The method of intervals

The method of intervals solves inequalities that are products or fractions. The procedure: 1) factor the expression into factors (x − a); 2) mark the zeros of the factors on the number line, which splits it into intervals; 3) in the rightmost interval every (x − a) is positive, so the expression is positive (if the leading coefficient is positive); 4) moving right to left the expression changes sign at each zero; 5) write down the intervals with the required sign. If a factor appears with an even power, such as (x − 3)², it does not change sign at its zero, but x = 3 may still be a solution of a non-strict inequality. For a fraction we mark the zeros of numerator and denominator; zeros of the denominator are never solutions, because the fraction is not defined there.