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.

How to go from (1-2x)(1+x)<0 to (2x-1)(1+x)>0:

You multiply both sides of the inequality by -1 and flip the direction of the inequality:

-1*(1-2x)*(1+x) <0

You can multiply the -1 with one of the factors not both, like in 2*3*4 you can multiply 2 and 3 to get an equal expression 6*4 but you don't multiply both: 6*8 is wrong. You are mixing distribution of multiplication over addition like in 2*(3+4) = 2*3 + 2*4 with associativity of multiplication like 2*3*4 = (2*3)*4.

Anyways, choose to multiply the first factor: -1*(1-2x) = 2x-1

So you get (2x-1)(1+x)<0

thanks for the explanation tutorphd.

1 more quick question:

when you started out by writing: x<-1, -1<x<1/2, x>1/2, why did we go with the assumption x>1/2, why cant we start with x<1/2? If it's b/c the original equation had a > sign, then why not x>-1, why x<-1?

Sorry if this sounds a bit elementary.

Because you have to draw a number line and take the intervals into which the roots split it.

Draw the line, put the roots on it, and you will see it is split in exactly 3 intervals I mentioned. You consider those because the sign of the product (2x-1)(x+1) remains the same inside each interval but flips once you cross over a root to the next interval.

The solutions of the inequality (2x-1)(x+1)>0 are the intervals where the product of factors (2x-1)(x+1) is positive. That is the only place where the direction of the original inequality plays a role: to select which intervals form the solution.

that clears it up. thanks!