Posted by Alex on Sunday, August 12, 2012 at 11:38pm.
Find all intervals on which the given expressions are positive.
(x1)(x+2)
I know the answer is : x<2, x>1
please show me the steps on how to solve this math problem.

Algebra  Steve, Sunday, August 12, 2012 at 11:59pm
to have the product be positive, either both factors are negative, or both are positive.
Both negative:
x1 < 0 AND x+2 < 0
x < 1 AND x < 2
==> x < 2
Both positive:
x1 > 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 xaxis 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 xaxis (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.
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