 + 20

x² -x ≤ 20

But everyone else has got difference answers to me......so I wouldn't trust me but that's how we do it in my school........good luck! =)

Boy am I glad I'm done with school!