In this Section:
In this section we learn how to solve quadratic and rational inequalities. When solving a quadratic inequality, we must lean on our knowledge of quadratic equations along with linear inequalities. One method we can use to solve a quadratic inequality is graphing. As we have seen before, graphing is not a practical method for finding solutions. Alternatively, we can turn to a more practical approach which involves using test numbers. For this approach, we write our quadratic inequality as an equation and solve. Once this is done, we then use the solutions to divide the number line into intervals. We then test inside each interval to find which intervals satisfy the inequality. If one number inside the interval satisfies the inequality, then the whole interval does. Lastly, we must consider the endpoints. These are included in the solution set if we have a non-strict inequality and not included if we have a strict inequality. We use a similar approach to solve a rational inequality. We write our rational inequality so that zero is on one side and there is a single fraction on the other. Then we determine the numbers that will make the numerator or denominator equal to zero. After this we use these numbers to divide the number line into intervals. We find the intervals that satisfy our inequality by testing a number from each interval. Lastly, we consider our endpoints. These are included in the solution set if we have a non-strict inequality and not included if we have a strict inequality. We must exclude any values that make the denominator zero.