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C. Solving Quadratic Inequalities Using Sign Analysis

If one side of a quadratic inequality is zero and the other side contains first degree factors, sign analysis can be used to solve the inequality.



 

Investigation

When is a Factored Expression Positive?


  1. Consider the equation \(AB = C\)

    1. List the possible sign combinations for \(A\) and \(B\) that will make \(C\) positive.


    2. List the possible sign combinations for \(A\) and \(B\) that will make \(C\) negative.


  2. Consider the expression \((x + 3)(x - 4)\).


      1. For what values of \(x\) is \((x + 3)\) positive?

      2. For what values of \(x\) is \((x + 3)\) negative?

      3. For what values of \(x\) is \((x - 4)\) positive?

      4. For what values of \(x\) is \((x - 4)\) negative?

    2. For what values of \(x\) is \((x + 3)(x - 4)\) positive?


    3. For what values of \(x\) is \((x + 3)(x - 4)\) negative?
When multiplying two values, there are four possible sign combinations.

positive \(\times\) positive = positive

positive \(\times\) negative = negative

negative \(\times\) positive = negative

negative \(\times\) negative = positive

By determining the conditions that make factors positive or negative, one can determine the variable values that will make a factored expression positive or negative. Solving an inequality using this type of thinking is often called sign analysis.