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Find all intervals on which the given expressions are positive.


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

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

  • Algebra - ,

    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.

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