Rational Inequalities

Rational Inequalities - Rational Inequalities A rational...

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Rational Inequalities A rational inequality is one that can be written in one of the following standard forms: or or or Q does not equal 0. In other words, a rational inequality is in standard form when the inequality is set to 0. Solving Rational Inequalities Using a Sign Graph of the Factors This method of solving rational inequalities only works if the numerator and denominator factor. If at least one of them doesn't factor then you will need to use the test-point method shown later on this page. This method works in the same fashion as it does with quadratic inequalities. If you need a review on solving quadratic inequalities, feel free to go to Tutorial 23A: Quadratic Inequalities.
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Be careful , it is really tempting to multiply both sides of the inequality by the denominator like you do when solving rational equations. The problem is the expression in the denominator will have a variable, so we won't know what the denominator is equal to. Remember that if we multiply both sides of an inequality by a positive number, it does not change the inequality. BUT if we multiply both sides by a negative, it does change the sign of the inequality. Since we don't know what sign we are dealing with we need to go about it the way described below. Step 1: Write the rational inequality in standard form. It is VERY important that one side of the inequality is 0. 0 is our magic number. It is the only number that separates the negatives from the positives. If an expression is greater than 0, then there is no doubt that its sign is positive. Likewise, if it is less than 0, its sign is negative. You can not say this about any other number. Since we are working with inequalities, this idea will come in handy. With this technique we will be looking at the sign of a number to determine if it is a solution or not. Step 2: Factor the numerator and denominator and find the values of x that make these factors equal to 0 to find the boundary points. The boundary point(s) will mark off where the rational expression is equal to 0. This is like the cross over point. 0 is neither positive or negative. As mentioned above, this method of solving rational inequalities only works if the numerator and denominator factor. If at least one of them doesn't factor then you will need to use the test-point method shown later on this page. Step 3: Use the boundary point(s) found in step 2 to mark off test intervals on the number line. The boundary point(s) on the number will create test intervals.
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Step 4: Find the sign of every factor in every interval. You can choose ANY value in an interval to plug into each factor. Whatever the sign of the factor is with that value gives you the sign you need for that factor in that interval. Make sure that you find the sign of every factor in every interval. Since the inequality will be set to 0, we are not interested in the actual value that we get when
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This note was uploaded on 02/18/2011 for the course MATH 101 taught by Professor Kindle during the Spring '11 term at South Texas College.

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Rational Inequalities - Rational Inequalities A rational...

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