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Mastering Rational Inequalities: A Step-by-Step Approach with Examples, Lecture notes of Algebra

University of the Philippines Mindanao (UPM)Algebra

Step 4: Determine if the ratio is positive/negative in each interval formed by the critical points. Step 5: Determine the solution based on the inequality. Page ...

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SolvingRationalInequalities

ToBracketorNottoBracket...

SolvingRationalInequalities

Solvingrationalinequalitiesisverysimilartosolvingpolynomial

inequalities.

Thismeansthatwearetryingtofigureoutwhenthefunctionitself

hasavaluegreaterthanorlessthan0.

Butbecauserationalexpressionshavedenominators(andtherefore

mayhaveplaceswherethey'renotdefined),youhavetobealittle

morecarefulinfindingyoursolutions.

Tosolvearationalinequality:

1.findthe

zeroes

(fromthenumerator)

2.findthe

undefinedpoints

(fromthedenominator).

3.usethesezeroesandundefinedpointstodivide

the

numberline

intointervals.

4.

testeachinterval

tofindthesignoftherational

functiononeachinterval.

5.writethefinalsolutionoftheinequalityusing



intervalnotation

.

**Theundefinedpointswillalwayshaveopen

intervals,whiletheintervalsofthezerosdependon

theinequalitysign.**

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To Bracket or Not to Bracket ...

Solving Rational Inequalities

Solving rational inequalities is very similar to solving polynomial inequalities. This means that we are trying to figure out when the function itself has a value greater than or less than 0. But because rational expressions have denominators (and therefore may have places where they're not defined), you have to be a little more careful in finding your solutions.

To solve a rational inequality:

1. find the zeroes^ (from the numerator)

2. find the undefined points (from the denominator).

3. use these zeroes and undefined points to divide

the number^ line^ into intervals.

4. test each interval to find the sign of the rational

function on each interval.

5. write the final solution of the inequality using

interval notation.

** The undefined points will always have open

intervals, while the intervals of the zeros depend on

the inequality sign. **

Ex. 1:

zeroes 1

und. pts. 2

# line 3

test int. 4

solution 5

Ex. 2: Now that we've worked through the entire process think about this problem:. How is this problem different from our first problem? What should we do so that we can solve it? Now complete the problem.

Ex. 4: Mrs. Robinson thinks the answer to is [3,5] and Mrs. Fuston thinks the answer is [3,5). Who is correct and why?