MODULE
9
MODULE
STUDY GUIDE REVIEW
Study Guide Review
Essential Question: How can you use rational expressions
and equations to solve real-world problems?
ASSESSMENT AND INTERVENTION
KEY EXAMPLE
(Lesson 9.1)
3 - x , simplify the result, and
1 and _____
Add _____
x
3+x
note the excluded values.
(3 - x)(3 + x)
3 - x = ________
1 +_
1x
_
+ __
x
3+x
(3 + x)x
x(3 + x)
KEY EXAMPLE
3
© Houghton Mifflin Harcourt Publishing Company
2
Factor the numerators and denominators.
2(x + 3)(x - 2)
= __
(x + 2)(x + 3)(x - 3)
SUPPORTING STUDENT REASONING
1
(Lesson 9.2)
2(x - 2)
x+3
= _ ∙ __
x + 2 (x + 3)(x - 3)
Mathematical Practices: MP.1, MP.2, MP.3, MP.4, MP.6, MP.7
A-APR.D.7
T
Simplify.
x+3
x 2 - 9 and note any excluded values.
Find the quotient _____ ÷ ______
x+2
2x - 4
x+3 _
x+3
x2 - 9 = _
_____
Multiply by the reciprocal.
∙ 2x - 4
÷_
x+2
2x - 4
x + 2 x2 - 9
COMMON
CORE
• What is the relationship for total resistance for
resistors in parallel?
1
1
1
1
1
__
= __
+ __
+ __
+ ... + __
, for n resistors.
R
R
R
R
Rn
closure (cerradura)
extraneous solution
(solución extraña)
rational expression
(expresión racional)
reciprocal (recíproco)
Add.
-x 2 + x + 9
= ___________ , x ≠ -3, 0
x(x + 3)
MODULE
PERFORMANCE TASK
Key Vocabulary
Write with like denominators.
x + (9 - x 2)
= ___________
x(x + 3)
Assign or customize module reviews.
Students should begin by focusing on how to set up
the problem. They can then do research, or you can
provide them with specific information. Here is some
of the information they may ask for.
9
Rational Expressions
and Equations
Multiply and cancel the common factors.
2(x - 2)
= __ ; x ≠ ±2, ±3 Simplify.
(x + 2)(x - 3)
KEY EXAMPLE
(Lesson 9.3)
Solve the rational equation algebraically.
6x
x + __
x =_
_
2
x-3
2x - 6
Multiply each term by the LCD and divide out
common factors.
x = 2(x - 3)_
6x
x + 2(x - 3)_
2(x - 3)_____
x-3
2
2x - 6
2x + x(x - 3) = 6x
Simplify.
x 2 - 7x = 0
Write in standard form.
x(x - 7) = 0
Factor.
x = 0 or x = 7
Module 9
Solve for x.
467
Study Guide Review
SCAFFOLDING SUPPORT
A2_MNLESE385894_U4M09MC 467
• If students are having difficulty adding long series of fractions, you may wish
to suggest that they try using decimals instead.
467
Module 9
7/7/14 9:55 AM
EXERCISES
Add or subtract the given expressions, simplify the result, and note the excluded
values. (Lesson 9.1)
1.
6x + 6
-3x + 3
_
+ ________
x2 - 9
x2 - 9
2.
3
_
SAMPLE SOLUTION
The available resistance values are 20, 50, 80, and
200 ohms. Use the equation for the total resistance
1
1
1
1
1
of parallel resistors, __
= __
+ __
+ __
+ ... + __
, to
RT
R1
R2
R3
Rn
find a combination of resistors that results in
R T = 10 ohms. One way to do this is to find the
reciprocal of each resistor value, and use these to
find the combinations which, when added, result in
the reciprocal of the total resistance:
x+2
4 -_
______
2
x -1
x -3
x-1
-x - 3x + 2
__
2
(x - 1)(x +1)
Excluded values: x ≠ ±3
Excluded values: x ≠ ±1
Multiply or divide the given expressions, simplify the result, and note the excluded
values. (Lesson 9.2)
3.
2
4
x___________
- 4x - 5 ∙ __
3x - 15
x 2 - 2x - 3
4.
4
_
3x - 9
Excluded values: x ≠ -1, 3, 5
x+2
x
_____
÷_
x-4
3x - 12
3(x + 2)
_
x
1
__1 = ___1 = 0.1, ______
= 0.05,
10
R
20 ohm
1
1
______
= 0.02, ______
= 0.0125, and
50 ohm
80 ohm
1
_______
= 0.005.
Excluded values: x ≠ 0, 4
T
Solve each rational equation algebraically. (Lesson 9.3)
5.
10 = 3
x - ___
x
6.
5 = _____
2
_
x+4
x +1
200 ohm
One possible configuration is one 20-ohm, one
50-ohm, two 80-ohm, and one 200-ohm resistor:
x = -6
x = -2 or x = 5
__1 = 0.1 = 0.05 + 0.02 + 2(0.0125) + 0.005
RT
MODULE PERFORMANCE TASK
Robots and Resistors
Another possible configuration is four 80-ohm and
one 20-ohm resistor:
An engineer is designing part of a circuit that will control a robot. The circuit must have
a certain total resistance to function properly. The engineer plans to use several resistors in
parallel, which means each resistor is on its own branch of the circuit. The resistors available
for this project are 20, 50, 80, and 200-ohm.
__1 = 0.1 = 4(0.0125) + 0.05
How can the engineer design a parallel circuit with a total resistance of 10 ohms using
a maximum of 5 resistors, at least two of which must be different values? Find at least two
possible circuit configurations that meet these criteria.
A third possible configuration is one 20-ohm, two
50-ohm, and two 200-ohm resistors:
RT
© Houghton Mifflin Harcourt Publishing Company
For another part of the circuit, the engineer wants to use resistors in parallel to create a total
resistance of 6 ohms. Can she do it using the available resistor values? If so, how? If not, explain
why not.
Begin by listing in the space below all of the information you will need to solve the problem.
Then use your own paper to complete the task. Be sure to write down all your data and
assumptions. Then use graphs, numbers, words, or algebra to explain how you reached your
conclusion.
Module 9
468
__1 = 0.1 = 0.05 + 2(0.02) + 2(0.005)
RT
However, it is not possible to use these resistors in
parallel to create 6 ohms of total resistance.
One way to see this is to examine the reciprocal of
_
1
6: __
= 0.16, a repeating decimal. No sum of
6
non-repeating decimals will create a repeating
decimal.
Study Guide Review
DISCUSSION OPPORTUNITIES
A2_MNLESE385894_U4M09MC.indd 468
29/03/14 7:48 PM
• Lead students in discussing the meaning of electrical resistance. What do they
think happens when total resistance increases?
• Will adding more resistors in parallel ever cause total resistance to increase?
Assessment Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain.
0 points: Student does not demonstrate understanding of the problem.
Study Guide Review 468
Ready to Go On?
Ready to Go On?
ASSESS MASTERY
9.1–9.3 Rational Expressions
and Equations
Use the assessment on this page to determine if
students have mastered the concepts and standards
covered in this module.
• Online Homework
• Hints and Help
• Extra Practice
Perform the indicated operations, simplify the result, and note any excluded values.
(Lessons 9.1, 9.2)
2x
4 +_
_
x+5
x 2 - 25
1.
ASSESSMENT AND INTERVENTION
2.
2x - 3
_
;x≠ 2
2(3x - 10)
_____________
; x ≠ ±5
(x + 5)(x - 5)
x-2
x+3 _
_____
∙ 2x2 - 4
3.
x+2
4.
x -9
Solve each rational equation. (Lesson 9.3)
3
3 +_
x
= ______
5. _____
x+2
2x + 4
2x + 4
x+2
Differentiated Instruction Resources
• Reading Strategies
• Success for English Learners
• Challenge Worksheets
Assessment Resources
x
24 - 2x
_
=_
x-8
x-8
8.
x -4
3x
7
6 =_
_
+_
x
x+1
2x
2, 2
x = -__
3
x=0
ESSENTIAL QUESTION
© Houghton Mifflin Harcourt Publishing Company
• Reteach Worksheets
x 2 - 16
no solution
8x - _____
8
4 = ______
______
2
2
ADDITIONAL RESOURCES
Response to Intervention Resources
6.
x=9
x -4
x-4
(x - 2)
(x + 2)(x - 3)
7.
x-3
x-2
_
÷_
(x - 3)(x + 4)
__
; x ≠ 2, ±4
2(x - 2)
__
; x ≠ -2, ±3
Access Ready to Go On? assessment online, and
receive instant scoring, feedback, and customized
intervention or enrichment.
3x + 2
x+5
_
-_
x-2
x-2
How do you add or subtract rational expressions and identify any excluded values?
9.
Possible Answer: To add or subtract rational expressions, find and
apply the LCD and then simplify. For each situation, the excluded
values are the values that make the denominator 0 in the original
form or in the simplified form.
• Leveled Module Quizzes
Module 9
COMMON
CORE
A2_MNLESE385894_U4M09MC 469
469
Module 9
Study Guide Review
469
Common Core Standards
3/29/14 12:57 AM
Content Standards Mathematical Practices
Lesson
Items
9.1
1
A-APR.D.7, A-SSE.A.2
MP.2
9.1
2
A-APR.D.7, A-SSE.A.2
MP.2
9.2
3
A-APR.D.7
MP.2
9.2
4
A-APR.D.7
MP.2
9.3
5–8
A-REI.A.2
MP.1
MODULE
MODULE 9
MIXED REVIEW
MIXED REVIEW
Assessment Readiness
Assessment Readiness
1. Look at each expression. Is it equivalent to x - 3? Select Yes or No for A–C.
(x - 3) (x + 5) x + 3
A. __ + _
x+5
x+3
x+3 _
x-3
_
+
B.
x+5
x+5
(x + 3) (x + 5) x + 5
C. __ ÷ _
x-5
x-5
9
Yes
No
Yes
No
Yes
No
ASSESSMENT AND INTERVENTION
2. Consider finding how many roots a quadratic equation has. Choose True or False for
each statement.
A. The quadratic equation x 2 - 12 = 0
has real roots.
True
False
B. The quadratic equation x 2 + 25 = 0
has imaginary roots.
True
False
C. The quadratic equation -8x 2 + 20 = 0
has one real root, and one imaginary root.
True
False
Assign ready-made or customized practice tests to
prepare students for high-stakes tests.
ADDITIONAL RESOURCES
3. A hiker averages 0.6 mile per hour walking up a mountain trail and 1.3 miles per hour
walking down the trail. Find the total time in terms of d. Explain your answer.
Assessment Resources
Total time is approximately 2.44d. The time to go up or down is found using
_d
• Leveled Module Quizzes: Modified, B
d
d
t = r . The total time is given by adding the time up to the time down: ___
+ ___
0.6
1.3
1.9d
= ___
= 2.44d
0.78
12 hours; (rate of large oven · hours worked) + (rate of small oven · hours
_
_
1
1
worked) = 1 complete job: (3) + (3) = 1; 3h + 12 = 4h, h = 12
4
Module 9
COMMON
CORE
h
AVOID COMMON ERRORS
Item 3 Some students may have difficulty with this
problem because the substituted value is in the
denominator. Remind students to divide 1 by the
denominator to get the non-fractional coefficient of d.
© Houghton Mifflin Harcourt Publishing Company
4. A restaurant has two pastry ovens. When both ovens are used, it takes about 3 hours to
bake the bread needed for one day. When only the large oven is used, it takes about 4
hours to bake the bread for one day. About how long would it take to bake the bread
for one day if only the small oven were used? Explain how you got your answer.
Study Guide Review
470
Common Core Standards
A2_MNLESE385894_U4M09MC 470
3/29/14 12:57 AM
Content Standards Mathematical Practices
Lesson
Items
9.1, 9.2
1
A-APR.D.6, A-APR.D.7
MP.1
3.1, 3.2
2*
N-CN.A.1, N.CN.A.2
MP.2
1.2, 9.1
3*
A-CED.A.2, A-APR.D.7,
N-Q.A.1
MP.6
9.3
4
A-REI.A.2, N-Q.A.1
MP.4
* Item integrates mixed review concepts from previous modules or a previous course.
Study Guide Review 470