3.6 Polynomial & Rational Inequalities

Graph of a Rational Inequality

Graph of f(x)=3x+3/2x+4

How to solve a rational inequality?

1. Bring all terms to one side, obtaining zero on the other side. 2. Express the rational function on the nonzero side as a single quotient. 3. Set the numerator and the denominator of the rational function f equal to zero. The solutions of these equations serve as boundary points that separate the real number line into intervals. From here the procedures for solving a rational inequality are the same as solving a polynomial inequality (testing values and writing the solution set).

5 Step Procedure for Solving Polynomial Inequalities

1. Express the inequality in the form: f(x) < 0 or f(x) > 0, where f is a polynomial function. *Rewrite the inequality so that 0 is on the right side if necessary; do this by adding or subtracting* 2. Find the x-intercepts by solving the equation f(x) = 0. *Don't forget to factor and solve for x. Use the quadratic formula if you can't factor* The real solutions are boundary points. 3. Locate these boundary points on a number line, thereby dividing the number line into intervals. 4. Choose one representative number, called a test value, within each interval and evaluate f at that number. a. If the value of f is positive, then f(x) > 0 for all numbers, x, in the interval. b. If the value of f is negative, then f(x) < 0 for all numbers x in the interval. 5.Write the solution set, selecting the interval or intervals that satisfy the given inequality. *This procedure is valid if < is replaced by ≤ or > greater than is replaced by ≥. However, if the inequality involves ≤ or ≥, include the boundary points [the solutions of f(x)=0] in the solution set.

Example Using the Position Function

A ball is thrown vertically upward from the top of the Leaning Tower of Pisa (190 feet high) with an initial velocity of 96 feet per second. During which time period will the ball's height exceed that of the tower?

What is a Polynomial Inequality?

A polynomial inequality is any inequality that can be put into one of the forms: f(x) < 0, f(x) > 0, f(x) ≤ 0, or f(x) ≥ 0, where f is a polynomial function.

What is a rational inequality?

Any inequality that can be put into one of the forms: f(x) < 0, f(x) > 0, f(x) ≤ 0, or f(x) ≥ 0 where f is a rational function.

Which boundary points must I always exclude from the solution set of a rational inequality?

Any values that cause a function's denominator to equal zero. Division by zero is undefined.

How to find where f is undefined in a rational inequality

Set the denominator equal to 0. The function's vertical asymptote is x=-2. Remember to find the vertical asymptote, p(x) and q(x) cannot have any common factors.

How to find the x-intercept of f in a rational inequality?

Set the numerator and denominator equal to 0 and solve for x.

Graph showing the solutions of a polynomial inequality

f(x)= x^2-7x+10. The x-intercepts, 2 and 5, are boundary points between where the graph lies above the x-axis, shown in blue, and where the graph lies below the x-axis, shown in red. The boundary points divide the real number line into intervals. On each interval, the graph of f is either above the x-axis [f(x)>0] or below the x-axis [f(x)<0].