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# 3.8: Polynomial and Rational Inequalities

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## Solving Polynomial Inequalities

A polynomial inequality is a mathematical statement that relates a polynomial expression as either less than or greater than another. We can use sign charts to solve polynomial inequalities with one variable.

Graphs are helpful in providing a visualization to the solutions of polynomial inequalities. Examine the graph below to see the relationship between a graph of a polynomial and its corresponding sign chart.

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In this graph, the (x)-intercepts, -3, 0 and 4, are critical points, which are the only places where the graph may possibly change from being above the (x)-axis (where we say that (f(x) > 0) ) ), to below the (x)-axis (where we say that (f(x) < 0) ). Within each interval between two adjacent critical points, the graph is either always above the (x)-axis, or always below the (x)-axis. Thus, finding the critical points of a polynomial inequality plays a fundamental role in solving a polynomial inequality.

Graphs can help us to visualize solutions of polynomial inequalities. Below is the graph of the function in the above example. Compare the graph to its corresponding sign chart. Notice that the sign chart is positive when the graph is above the (x)-axis and negative when the graph is below the (x)-axis. The graph crosses or touches the (x)-axis at the critical points.

Certainly it may not be the case that the polynomial is factored nor that it has zero on one side of the inequality. To model a function using a sign chart, all of the terms should be on one side and zero on the other. The general steps for solving a polynomial inequality are listed in the following example.

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