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Algebra I, Chapter 11
Rational Expressions and Equations
Standard (Chapter section):
1.
2.
3.
Alg. 12.0 – Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest
terms (11.2)
Alg. 13.0 – Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually
challenging problems by using these techniques (11.3-11.5)
Alg. 15.0 – Students apply algebraic techniques to solve rate problems, work problems, an percent mixture problems (11.1, 11.6)
Targets:
1. I can identify direct and inverse variation
(11.1)
2. I can graph an inverse variation equation, both positive and negative
(11.1)
3. I can match direct and inverse variation graphs with their equations
(11.1)
4. I can tell whether or not a table represents inverse variation
(11.1)
5. I can write an inverse variation equation from a data in a table
(11.1)
6. I can find excluded values
(11.2)
7. I can simplify rational expressions by dividing out monomials
(11.2)
8. I can recognize opposites and use that information to simplify a rational equation
(11.2)
9. I can multiply a rational expressions involving monomials
(11.3)
10. I can multiply a rational expression by a polynomial
(11.3)
11. I can divide a rational expression involving polynomials
(11.3)
12. I can add or subtract rational expressions with common denominators
(11.4)
13. I can find the LCD of two rational expressions
(11.4)
14. I can add or subtract rational expressions with different denominators
(11.4)
15. I can find the add or subtract two rational expressions by first factoring both expressions’
denominators in order to find the LCD
(11.4)
16. I can use the Distance Traveled formula to solve a rational expression
(11.4)
17. I can use cross-products to solve rational equations
(11.5)
18. I can use properties of equality and the LCD to solve a rational equations
(11.5)
19. I can factor to find the LCD and use properties of equality to solve a rational equation (11.5)
20. I can solve a Mixture problem involving rational expressions
(11.5)
21. I can use the Work Rate formula to find the time it takes to complete a job
(11.6)
Vocabulary: (italics indicate a vocabulary word previously learned in an earlier chapter)
Inverse variation
Constant of variation
Rational expression
Excluded value
Hyperbola
Simplest form of a rational
expression
Rational function
Multiplicative inverse
Branches of a hyperbola
Asymptotes of a hyperbola
Polynomial
Least common denominator
(LCD) of rational expressions
Rational equation
Extraneous solution
Cross product
NOTES:
Graphs of Direct Variation - y  kx
Graphs of Inverse Variation - y 
y
y
2
2
1
–2
k
x
1
–1
–1
1
x
2
–2
–1
–1
–2
1
x
2
–2
Positive (k>0):
Positive (k>0):
y
y
2
2
1
–2
–1
–1
1
1
2
x
–2
–2
–1
–1
1
2
x
–2
Negative (k<0):

Negative (k<0):
Simplifying Rational Expressions
By finding the GCF of the numerator and denominator and then dividing both the numerator and denominator
by that GCF:

4 ( x  7)
2( x  3)
2 ( x  3)
x3


6( x  2) 2 3( x  2) 3( x  2)
5 ( x  7)

4
5
By factoring both the numerator and denominator and then dividing both the numerator and denominator
by the same binomial:
2 x  6 2 ( x  3) 2


3x  9 3 ( x  3) 3
Multiplying Rational Expressions
Cross-cancel any common factors or Binomials
Multiply the Numerators then multiply the
denominators
 Check to see that it is completely simplified
Adding & Subtracting Rational Expressions with
LIKE denominators:
 Add numerators
 Use the LIKE denominator as the new denominator
 Simplify the rational expression by whatever
method is appropriate
Solve Rational Expressions with Cross-Products:


6
x

x4 2
2(6)  x( x  4)
12  x 2  4 x
( x  4) ( x  4) x  4
x 2  8 x  16
( x  4) 2



x 2  16
( x  4)( x  4) ( x  4) ( x  4) x  4
Dividing Rational Expressions
Rewrite as a multiplication problem by:
 replacing the  with
 taking the reciprocal of the divisor (2nd expression)
 Multiply
Adding & Subtracting Rational Expressions with
UNLIKE denominators:
 Find the LCD (least common denominator)
 Multiply the numerator & denominator of each rational
expression to create expressions with LIKE denominators
 Then add or subtract as appropriate
Work Problems:
Work Rate Time = Work Done (usually 1 job)
Work Rate = Work Done  Time
Bob takes three hours to mow one lawn.
Bob’s Work Rate is 1 job per hour.
3
Sue takes 2 hours to mow one lawn.
Sue’s Work Rate is 1 job per hour.
2
If they work together then:
0  x 2  4 x  12
0  ( x  6)( x  2)
x  6 or x  2
t t
  1 , multiply by LCD (6)
3 2
2t  3t  6
5t  6
t
6
5
Bob and Sue can mow one lawn together in
1 hour and 12 minutes.