Solving Rational Equations and Inequalities
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Transcript Solving Rational Equations and Inequalities
Solving Equations Containing
Rational Expressions
Unit 4
Lesson 9.6 text book
CCSS: A.CED.1
Standards for Mathematical
Practice
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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
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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):
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How do I solve a rational equation?
How do I use rational equations to solve
problems?
Recap of This Unit
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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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Example 1
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Solve the equation below:
3
2
5x x 7
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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.
Example 2
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Solve the equations below by crossmultiplying. Check your solution(s).
4
5
x3 x3
1
x
2x 5 11x 8
LCDs: PerformanceEnhancing Math Term?
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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
7-time Cy Young Award
winner Roger Clemens
by the same quantity on
never “knowingly used
both sides of the equation.
LCDs” in his career.
“One less thing to worry
about”
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If your solution doesn’t
If one or more of your solutions are not
work in the original
valid in the original equation, they are
equation, well, I guess
called “extraneous solutions” and
should not be included in your list of that’s just one less thing
you has to worry about.
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.
Solving rational equations
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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
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Solve the equation by using the LCD.
Check for extraneous solutions.
7 3
3
2 x
Example 4
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Solve the equation by using the LCD.
Check for extraneous solutions.
3x
5
3
x 1 2x 2x
Example Real life 5
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Suppose the population density in the
Wichita, Kansas, area is related to the
distance from the center of the city.
4500x
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x 32
where D is the population density (in people
per square mile) and x is the distance (in
miles) from
the center of the city.
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Find the distance(s) where the population
density is 375 people per square mile.
INEQUALITIES
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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)
1
2 2
3x
9x
3
Solve the Inequalities
Multiply by the LCM which is 9x.
2 2
1
9x
9x
3x 9 x 3
3 2 6x
5 6x
5
x
6
Solve the Inequalities
Was there any excluded values?
2 2
1
9x
9x
3x 9 x 3
3 2 6x
5 6x
5
x
6
Solve the Inequalities
Was there any excluded values? YES
2 2
1
9x
9x
3x 9 x 3
3 2 6x
5 6x
5
x
6
x0
Solve the Inequalities
5
Using the exclude value and the solution
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Make a Number line. Test the values between
the dotted lines.
5
6
Solve the Inequalities
5
Using the exclude value and the solution
6
Make a Number line. Test the values between
the dotted lines.
Let x 1
1
2 2
3x 9 x 3
1 2 2
3 9 3
5
6
3 2 2
9 9 3
5 2
5
yes x works
9 3
6
Solve the Inequalities
5
Using the exclude value and the solution
6
Make a Number line. Test the values between the
dotted lines.
Let x 1
1
2 2
3 9 3
5 2
9 3
5
6
Yes x 0 works
Solve the Inequalities
5
Using the exclude value and the solution
6
Make a Number line. Test the values between the
dotted lines.
1
Let x
3
1
2
2
1
1 3
3 9
3
3
1 2 2
1 3 3
5
6
2 2
1 Does not work
3 3
So final solution will be:
x0
5
x
6
Real Life example:
The function
20t
P (t )
t 1
models the
population, in thousands, of Nickelford, t
years after 1997. The population, in
thousands, of nearby New Ironfield is
modeled by
240
Q(t )
t 8
Determine the time period when the
population of New Ironfield exceeded the
population of Nickelford.
Continue:
**Solve for the interval(s) of t where Q(t) > P(t)
Q(t ) P (t )
240
20t
t 8 t 1
240 20t
0
t 8 t 1
240(t 1)
20t (t 8)
0
(t 8)(t 1) (t 1)(t 8)
20t 2 80t 240
0
(t 8)(t 1)
20(t 6)(t 2)
0
(t 8)(t 1)
20(t 6)(t 2)
Let f (t )
(t 8)(t 1)
Solving the Inequality Graphically
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.
240
20t
t 8 t 1
Solving Rational Equations &
Inequalities Practice
1.
2x
x3
6 x 5 3x 1
2.
x 2 9 x 14
0
2
x 6x 5
3.
x3
0
x4
4.
5
3
x3 x2
5.
5
2
x 4 x 1
x 2 5x 6
6.
0
x4