This is solving inequality by factorizing it.
(2x-1)(x+1)>0
You set each factor to zero to find the roots. You get root x=1/2 that makes the first factor zero and x=-1 that makes the second factor zero.
Then you draw a number line and put the two roots -1 and 1/2 on it. The roots separate the line into 3 intervals: x<-1, -1<x<1/2, x>1/2.
Start with the right-most interval: x>1/2
If x>1/2, both factors (2x-1) and (x+1) are positive and their product is positive, so the interval x>1/2 is a solution to the inequality.
Move to the next interval to the left: -1<x<1/2
Moving to the left of the root x=1/2 flips the sign of the factor (2x-1) to negative, the other factor (x+1) has the same sign as before, positive. The product of the two factors is negative, so this interval is not a solution of the inequality.
Move to the next interval to the left: -1<x
Moving on the left of the root x=-1, flips the sign of the factor (x+1) to negative and keeps the previous sign of (2x-1) to negative. The product (x+1)(2x-1) becomes positive so -1<x is a solution of the inequality.
In summary, the solution is -1<x OR x>1/2.
The principle is always the same, when you cross from one side of a root to the other, one of the factors flips sign which flips the sign of the product of all factors. So the sign of the product alternates pos, neg, pos, neg .... etc.
The exception of that pattern is when your factors are raised to even power like (x+1)^2. If you have such a case, crossing on the other side of the root on the number line does NOT flip the sign of the factor and of the whole expression.
Last edited by
tutorphd on Tue Jun 26, 2012 8:38 pm, edited 2 times in total.
