rational functions
MHF4U: Advanced Functions
Solving Rational Inequalities
Part 1: Simple Inequalities
J. Garvin
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rational functions
Rational Inequalities
Rational inequalities can be solved using similar techniques
for solving polynomial inequalities: cases or intervals.
Recall the rules for solving inequalities.
Rules for Solving Inequalities
•The same value may be added to, or subtracted from,
both sides of an inequality.
•Each side of an inequality may be multiplied, or divided,
by the same positive value.
•Each side of an inequality may be multiplied, or divided,
by the same negative value if the inequality is reversed.
•If each side of an inequality has the same sign, the
reciprocal of each side may be taken if the inequality is
reversed.
J. Garvin — Solving Rational Inequalities
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rational functions
Solving Rational Inequalities Using Cases
Example
Solve 3
x−2>−4 using cases.
Since x−26= 0, there are two cases to consider.
Case 1: x−2>0, or x>2.
3
x−2>−4
3>−4(x−2)
3>−4x+ 8
−5>−4x
5
4<x
J. Garvin — Solving Rational Inequalities
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Solving Rational Inequalities Using Cases
Consider the two intervals on a number line.
Since x>2 is common, it is a solution to the inequality.
J. Garvin — Solving Rational Inequalities
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Solving Rational Inequalities Using Cases
Case 2: x−2<0, or x<2.
3
x−2>−4
3<−4(x−2)
3<−4x+ 8
−5<−4x
5
4>x
J. Garvin — Solving Rational Inequalities
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rational functions
Solving Rational Inequalities Using Cases
Consider the two intervals on a number line.
Since x<5
4is common, it is a solution to the inequality.
The solution, then, is 3
x−2>−4 on −∞,5
4∪(2,∞).
A graph confirms these intervals.
J. Garvin — Solving Rational Inequalities
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