Solving Equations Containing Rational Expressions

Unit 4Lesson 9.6 text book

CCSS: A.CED.1

Standards for Mathematical Practice

• 1. Make sense of problems and persevere in solving them. • 2. Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the reasoning

of others.  • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6. Attend to precision. • 7. Look for and make use of structure. • 8. Look for and express regularity in repeated reasoning.

CCSS: A.CED.1• Create equations and inequalities in one

variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Essential Question(s): • How do I solve a rational equation? •  How do I use rational equations to solve

problems?

Recap of This Unit• So far in this unit we have:

Talked about Polynomial Functions with one variable.

Graphed polynomial functions with one variable.Learned how to use Quadratic Techniques.Talked about the Reminder and Factor Theorem.Roots and Zeros.

Next up…

• In this section, we

will apply our

knowledge of solving polynomial equations to solving rational equations and inequalities.

Please try to control your excitement.

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section

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Example 1• Solve the equation below:

• When each side of the equation is a single rational expression, we can use cross multiplication.

• It is VERY important to check your answer in the original equation.

3

5x

2

x 7

Example 2• Solve the equations below by cross-

multiplying. Check your solution(s).

4

x 3

5

x 3

1

2x 5

x

11x 8

LCDs: Performance-Enhancing Math

Term?• When a rational equation is not

expressed as a proportion (with one term on each side), we can solve it by multiplying each side of the equation by the least common denominator (LCD) of the rational equation.

• NOTE: To balance the equation, we must be sure to multiply by the same quantity on both sides of the equation.

7-time Cy Young Awardwinner Roger Clemensnever “knowingly used

LCDs” in his career.

“One less thing to worry about”

• If one or more of your solutions are not valid in the original equation, they are called “extraneous solutions” and should not be included in your list of actual solutions to the equation.

• The graphs of rational functions may have breaks in continuity. Breaks in continuity may appear as asymptote (a line that the graph of the function approaches, but never crosses) or as a point of discontinuity.

If your solution doesn’twork in the original

equation, well, I guessthat’s just one less thing you has to worry about.

Solving rational equations• Find the LCM for the denominators• Any solution that results in a zero in denominator

must be excluded from your list of solutions.• Multiply both sides of the equation by the LCM to

get rid of all denominators• Solve the resulting equation (may need quadratic

techniques, etc.)• Always check your answers by substituting back

into the original equation! WHY????

Example 3• Solve the equation by using the LCD.

Check for extraneous solutions.

7

2

3

x3

Example 4• Solve the equation by using the LCD.

Check for extraneous solutions.

3x

x 1

5

2x

3

2x

Example Real life 5

• Suppose the population density in the Wichita, Kansas, area is related to the distance from the center of the city.

• This is modeled by

where D is the population density (in people per square mile) and x is the distance (in miles) from the center of the city.

• Find the distance(s) where the population density is 375 people per square mile.

D 4500x

x 2 32

INEQUALITIES• Recall that for inequalities, we often pretend we

are dealing with an equation, put the solutions on a number line, and then test a point from each region

• Same thing here!• 1st find the excluded values• Then solve the related equation• Put the solutions and excluded values on a number

line• Then test a point in each region to determine which

range(s) of values represent solutions!

Solve the Inequalities

Remember to be careful with multiply

(negative number changes the direction of the inequality)

3

2

9

2

3

1

xx

Solve the InequalitiesMultiply by the LCM which is 9x.

x

x

x

xxx

x

6

5

65

623

93

2

9

2

3

19

Solve the InequalitiesWas there any excluded values?

x

x

x

xxx

x

6

5

65

623

93

2

9

2

3

19

Solve the InequalitiesWas there any excluded values? YES

x

x

x

xxx

x

6

5

65

623

93

2

9

2

3

19 0x

Solve the InequalitiesUsing the exclude value and the solution

Make a Number line. Test the values between the dotted lines.

6

5

6

5

Solve the InequalitiesUsing the exclude value and the solution

Make a Number line. Test the values between the dotted lines.

6

5

6

5

3

2

9

2

3

1

3

2

9

2

3

1

xx

worksxyes6

5

3

2

9

5

3

2

9

2

9

3

1xLet

Solve the InequalitiesUsing the exclude value and the solutionMake a Number line. Test the values between the

dotted lines.

1xLet

6

5

6

5

3

2

9

5

3

2

9

2

3

1

worksxYes 0

Solve the InequalitiesUsing the exclude value and the solutionMake a Number line. Test the values between the

dotted lines.

6

5

6

5

3

2

3

2

1

1

3

2

31

9

2

31

3

1

worknotDoes3

2

3

21

3

1xLet

So final solution will be:

0x6

5x

Real Life example:

The function models the

population, in thousands, of Nickelford, t years after 1997. The population, in thousands, of nearby New Ironfield is

modeled by

1

20)(

t

ttP

8

240)(

ttQ

Determine the time period when the population of New Ironfield exceeded the population of Nickelford.

)()( tPtQ **Solve for the interval(s) of t where Q(t) > P(t)

1

20

8

240

t

t

t

01

20

8

240

t

t

t

0)8)(1(

)8(20

)1)(8(

)1(240

tt

tt

tt

t

0)1)(8(

2408020 2

tt

tt

0)1)(8(

)2)(6(20

tt

tt )1)(8(

)2)(6(20 )(Let

tt

tttf

Continue:

o Graph both curves on the same set of axes. o Find the POIs of the two curves.o Use the POI to determine the intervals of t that satisfy the inequality.

Solving the Inequality Graphically

1

20

8

240

t

t

t

Solving Rational Equations & Inequalities Practice

04

65.6

2

x

xx

056

149.2

2

2

xx

xx

04

3.3

x

x

2

3

3

5.4

xx

1

2

4

5.5

xx

13

3

56

2.1

x

x

x

x