Solve x^2 -3x is greater than or equal to 10
Would the solutions be x=5 and x= -2?
Here's how I figured it.
x^2-3x is greater than or equal to 10.
I subtracted 10 from both sides. This left me with x^2-3x-10 is greater than or equal to 0. Next, I factored and came up with (x-5)(x+2) is greater than or equal to 0. Logically, it would follow that x=5 and x= -2 would it not? And then in symbolic form my answer would read x is less than or equal to -2 OR x is greater than or equal to 5.
Where, exactly, did I go wrong?
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:
(-∞,-2]∪[5,∞), or
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
ARGLHHHH!!!
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.
Your reasoning is correct up until factoring the quadratic expression.
When you factor x^2 - 3x - 10, you actually get (x - 5)(x + 2) = 0. This means that either x - 5 = 0 or x + 2 = 0. So the correct solutions are x = 5 and x = -2.
However, when dealing with inequalities, you need to consider the signs of the factors. In this case, you have (x - 5)(x + 2) ≥ 0.
To determine the values of x that satisfy the inequality, you can use a sign chart or test values within each interval.
Using a sign chart:
- On the number line, mark the values x = -2 and x = 5.
- Choose a test point within each resulting interval.
* For x < -2, you can choose x = -3.
* For -2 < x < 5, you can choose x = 0.
* For x > 5, you can choose x = 6.
- Substitute these test points into the inequality to determine the sign of the expression (x - 5)(x + 2):
* For x = -3: (-3 - 5)(-3 + 2) = (-8)(-1) = 8, which is > 0.
* For x = 0: (0 - 5)(0 + 2) = (-5)(2) = -10, which is < 0.
* For x = 6: (6 - 5)(6 + 2) = (1)(8) = 8, which is > 0.
From the sign chart, you can see that the expression (x - 5)(x + 2) is positive (≥ 0) when x ≤ -2 or x ≥ 5.
Therefore, the correct solution to the inequality x^2 - 3x ≥ 10 is 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,
you had
(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
OR
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.