7.1 ‐ Rational Expressions and Their Simplification Learning Objectives: 1. Find numbers for which a rational expression is undefined. 2. Simplify rational expressions. Examples of rational expressions: 3 2 x and 2 2 1 x x and 2 1 2 3 x x x • Rational expressions are quotients of two polynomials. They indicate division and division by zero is undefined. We must always exclude any value(s) of the variable that make a denominator zero . Ex 1 Find all the numbers for which the rational expression is undefined. If the rational expression is defined for all real numbers, so state. a) 3 2 x b) 2 2 x x c) 2 1 2 x x x d) 9 2 6 2 x x x e) 7 7 x 1
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7.1 ‐ Rational Expressions and Their Simplification
Learning Objectives: 1. Find numbers for which a rational expression is undefined. 2. Simplify rational expressions.
Examples of rational expressions:
3
2x and
2
2
1
x
x
and 2
1
2 3
x
x x
• Rational expressions are quotients of two polynomials. They indicate division and division by zero is undefined. We must always exclude any value(s) of the variable that make a denominator zero.
Ex 1 Find all the numbers for which the rational expression is undefined.
If the rational expression is defined for all real numbers, so state.
a) 3
2x
b) 2
2
x
x
c)
2
1 2
x
x x
d) 9
26 2
x
x x
e) 7
7
x
1
• When simplifying rational expressions, first, factor the numerator and denominator completely, and then divide both the numerator and the denominator by any common factors. A rational expression is simplified if its numerator and denominator have no common factors other than 1 and –1. • When reducing rational expressions, only factors (not common terms) that are common to the entire numerator and the entire denominator can be divided out. • A monomial can ONLY reduce with a monomial. A binomial can ONLY reduce with an identical binomial.
Ex 2 Simplify.
a) 46
b) 22
18xx
c) 7 21
49x
d) 217 14x
2
e) 5 10
2
x
x
f) 4
2 16
x
x
• NOTE: 1a b b a
g) 9 15
5 3
x
x
h) 23 4
26 2
x x
x x
4
3
i) 3 24 3 1
4
x x x
x
2
j) 5
2 25
x
x
Recall: There are a few polynomials that have special factoring
1) The Difference of Two Squares 2 2a b a b a b
2) The Sum of Two Squares 2 2 Does not factora b
3) If asked to factor 3 3 2 2a b a b a ab b
The Difference of Two Cubes on an exam this formula will be given.
4) If asked to factor 3 3 2 2a b a b a ab b
The Sum of Two Cubes on an exam this formula will be given.
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7.2 ‐ Multiplying and Dividing Rational Expressions
Learning Objectives: 1. Multiply rational expressions. 2. Divide rational expressions.
• “When in doubt, factor it out.” Factor first before multiplying. • You can ONLY cross cancel (reduce on the diagonal) if you are multiplying two rational expressions. • A monomial can ONLY reduce with a monomial. A binomial can ONLY reduce with an identical binomial.
Ex 1 Multiply as indicated.
a) 30
6 4
x
x
b) 7 5
35
x
x
c) 2 9 18 1
6 3
x x
x x
d) 2
2
49 3
4 21
x x
x x x
5
e) 2 2
2 2
3 17 10 4 3
3 22 16 8 4
x x x x
x x x x
2
8
f) 2 316
4x
x
g) 2
3
6 9 1
27 3
x x
x x
• NOTE: 1a b b a
h) 2 2
8 2 3
9 4
x x
x x x
Class Practice
a) 2
9 21 2
2 3 7
x x
x x x
b)
2
2
2 2 9
3 8 1
y y y
y y y
9
2
6
• Recall: a c a d ad
b d b c bc
• You cannot cross cancel (reduce on the diagonal) when you are dividing two rational expressions. You can ONLY cross cancel (reduce on the diagonal) if you are multiplying two rational expressions.
Ex 2 Divide as indicated.
a) 9 3
4x x
b) 5 4 20
7 9
x x
c) 2 2 12 24 5
x x x6x x x
7
d) 2 24 21 14 48
22 4 323 28
x x x x
x xx x
e) 23 12 12
32 93
y y y
y yy y
Class Practice
a) 4 40
7 426 xx
b)
2 24 5 6
22 8 153 10
x x x
x xx x
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7.3 ‐ Adding and Subtracting Rational Expressions with the Same Denominator
Learning Objectives: 1. Add rational expressions with the same denominator. 2. Subtract rational expressions with the same denominator. 3. Add and subtract rational expressions with opposite denominators. • To add rational expressions with the same denominator, add numerators and place the sum over the common denominator. Simplify the answer if possible. • To subtract rational expressions with the same denominator, subtract numerators and place the difference over the common denominator. Simplify the answer if possible. • When subtracting numerators with a common denominator, make sure to subtract every term in that expression. Ex 1 Add or Subtract.
a) 3 8
1717
x x
b) 9
2424
x x
c) 8 10
66 xx
d) 9 7
44
x xxx
9
e) 3 2 3 6
3 43 4
x xxx
f) 2
2 72 7
xxx
g) 2 1 8
3 73 7
y yyy
h) 2 23 12
22 1212
y y y
y yy y
Class Practice
a) 4 2 25
9 9
x x b)
2 22
22 33
y y y y
y yy y
10
• RECALL: a b b a • RECALL: a b b a
• RECALL: a ab b
ab
Ex 2 Add or Subtract.
a) 6 2
55 xx
b) 2
933
xxx
c) 10 6
22 xx
11
d) 11 5
77 xx
e) 2 2x y x y
y xx y
Class Practice
12
a) 6 5 4
22
x b)
9 1 6 23 77 3
x xxx
x
x
x
7.4 ‐ Adding & Subtracting Rational Expressions with Different Denominators
Learning Objectives: 1. Find the least common denominator. 2. Add and subtract rational expressions with different denominators. Ex 1 Find the least common denominator for the rational numbers or rational expressions. Factor
the denominators first, then build the least common denominator from those factors.
a) 11 17
35225
andx
x
LCD: ____________________
b) 7 11
52 2415
andxx
LCD: ____________________
c) 2 3
75and
xx LCD: ____________________
d) 3 4
26 36and
x x LCD: ____________________
e)
14 12
2 749
andy yy
LCD: ____________________
f) 7
22 4 55 6
xandx xx x
LCD: ____________________
13
Ex 2 Add or Subtract.
a) 2
5 7
6 8x x
b) 3 4
6x x
c) 3 4
2 3x x
d) 16
x
x
14
Ex 3 Add or Subtract.
a) 2
5 2
3 3x x
b) 2 22 24 7 6
x x
x x x x
c) 2
7 2
5 5 5x x x
5
15
d) 2 2
7 3x
x y y
x
Class Practice
a) 4 3
2x x
b)
6 2
224 2x x
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7.5 ‐ Complex Rational Expressions
Learning Objectives: 1. Simplify complex rational expressions by dividing. 2. Simplify complex rational expressions by multiplying by the LCD.
• Complex rational expressions are called complex fractions. They have numerators and/or denominators containing one or more rational expressions. • One method for simplifying a complex rational expression is to add or subtract to get a single rational expression in the numerator and add or subtract to get a single rational expression in the denominator. Then divide by multiplying by the reciprocal of the term in the denominator.
Ex 1 Simplify by dividing.
a)
1 22 37 5
12 6
b)
1 1
1x y
xy
17
c)
14
14
x
x
d)
8 2
2
10 6
2
xx
x x
Class Practice
a) Simplify by dividing.
3
3
3
3
x
x
x
x
18
• A second method for simplifying a complex rational expression is to multiply each term in the numerator and denominator by the least common denominator (LCD). This will produce an equivalent expression that does not contain fractions in the numerator or denominator.
Ex 2 Simplify by the LCD method.
a)
1 1
3 41 1
3 6
b)
2
32
3
x
x
c)
1 1
x y
x y
19
d)
1
21
12
x
x
• Both methods for simplifying complex rational expressions produce the same answer. See which method you prefer.
Class Practice
a) Simplify by the LCD method.
1 1
5 6
1 3
10 15
b) Simplify by the LCD method.
1 1
1 1
x y
x y
Note: To be ready for 7.6 lecture, please review how to solve linear equations in 2.3 and how to solve quadratic equations in 6.6 in our textbook! VERY important!
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7.6 ‐ Solving Rational Equations
Learning Objectives: 1. Solve rational equations. 2. Solve problems involving formulas with rational expressions. 3. Solve a formula with a rational expression for a variable. Ex 1 What is the LCD in each problem? Remember to FACTOR the denominator first, if possible!
a) 3 1
2 4
5
x x LCD: ___________________
b) 7 2 4
24 3y y
LCD: ___________________
c)
3 1 5
21 1x x x
LCD: ___________________
d) 3 1 5
22 3 4 4x x x x
LCD: ___________________
21
• Steps to solving a Rational Equation: 1. Factor the denominator first if possible! 2. When all denominators are in factored form – list the restricted values. Restricted values are any numbers that would make any denominator
zero and would, therefore, make the rational expression undefined. 3. Then multiply each term on both sides of the equation by the LCD to get rid of the denominators. 4. Last, check solutions for restricted values.
Ex 2 Solve. Clear each equation of fractions first.
a) 82 3x x LCD: __________________
Restricted values: _______ Solution: _______________
b) 3 2 1
3 3x x 4
LCD: _________________
Restricted values: _______ Solution: ______________
22
• Some rational equations can be solved using cross products, BUT this method can only be used when there is only one rational expression on each side of the equation.
Ex 3 Solve. Clear each equation of fractions first.
a) 25 5
a aa a
3
Restricted values: _________ Solution: __________
b) 134 31
aaa
Restricted values: _________ Solution: __________
23
• Remember you can only cross multiply when there is only one rational expression on each side of the equation, otherwise you have to clear out the denominator by multiplying each term by the LCD.
Ex 4 Solve. Clear each equation of fractions first.
a) 3 5 2 2 3
2 3 14 3
x x xx xx x
Restricted values: _________
Solution: __________
b) 510 3
2 2y
y y
Restricted values: _________
Solution: __________ Class Practice
a) Solve: 1 5
22 6x
b) Solve: 4 5
2 3
a a
a a
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7.7 ‐ Applications Using Rational Equations & Proportions
Learning Objectives: 1. Solve problems involving motion. 2. Solve problems involving work. 3. Solve problems involving proportions.
• Time in motion equation: dtr
distance traveledtime traveledrate of travel
Ex 1 You can travel 40 miles on a motorcycle in the same time that it takes to travel 15 miles on a bicycle. If your motorcycle’s rate is 20 miles per hour faster than your bicycle’s, find the average rate for each.
Ex 2 Nina can run 1.5 times as fast as she can walk. If she runs for 6 miles and walks for 8 miles, the total time to complete the trip is 4 hours. How many hours did she walk?
25
26
Ex 3 A boat travels 5 km upstream in the same amount of time that the boat covers 15 km downstream. The current in the stream moves at a speed of 2 km/h. What is the speed of the boat in still water?
Ex 4 In still water, a boat averages 18 mph. It takes the same amount of time to travel 33 miles downstream, with the current, as it takes to travel 21 miles upstream, against current. What is the rate of the water’s current?
• Work problem equation:
fractional part of job done fractional part of job done 1 job completedby one person by the second person
Let a = time it takes person A to do a job working alone b = time it takes person B to do a job working alone x = time it takes A and B to complete the entire job working together
Giving us the following equation for work problems in this section 1x xa b
Ex 5 John working alone can paint a room in 4 hours. His helper, Luke, would need 6 hours to do the job by himself. If they work together, how long will it take to complete the job?
Ex 6 Shannon can clean the house in 4 hours. When she worked with Rory, it took 3 hours. How long would it take Rory to clean the house if he worked alone?
27
28
Ex 7 A pool can be filled by one pipe in 3 hours and by a second pipe in 6 hours. How long will it take using both pipes to fill the pool? Ex 8 A hurricane strikes and a rural area is without food or water. Three crews arrive. One can dispense needed supplies in 10 hours, a second in 15 hours, and a third in 20 hours. How long will it take all three crews working together to dispense food and water?
Class Practice a) pg. 556 #4 b) pg. 556 #10 c) pg. 557 #12
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