Solving Polynomial and Rational Inequalities: A Step-by-Step Guide, Papers of Mathematics

Download Solving Polynomial and Rational Inequalities: A Step-by-Step Guide and more Papers Mathematics in PDF only on Docsity! Section 3.8 - Steps in Solving Polynomial and Rational Inequalities STEP 1 : Write the inequality so that a polynomial or rational expression f is on the left side and zero is on the right side in one of the following forms: f(x) > 0 f(x) ≥ 0 f(x) < 0 f(x) ≤ 0 For rational expressions, be sure that the left side is written as a single quotient. This step converts the problem of solving an inequality into an equivalent (i.e., the same solution) problem of determining where a function is positive (or negative). STEP 2 : Factor f(x) to determine the numbers at which the expression f (x) on the left side equals zero and, if the expression is rational, the numbers at which the expression f on the left side is undefined. We will call these numbers partition values. The Intermediate Value Theorem tells us that a continuous function (graph can be drawn without raising the pencil from the paper) cannot change signs on an interval without having a zero in that interval. So the above partition values divide the x-axis into intervals on which the sign of f(x) CANNOT change. STEP 3 : Use the numbers found in STEP 2 to separate the real number line into intervals. We will construct a sign chart for the function f(x). STEP 4 : Determine the sign of each factor of f(x) on the intervals found in step 3. STEP 5: Multiply the sign of the factors to determine the sign of f(x) in each interval. STEP 6: The solution of the inequality includes all intervals with the correct sign (positive or negative). If the inequality is not strict, include the numbers at which f (x) is zero in the solution set. Be careful: The numbers which make f(x) undefined (i.e., the zeroes of the denominator in a rational function) are never included in the solution set.