You can start with:
"(x-5)(x+2) is greater than or equal to 0"
The part " x=5 and x= -2" is only part of the solution, namely the equality part.
You still have to solve the inequality part.
There are two ways to look at this.
First, the easy way witout any calculatoins. For a parabola with the leading coefficient (1)x² >0, we know that the curve is concave upwards, so the values of x between these two zeroes must be less than zero. The answer should therefore be:
x≤-2 or x≥5.
The second way is to assign values of x close to but not equal to the zeroes.
f(-10) = 120
f(-2.1) = 0.71
f(-1.9) = -0.69
f(0) = -10
f(4.9) = -0.69
f(5.1) = 0.71
f(10) = 60
It is clear then that for f(x) = x²-3x-10 ≥ 0,
x ≤ -2 or x ≥ 5
Please don't start a new post for a question if you have a second part to the same problem.
Also give yourself a "First Name", it is much easier to keep track of replies.
Now to your question,
(x-5)(x+2) < 0
I showed you a simple way to proceed from there, but apparently you did not understand it.
so here is the "formal" way
We are multiplying two factors and the result is negative ( < 0)
so one must be positive , the other negative but we don't know which one, so
1. x-5 > 0 AND x+2< 0
x>0 AND x < -2
2. x-5<0 AND x+2>0
x<5 AND x>-2
clearly the first case is not possible,
Can you think of a number which is greater than 2 AND less than -2 ? There is none
Since the two cases are separated by OR, the second case is true.
now recall that the symbold was ≤ and ≥ so we have to include the -2 and 5
and my solution would be all values between -2 and 5 inclusive or
-2 ≤ x ≤ 5, as I stated earlier.
After all that I messed up by reading your question as
≤ instead of ≥
Please follow MathMate's solution, he is right.
Haha... I was about to say, I'm a bit confused...
In other words, I'm correct?
I said x is less than or equal to -2 OR x is greater than or equal to 5.
MathMate concluded with x ≤ -2 or x ≥ 5 ; both mean the same.. or so I thought.
Yes, your last sentence is correct.
I have mis-read your answer as:
"it would follow that x=5 and x= -2 would it not" meaning only equality.