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Solve a Rational Inequality
You are asked to find the points to make a rational inequality true.
Rational inequalities include finding the
~ zeros of the function (numerator)
~ the vertical asymptote(s) (denominator).
The analysis is the same as finding the solution set of a polynomial inequality.
In addition to finding the behavior around the zeros (numerator), there is a restriction on the denominator, so the analysis will also include the behavior around its vertical asymptote(s).
Plugging the test points (values around the zeros and vertical asymptote(s)) in the original inequality will determine if the region around the zero is positive or negative.
The regions are used
~ to determine the solution set, finding the solutions to the inequality
~ to help graph the inequality.
Example 9: Graph Solution
The graph of the rational function and its positive/negative regions are below. You should notice that the blue regions include the solution set for the inequality: x<=0.
Because x = 1 is a vertical asymptote, the solution set does not include x=1. Solution Set: [-2,0] u (1, 2]
Making a connection between the solution set and the graph of the inequality
~ When the region is negative, the function's graph is located below the x-axis.
~ When the region is positive, the function's graph is located above the x-axis.
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