Question

Find all intervals on which the given expressions are positive.

(x-1)(x+2)

I know the answer is : x<-2, x>1

please show me the steps on how to solve this math problem.

Answer 1

to have the product be positive, either both factors are negative, or both are positive.

Both negative:
x-1 < 0 AND x+2 < 0
x < 1 AND x < -2
==> x < -2

Both positive:
x-1 > 0 AND x+2 > 0
x > 1 AND x > -2
==> x > 1

or, you can consider what you know about polynomials. If there are no repeated roots, then the graph crosses the x-axis at various points. In between those points, the function does not change sign. When crossing the axis, the function changes sign.

So, consider x very large negative. Both factors are negative, so the product is positive. Each time it crosses the x-axis (at one of the roots), it changes sign. So, looking at the number line, and marking + or -, we start way out on the left with +, and change sign at each root:

++++++ -2 ----- 1 +++++++

Repeated roots modify this algorithm some, as the graph may just touch the axis and not change sign.