I'm having trouble with inequalities. Basically my problem is about which way the sign goes. Here's an example..
X(X+1)>2
so then it becomes
X^2 + X>2
Correct? Then
X^2 + X - 2>0
then factor so it becomes
(X+2)(X-1)>0
And so I end up with X>-2 and X>1 as my answer, but the book says it should be X<-2. Why did the sign switch? I didn't divide by a negative did I? Can someone help me please?
Here is the logical analysis:
(X+2)(X-1)>0 says that the answer is positive, so either
x+2 > 0 AND x-1 > 0
x > -2 AND x > 1 ----> x > 1
or
x+2 < 0 AND x-1 < 0
x < -2 AND x < 1 ----x < -2
so x < -2 OR x > 1
I use the following quick way to solve these kind of problems.
from (X+2)(X-1)>0
the "critical values", or the solution to the corresponding "equation", are
x = -2 and x = 1
put these values on a number line, this will split the number line into 3 regions
1. x < -2
2. x between -2 and 1
3. x > 1
select any value of x in those regions and test it in your factored inequation.
e.g. for
1. I picked x=-5 that works
2. let x = 0 , does NOT work
3. let x = 10 , that also works
so x > 1 OR x < -2
When solving inequalities, it's important to keep track of the sign changes and to consider the cases where the sign switch occurs. Let's walk through solving the inequality X(X+1) > 2 step by step.
First, you correctly rewrote the inequality as X^2 + X > 2.
To solve this quadratic inequality, we typically want to set it equal to zero by subtracting 2 from both sides:
X^2 + X - 2 > 0
Then, you factored the quadratic expression:
(X+2)(X-1) > 0
Now, here comes the crucial step. To determine the sign of the inequality for different values of X, we need to consider the sign of each factor separately and their combination:
1. When (X+2) > 0 and (X-1) > 0:
- (X+2) > 0: X > -2
- (X-1) > 0: X > 1
2. When (X+2) < 0 and (X-1) < 0:
- (X+2) < 0: X < -2
- (X-1) < 0: X < 1
3. When (X+2) > 0 and (X-1) < 0:
- (X+2) > 0: X > -2
- (X-1) < 0: X < 1
4. When (X+2) < 0 and (X-1) > 0:
- (X+2) < 0: X < -2
- (X-1) > 0: X > 1
Now, looking at the resulting combinations:
- In case 1, both factors are positive, so the inequality holds true for all X values satisfying X > 1.
- In case 2, both factors are negative, so the inequality does not hold true for any X values.
- In cases 3 and 4, one factor is positive and the other is negative, so the inequality doesn't hold true for any X values in these cases.
Therefore, the solution is X > 1 since it is the only case where the inequality holds true.
The book stating X < -2 as the answer seems to be incorrect. It's possible that there was a mistake or a typo in the book. Always make sure to carefully examine the solution steps, check for sign changes, and pay attention to the combinations of positive and negative factors to determine the correct answer.