reteaching transforming linear functions 6 - … · 2015-06-08 · you have graphed linear...

Name Date Class

© Saxon. All r ights reserved. 131 Saxon Algebra 1

You have graphed linear functions on the coordinate plane. Now you will investigate transformations of the parent function for a linear function, f(x) � x.

Graph f(x) � x � 2 and f(x) � x � 4 on a coordinate plane with the parent function f(x) � x. Then describe the transformations.

Step 1: Graph both functions on a coordinate plane with the parent function.

Step 2: Describe the transformations.

The graph of the line f(x) � x � 2 shifts up 2 units from the graph of the line f(x) � x. The y-intercept shifts from (0, 0) to (0, 2).

The graph of the line f(x) � x � 4 shifts down 4 units from the graph of the line f(x) � x. The y-intercept shifts from (0, 0) to (0, �4).

2

4 6

-4

x

y

O

4

2

-2

-2-4-6

f(x) = x

f(x) = x – 4

f(x) = x + 2

Changing the y-intercept will shift the line up or down. This is called a translation.

PracticeComplete the steps to describe the transformations.

1. Graph f(x) � x � 3 and f(x) � x � 2 on a coordinate plane with the parent function f(x) � x. Then describe the transformations.

The graph of the line f(x) � x � 3 shifts up 3 units from the graph of the line f(x) � x. The y-intercept shifts from (0, 0) to (0, 3).

The graph of the line f(x) � x � 2 shifts down 2 units from the graph of the line f(x) � x. The y-intercept shifts from (0, 0) to (0, �2).

The transformations are translations.

2. Graph f(x) � x � 3 on a coordinate plane with the parent function f(x) � x. Then describe the transformation.

The graph of the line f(x) � x � 3 shiftsdown 3 units from the graph of the linef(x) � x. The y-intercept shifts from(0, 0) to (0, �3). The transformationis a translation.

2

4 6

-4

x

y

O

4

2

-2

-2

f(x) = x

f(x) = x – 2

f(x) = x + 3

-4-6

2

4 6

-4

x

y

O

4

2

-2

-2

f(x) = x

f(x) = x – 3

-4-6

ReteachingTransforming Linear Functions

INV

6


Reteachingcontinued

INV

6

Graph f(x) � 4x and f(x) � �4x and on a coordinate plane with the parent function f(x) � x. Then describe the transformations.

Step 1: Graph both functions on a coordinate plane with the parent function.

Step 2: Describe the transformations.

The graph of the line f(x) � 4x is steeper than the graph of the line f(x) � x. When the slope changes, it causes a stretch or compression of the graph of the parent function.

The graph of the line f(x) � �4x is steeper than the graph of the line f(x) � x and is reflected over the x-axis. The transformation is both a stretch and a reflection.

PracticeComplete the steps to describe the transformation.

3. Graph f(x) � 3x on a coordinate plane with the parent function f(x) � x. Then describe the transformation. The graph of the line f(x) � 3x

is steeper than the graph of the line f(x) � x. When the slope changes,

it causes a stretch of the graph of the parent function.

4. Graph f(x) � �x on a coordinate plane with theparent function f(x) � x. Then describe thetransformation.

The graph of the line f (x) � �xis a reflection of the line f(x) � xover the x-axis.

2

4 6

-4

x

y

O

4

2-2

f(x) = x

f(x) = –4x

f(x) = 4x

-4-6

-2

2

4 6

-4

y

O

4

2-2

f(x) = x

f(x) = 3x

-4-6

-2

2

4 6

-4

y

O

4

2-2

f(x) =-x

f(x) = x

-4-6

-2

Name Date Class


61You have found square roots of perfect squares. Now you will simplify radical expressions using the product of radicals rule.

Product of Radicals Rule

If a and b are non-negative real numbers, then

n �� a n

�� b �

n ��

ab and n ��

ab � n �� a n

�� b .

Simplify using perfect squares: ��

64 .

Step 1: Find factors of 64 that are perfect squares.

��

64 � ��

4 � 16 4 and 16 are perfect squares whose product is 64.

Step 2: Apply the Product of Radicals Rule.

� ��

4 � ��

16 Use n ��

ab � n ��

a n ��

b .

Step 3: Simplify the perfect squares.

� 2 � 4 The square root of 4 is 2. The square root of 16 is 4.

� 8 Multiply.

PracticeComplete the steps to simplify using perfect squares.

1. ��

144 2. ��

108

��

144 � ��

4 � 36 ��

108 � ��

9 � 4 � 3

� ��

4 � ��

36 � ��

9 � ��

4 � ��

3

� 2 � 6 � 3 � 2 ��

3

� 12 � 6 ��

3

Simplify using perfect squares.

3. ��

36 4. ��

196

6 14

5. ��

256 6. ��

324

16 18

7. ��

180 8. ��

128

6 ��

5 8 ��

2

9. ��

192 10. ��

450

8 ��

3 15 ��

2

ReteachingSimplify Radical Expressions


Reteachingcontinued

The rules of radicals and exponents also apply to variable expressions.

Simplify: ��

100 x 2 y 5 .

Step 1: Find factors of 100 x 2 y 5 that are perfect squares.

��

100 x 2 y 5 � ��

10 2 � x 2 � y 4 � y Use Rules of Exponents to write y 5 as y 4 � y 1 .

Step 2: Apply the Product of Radicals Rule.

� ��

10 2 � ��

x 2 � ��

y 4 � ��

y Use n ��

ab � n �� a n ��

b .

Step 3: Simplify the perfect squares.

� 10 � x � y 2 � ��

y

� 10x y 2 ��

y Multiply.

PracticeComplete the steps to simplify each expression.

11. ��

100 x 3 y 4 12. ��

10 x 4 y 5

��

100 x 3 y 4 � ��

10 2 � x 2 � x � y 4 ��

10 x 4 y 5 � ��

10 � x 4 � y 4 � y

� ��

10 2 � ��

x 2 � �� x � �

� y 4 � �

� 10 � �

� x 4 � �

� y 4 � ��

y

� 10 � x � �� x � y 2 � �

� 10 � x 2 � y 2 � �

� y

� 10x y 2 �� x � x 2 y 2 �

� 10y

Simplify each expression.

13. ��

100x 2 14. ��

144 x 2 y 2

10x 12xy

15. ��

36 x 3 16. ��

49 x 3 y 3

6x �� x 7xy �

� xy

17. ��

1000x 2 18. ��

10,000 y 4

10x ��

10 100 y 2

19. ��

75 x 2 20. ��

72 x 3 y 2

5x ��

3 6xy ��

2x

21. Find the length of one side of a square room with an area of 96 square meters.

4 ��

6 m

61

Name Date Class


Now you will display data in a stem-and-leaf plot.

The list below shows the daily low temperatures (°F) for a town in the Northeast. Create a stem-and-leaf plot of the data.40, 56, 50, 60, 62, 63, 49, 48, 49, 40, 36, 59, 39

Step 1: There are two place values in each temperature in the data set, tens and ones. Organize the data by each tens value. Write each place-value group in ascending order. Include any values that repeat.

30’s: 36, 39

40’s: 40, 40, 48, 49, 49

50’s: 50, 56, 59

60’s: 60, 62, 63

Step 2: Use the tens digit of each group as the stem of a row on a stem-and-leaf plot. Write each ones digit as the leaf for the corresponding tens digit.

Step 3: Create a key to show how to read each entry in the plot.

Daily Low Temperatures (°F)

Stem Leaf

3 6, 9

4 0, 0, 8, 9, 9

5 0, 6, 9

6 0, 2, 3 Key: 6|2 � 62�F

PracticeComplete the steps to create a stem-and-leaf plot of the data.

1. The list below shows the daily high temperatures Daily High Temperatures (°F)(°F) for a town in the Northeast. Create a stem-and-leaf plot of the data.

70, 84, 71, 73, 71, 70, 73, 78, 76, 65, 65, 67, 66

Key: 6|5 � 65�FCreate a stem-and-leaf plot of the data.

2. The list below shows the test scores from Mr. Clark’s History class.

75, 81, 96, 63, 78, 94, 88, 82, 87, 99, 94, 68, 70

Key: 8|1 � 81

ReteachingDisplaying Data in Stem-and-Leaf Plots and Histograms 62

Stem Leaf

6 5, 5, 6, 77 0, 0, 1, 1, 3, 3, 6, 88 4

Stem Leaf

6 3, 87 0, 5, 88 1, 2, 7, 89 4, 4, 6, 9


Stem-and-leaf plots are used to find measures of central tendency.

Use the stem-and-leaf plot to find the median, mode, range of the data, and the relative frequency of $8.00. Cash Sales for Last 15 Transactions

Stem Leaf

6 2, 4

7 2, 7, 8

8 0, 0, 0, 1, 3, 6, 7

9 2, 5

10 2 Key: 9|2 � $9.20

Step 1: Find the median.

Find the middle value(s). For this data there

is one middle value. The 8th value is $8.00.

The median of the data is $8.00.

Step 2: Find the mode.

The mode is the value or values that occurs the most frequently. There is one value that occurs 3 times: $8.00.

The mode of the data is $8.00.

Step 3: Find the range.

The range is the difference between the greatest and the least value in the set. The greatest data value is $10.20; the least data value is $6.20. The difference is $4.00.

The range of the data is $4.00.

Step 4: Find the relative frequency of $8.00.

The relative frequency is the number of times the data value occurs divided by the total number of data values.

Number of times $8.00 occurs _________________________ Total number of data values

� 3 ___ 15

� 0.20 � 20%

PracticeComplete the steps to find the median, mode, and range of the data using the stem-and-leaf plot. Find the relative frequency of 37.

3. 22 � 25

___________ 2 � 47 ___

2 � 23.5.

The median is 23.5.

The data value 37 occurs most frequently.

The mode is 37 .

Find the difference between the greatest and least data value.

37 � 1 � 36

The range is 36 .

3 _____

12 � 1 __

4 � 0.25 � 25%

The relative frequency of 37 is 25%.

Reteachingcontinued

Stem Leaf

0 1, 1, 7

1 4, 6

2 2, 5, 5

3 6, 7, 7, 7Key: 3|6 � 36

62

Name Date Class


Now you will solve systems of linear equations by elimination.

Solve the system of equations by elimination and check the solution.

3x � y � 6

5x � y � 10

Add the equations to eliminate y.

Step 1: Add the equations vertically.

3x � y � 6

_ 5x � y � 10

8x � 16 Combine like terms.

x � 2 Divide by 8.

Step 2: Substitute 2 for x in one of the original equations.

3x � y � 6 First Equation.

3(2) � y � 6 Substitute 2 for x.

6 � y � 6 Multiply.

y � 0 Subtract 6 from both sides.

The solution to the system is (2, 0).

Step 3: Check the solution.

Substitute the ordered pair into one of the original equations. If the solution is true for the first equation, then check the second equation.

5x � y � 10

5(2) � (0) � 10 Substitute (2, 0) for x and y.

10 � 10 ✓ Simplify.

PracticeSolve the system of equations by elimination. Check the solution.

1. 3x � 2y � 10 3x � 2y � 10 3x � 2y � 10 The solution is (4, �1).

3x � 2y � 14 __ 3x � 2y � 14 3( 4 ) � 2y � 10 Check: 3x � 2y � 14

6x � 24

x � 4 12 � 2y � 10 3( 4 ) � 2(�1) � 14

2y � �2 12 � 2 � 14

y � �1 14 � 14

2. x � y � 1 3. x � 4y � 11

x � y � 3 (2, 1)

x � 6y � 11 (11, 0)

4. 6x � 2y � 4 5. 2x � 3y � 6

x � 2y � 3 (1, 1)

�2x � 4y � �10 (�3, 4)

6. 3x � 4y � 2 (2, �1)

7. 5x � y � �6 (�1, 1)

4x � 4y � 12 �x � y � 2

8. A garden is 4 feet longer than 3 times its width. If the perimeter of the garden is 64 feet, what is the length? (Hint: Use 3w � 4 and w to represent the length and width.) 25ft

ReteachingSolving Systems of Linear Equations by Elimination 63


Solve the system of equations by elimination and check the solution.

2x � 5y � 3

�x � 3y � �7

If the second equation is multiplied by 2, the equations will have equal and opposite coefficients for the variable x.

Step 1: Multiply the second equation by 2.

2(�x � 3y � �7) → �2x � 6y � �14

Step 2: Add the equations vertically.

2x � 5y � 3

__ �2x � 6y � �14

11y � �11 Combine like terms.

y � �1 Divide by 11.

The value of y in the solution is �1.

Step 3: Substitute �1 for y in one of the original equations.

2x � 5y � 3 First Equation

2x � 5(�1) � 3 Substitute �1 for y.

2x � 5 � 3 Multiply.

2x � 8 Add 5 to both sides.

x � 4 Divide both sides by 2.

The solution to the system is (4, �1).

Step 4: Check the solution.

Substitute the ordered pair into one of the original equations.

�x � 3y � �7 Second Equation

�(4) � 3(�1) � �7 Substitute (4, �1) for x and y.

�4 � 3 � �7

�7 � �7 ✓ Simplify.

PracticeSolve each system of equations by elimination.

9. 5x � 2y � �10 3(5x � 2y � �10) → 15x � 6y � �30

3x � 6y � 66 __ 3x � 6y � 66 3x � 6y � 66

18x � 36 3( 2 ) � 6y � 66

The solution is (2, 10). x � 2 6 � 6y � 66

6y � 60 y � 10

10. 2x � y � 3 (1, 1)

11. 2x � y � 0 (�2, 4)

�4x � 4y � �8 5x � 3y � 2

12. Amanda has dimes and quarters in her pocket totaling $2.95. She has 16 coins in all. Write and solve a system of equations to find the number of quarters. 9 quarters

Reteachingcontinued 63

Name Date Class


You have learned to recognize and write examples of direct variation. Now you will learn to recognize and write examples of inverse variation.

Direct variation: as the value of one variable increases, the value of the other variable also increases.

Inverse variation: as the value of one variable increases, the value of the other variable decreases.

Inverse variation can be represented by the equation y � k __ x . The product of xy is always a constant, k.

Is xy � 20 an example of inverse variation?

Solve the equation for y.

xy

__ x � 20 ___ x

y � 20 ___ x

Substitute a few values for x into the equation.

When x � 2, y � 20 ___ 2 � 10.

When x � 4, y � 20 ___ 4 � 5.

This is an inverse variation because as the value of x increases, the value of y decreases.

Write an inverse variation relating x and y when y � 15 and x � 3.

First, write the equation for inverse variation.

y � k __ x

Substitute the values for x and y and solve for k.

15 � k __ 3

k � 45

Use this value for k in the equation for inverse variation.

y � 45 ___ x

PracticeComplete the steps to solve.

1. Is the relationship y __

8 � x an inverse 2. Write an inverse variation relating x and y

variation? Explain. when y � 8 and x � 2.

y __

8 � 8 � 8 � x y � k __ x

y � 8x 8 � k _____ 2

This is not an inverse variation k � 16

because as x increases, y also y � 16 _____ x

increases.

Solve.

3. Is the relationship xy � 4 an inverse 4. Write an inverse variation relating x and y variation? Explain. when y � 0.5 and x � 10.

y � 4 __ x . This is an inverse y � 5 __ x

variation because as x increases y decreases.

ReteachingIdentifying, Writing, and Graphing Inverse Variation 64


Reteachingcontinued

For the inverse variation, xy � k, there can be several values for x and y as long as their product is equal to the constant k. For example, if k � 20, x and y could be any values that make the equation xy � 20 true. This is called the product rule for inverse variation.

You can use this rule to solve for a missing value in an inverse variation.

If y varies inversely as x and y � 3 when x � 32, find x when y � 4.Use the product rule for inverse variation, x 1 y 1 � x 2 y 2 . (32)(3) � x 2 (4) Substitute the values for x and y into the rule.

96 � 4 x 2 Multiply.

24 � x 2 Divide both sides by 4.

When y � 4, x � 24.

PracticeComplete the steps to solve.

5. If y varies inversely as x and y � 6 when 6. If y varies inversely as x and y � 8

x � 18, find x when y � 12. when x � �3, find y when x � �4.

x 1 y 1 � x 2 y 2 x 1 y 1 � x 2 y 2

(18)(6) � x 2 12 (�3)(8) � �4 y 2

108 � 12 x 2 �24 � �4 y 2

x 2 � 9 y 2 � 6 When y � 12, x � 9 . When x � �4, y 2 � 6 .

Solve.

7. If y varies inversely as x and y � 3 when 8. If y varies inversely as x and y � 3

x � 12, find x when y � 6. when x � 45, find y when x � 15.

When y � 6, x � 6. When x � 15, y � 9. 9. If y varies inversely as x and y � �5 10. If y varies inversely as x and y � 0.5 when x � 6, find y when x � �10. when x � 16, find x when y = 4 .

When x � �10, y � 3. When y � 4, x � 2.

11. Jan drives for 3 hours at an average rate of 50 miles per hour. How long would it take her to drive the same distance if she drove at an average rate of 60 miles per hour? Solve the following to find your answer: (50)(3) � (60) y 2 .

2.5 hours

12. One rectangle has a width of 3 cm and a length of 12 cm. A second rectangle has the same area as the first, but has a width of 4 cm. What is the length of the second rectangle? Solve the following to find your answer: (3)(12) � (4) y 2 .

9 cm

64

Name Date Class


You have learned to write the equation of a line using the slope and y-intercept. Now you will learn to write the equations of parallel and perpendicular lines.

Write an equation in slope-intercept form for the line that passes through the point (�3, 2) and is parallel to y � x � 1.

Use the point-slope form for the equation of a line.

(y� y 1 ) � m(x � x 1 )

Substitute the x- and y-values of the point given for x 1 and y 1 , and the slope of the line for m. Use the slope of the line from the original equation to create an equation of a line parallel to it. x 1 � �3, y 1 � 2, and m � 1

� y � 2 � � 1 � x � 3 �

y � 2 � x � 3 Distributive Property

y � x � 5 Add 2 to both sides.

PracticeComplete the steps to find the equation of the given line.

1. Write an equation in slope-intercept form for the line that passes

through the point (�2, 2) and is parallel to y � 1 __ 5 x � 3 __

5 .

� y � y 1 � � m � x � x 1 � � y � 2 � � 1 __

5 � x � 2 �

y � 2 � 1 __ 5 x � 2 __

5

y � 1 __ 5 x � 2 2 __

5

Find the equation of the given line.

2. Write an equation in slope-intercept form for the line that passes through the point (1, �2) and is parallel to y � 2x � 3.

y � 2x � 4


through the point (1, 1) and is parallel to y � 1 __ 2 x � 2.

y � 1 __ 2 x � 1 __

2

65ReteachingWriting Equations of Parallel and Perpendicular Lines


Two lines are perpendicular if the slope of one line is the negative reciprocal of the slope of the other. For example, if the slope of a line is

2 __ 3

, the slope of a line perpendicular to it is � 3 __ 2 .

Write an equation in slope-intercept form for the line that passes through the point (�1, �2) and is perpendicular to y � 4x � 2.

Use the point-slope form for the equation of a line.

(y � y 1 ) � m(x � x 1 )

Substitute the x- and y-values of the point given for x 1 and y 1 . This time, use the negative reciprocal of the slope given to replace m in the point-slope form. The negative reciprocal

of 4 is � 1 __ 4 .

x 1 � �1, y 1 � �2, and m � � 1 __ 4

(y � 2) � � 1 __ 4 (x � 1)

y � 2 � � 1 __ 4 x � 1 __

4 Distributive Property

y � � 1 __ 4 x � 2 1 __

4 Subtract 2 from both sides.

PracticeComplete the steps to find the equation of the given line.

4. Write an equation in slope-intercept form for the line that passes through the point (4, 2) and is perpendicular to y � 3x � 3.

(y � y 1 ) � m(x � x 1 )

(y � 2) � � 1 __ 3 (x � 4)

y � 2 � � 1 __ 3 x � 4 __

3

y � � 1 __ 3 x � 3 1 __

3

Find the equation of the given line.


through the point (0, 1) and is perpendicular to y � � 1 __ 2

x � 1. y � 2x � 1


through the point (8, 5) and is perpendicular to y � 2x � 5. y � � 1 __ 2 x � 9

65Reteachingcontinued

Name Date Class


You have learned to solve equations by adding and subtracting. Now you will learn to solve inequalities by adding or subtracting.

Solve the inequality x � 8 � �3. Then graph and check the solution.

x � 8 � �3

_ �8 _ �8 Add 8 to each side.

x � 5 Simplify.

Now graph the solution on a number line.

2 31-1 0-2-3-4-5-6 4 5 6

The solution includes all values less than 5.

Check to see if the inequality symbol is pointing in the correct direction.

Choose a number less than 5 and substitute it for x in the inequality.

4 � 8 � �3

�4 � �3

This inequality is true, so the direction of the inequality symbol is correct.

Practice

1. Solve the inequality x � 1 � �1. Then graph the solution.

x � 1 � �1

� 1 � 1 Add 1 to each side.

x � 0 Simplify.

2 31-1 0-2-3-4 4

2. Solve the inequality x � 9 � � 11. Then graph the solution.

2 31-1 0-2-3-4-5-6 4

x � �2

3. Solve the inequality x � 4 � 0. Then graph the solution.

2 31-1 0-2 4 5 6

x � 4

ReteachingSolving Inequalities by Adding or Subtracting 66



You can also subtract the same number from both sides of an inequality to solve it.

Solve the inequality x � 12 � 15. Then graph the solution.

x � 12 � 15

_ �12 _ �12 Subtract 12 from each side.

x � 3 Simplify.

Now graph the solution on a number line.

2 31-1 0-2 4 5 6

The solution includes all values greater than or equal to 3.

Check to see if the inequality symbol is pointing in the correct direction.

Choose a number greater than 3 and substitute it for x in the inequality.

4 � 12 � 15

16 � 15

The inequality is true, so the direction of the inequality symbol is correct.

Practice

4. Solve the inequality x � 4 � �2. Then graph the solution.

x � 4 � �2

�4 � �4 Subtract 4 from each side.

x � �6 Simplify.

0 1-1-3 -2-4-5-6-7-8 2

5. Solve the inequality x � 16 � 21. Then graph and check the solution.

Sample: x � 5;

4 � 16 � 21, 20 � 21 2 31-1 0-2 4 5 6 7 8

6. Aidan wants to buy a skateboard that will cost at least $75. He has already saved $47. Write and solve an inequality to find out how much money Aidan still has to save.

x � 47 � 75; x � 28. Aidan needs to save at least $28.

Name Date Class


ReteachingSolving and Classifying Special Systems of Linear Equations

You have learned to solve systems of linear equations. Now you will learn to recognize and solve special systems of linear equations.

Solve the system of equations.

3x � 2y � �2 3 __ 2 x � y � �1

3x � 2y ��2 y � 3 __ 2

x � 1 Write the equations in slope-intercept form.

3 __ 2 x � y � �1

y � 3 __ 2

x � 1

The equations are identical. They will produce the same line, with an infinite number of solutions. A system of equations that has an infinite number of solutions is called consistent and dependent.

Solve the system of equations.�3x � y � 5 �3x � y � �7

y � 3x � 5 y � 3x � 7 Write the equations in slope-intercept form.

Both equations have the same slope, 3, but they have different y -intercepts. This means these equations form parallel lines. Since parallel lines never intersect, the system has no solution. A system of equations that produces parallel lines with no solution is called inconsistent.

PracticeSolve each system of equations. Classify each system as either consistent and dependent or inconsistent.

1. 6x � 2y � 9 3x � y � 12 2. 2x � 3y � 5 6x � 9y � 15

Write the equations in slope-intercept form.

6x � 2y � 9 → y � 3x � 4 1 __ 2

3x � y � 12 → y � 3x � 12

There is no solution(s).

The system is inconsistent.

Write the equations in slope-intercept form.

2x � 3y � 5 → y � � 2 __ 3 x � 5 __

3

6x � 9y � 15 → y � � 2 __ 3 x � 5 __

3

There is an infinite number of solution(s).

The system is consistent and

dependent.

3. 2x � y � �6 y � �6 � 2x 4. 5x � y � �4 10x � 2y � 4

y � �2x � 6 y � �2x � 6 infinite number of solutions; consistent and dependent

y � �5x � 4 y � �5x � 2 no solutions; inconsistent

67


Reteachingcontinued

Solve the system of equations.2x � y � �5x � 2y � 0

2x � y � �5 → y � 2x � 5 Step 1: Write the equations in slope-intercept form.

x � 2y � 0 → y � � 1 __ 2 x The slopes are not the same.

2x � � � 1 __ 2 x � � �5 Step 2: Substitute y � � 1 __

2 x into the first equation.

2 1 __ 2

x � �5 Remember that the expression for x or y obtained from an equation must never be substituted back into the same equation.x � �2

�2 � 2y � 0 Step 3: Substitute �2 for x in the second equation.

2y � 2

y � 1

The solution is (–2, 1). The lines of these two equations intersect at only one point (–2, 1). A system of equations that has exactly one solution is called consistent and independent.

PracticeSolve each system of equations. Classify each system as consistent and dependent, consistent and independent, or inconsistent. Showyour work.

5. y � x � �3 6. y � 2x � �2

y � x � 1 y � 6.

(�2, �1); consistent and independent (4, 6); consistent and independent

y � x � �3 → y � �x � 3

y � x � 1 → y � x � 1

x � 1 � x � �3

2x � 1 � �3

x � �2 y � (�2) � 1

7. Karen is 6 years older than her brother Ben. In 2 years, Karen will be twice as old as Ben. How old are Karen and Ben? Use this system of equations to solve the problem.

x � 2 � 2(y � 2) x � y � 6 Karen is 10 and Ben is 4.

67

Name Date Class


When two events cannot happen at the same time, they are called mutually exclusive.

For example, when rolling a number cube, the events rolling a 6 and rolling an odd number are mutually exclusive because 6 is not an odd number.

To find the probability of one or the other events occurring, add the probabilities of each.

Probability of Mutually Exclusive Events

If A and B are mutually exclusive events, then

P(A or B) � P(A) + P(B).

Marilyn spins the spinner shown. What is the probability that it lands on a vowel or on an odd number?

The spinner is divided into 6 sections. One section is a vowel, “A”, and two sections are odd numbers, “1” and “3”. Spinning a vowel and spinning an odd number are mutually exclusive.

P(A or B) � P(A) + P(B)

P(vowel or odd) = P(vowel) + P(odd)

� 1 __ 6 � 2 __

6

� 3 __ 6 � 1 __

2

The probability of spinning a vowel or an odd number is 1 __ 2 .

PracticeComplete the steps to find the probability of the events using the spinner above.

1. landing on “B” or on a number

P(B or number) � P(B) � P(number)

� 1 ___ 6

� 3 ___ 6

� 4 ___ 6

� 2 ___ 3

2. landing on a number less than 3 or on “C”

P(less than 3 or C) � P(less than 3) � P(C)

� 2 ___ 6 �

1 ___ 6

� 3 ___ 6

� 1 ___ 2

A marble is randomly chosen from a bag containing 2 blue, 3 red, 2 yellow, and 5 green marbles. Find the probability of the events.

3. pick a red marble or a green marble

2 __ 3

4. pick a yellow, blue, or green marble

3 __ 4

ReteachingMutually Exclusive and Inclusive Events 68

B

A

3

2

1

C



When two events can occur at the same time, they are called inclusive, or overlapping. If you roll a number cube, rolling an odd number and rolling a prime number are inclusive events because 3 and 5 are both odd and prime. To find the probability of inclusive events, find the sum of the probabilities of each, and subtract the probability when both events occur.

Probability of Inclusive Events

If A and B are inclusive events, then

P(A or B) � P(A) � P(B) � P(A and B).

Find the probability of rolling an odd number or a prime number on a number cube.

outcomes that are odd: 1, 3, 5 P(odd) � 3 __ 6

� 1 __ 2

outcomes that are prime: 2, 3, 5 P(prime) � 3 __ 6 � 1 __

2

outcomes that are odd and prime: 3, 5 P(odd and prime) � 2 __ 6 � 1 __

3

P(odd or prime) � P(odd) � P(prime) � P(odd and prime)

� 1 __ 2 � 1 __

2 � 1 __

3 � 2 __

3

The probability that the number rolled is odd or prime is 2 __ 3 .

PracticeComplete the steps to find the probability of the events when rolling a number cube labeled 1– 6. 5. a number less than 4 or an even number

less than 4: 1, 2 , 3 even: 2, 4 , 6 less than 4 and even: 2

P(less than 4) � 1 __ 2 P(even) � 1 __

2 P(less than 4 and even) � 1 __

6

P(less than 4 or even) � P (less than 4) � P (even) � P (less than 4 and even)

� 1 __ 2

� 1 __ 2 � 1 __

6 � 5 __

6

A letter is chosen from the word “MATHEMATICS”. Find the probability of the events. 6. choose a vowel or one of the last 5 letters

7 ___ 11

7. choose an M or one of the first 5 letters

6 ___ 11

8. A deck of 20 cards has 5 red cards, 5 blue cards, 5 green cards and 5 yellow cards. The cards in each color group are numbered 1 through 5. If a card is chosen at random, what is the probability that it is red or a five?

2 __ 5

Name Date Class


You have added and subtracted polynomials. Now you will add and subtract radical expressions.

You can add and subtract radical expressions just like you add and subtract expressions with variables.

4 _ x � 2 _ x � 6x

These are like terms. Add.

2 _ x � 4 _ y

These are unlike terms. Do not add.

4 _ ��

7 � 2 _ ��

7 � 6 ��

7

These are like radicals. Add.

2 _ ��

5 � 4 _ ��

3

These are unlike radicals. Do not add.

Add 8 ��

10 � 5 ��

10 .

These are like radicals.

8 ��

10 � 5 ��

10 � (8 � 5) ��

10 Combine

� 13 ��

10 coefficients.

Subtract 10 ��

7x � 12 ��

7x .

These are like radicals.

10 ��

7x � 12 ��

7x � (10 � 12) ��

7x . Combine

� �2 ��

7x coefficients.

Add 4 ��

2ac � 8 ��

3ac .

These are unlike radicals.

Do not add.

Subtract 9 �� 5 � 4 �

� 6 .

These are unlike radicals.

Do not subtract.

PracticeComplete the steps to add or subtract. All variables represent non-negative real numbers.

1. Add 3 ��

2y � 8 ��

2y . These are like radicals.

3 ��

2y � 8 ��

2y � ( 3 � 8 ) ��

2y Combine coefficients.

� 11 ��

2y

2. Subtract 5 ��

2 � 2 ��

5 . These are unlike radicals. Do not subtract.

Add or subtract. All variables represent non-negative real numbers.

3. 4 ��

13 � 2 ��

13 6 ��

13 4. 12 ��

3m � 2 ��

3m 10 ��

3m

5. 8 � ��

8 8 � ��

8 6. 5 ��

11 � 6 ��

11 � ��

11

7. Find the perimeter of a right triangle if the lengths of the two legs are 3 �

� 2 centimeters and 4 �

� 2 centimeters and the hypotenuse is

5 ��

2 centimeters. 12 ��

2 cm

ReteachingAdding and Subtracting Radical Expressions 69


Sometimes it is necessary to simplify radical expressions before adding or subtracting.

Simplify �

� 50 � �

� 18 .

��

50 � ��

18

� ��

25 � 2 � ��

9 � 2 Factor the radicands using perfect squares.

� ��

25 � ��

2 � ��

9 � ��

2 Product of Radicals Rule

� 5 ��

2 � 3 ��

2 Simplify.

� (5 � 3) ��

2 Factor out ��

2 .

� 8 ��

2 Combine like radicals.

Simplifya �

� 80 � �

� 45 a 2. .

a ��

80 � 2 ��

45 a 2

� a ��

16 � 5 � ��

9 � 5 � a 2 Factor the radicands using perfect squares.

� a ��

16 � ��

5 � ��

9 � ��

5 � ��

a 2 Product of Radicals Rule

� 4a ��

5 � 3a ��

5 Simplify.

� (4a � 3a) ��

5 Factor out ��

5 .

� a ��

5 Simplify.

PracticeSimplify. All variables represent non-negative real numbers.

8. ��

12 � ��

300

� ��

4 � 3 � ��

100 � 3 Factor the radicands using perfect squares.

� ��

4 � ��

3 � ��

100 � ��

3 Product of Radicals Rule

� 2 ��

3 � 10 �� 3 Simplify.

� ( 2 � 10 ) �� 3 Factor out �

� 3 .

� 12 ��

3 Simplify.

9. ��

50 p 2 � p ��

8

3p ��

2

10. ��

63 h 2 � h ��

7 2h ��

7 11. ��

54 g 2 � g ��

24 + ��

36 g g ��

6 � 6g

12. A rectangular mirror is ��

48 z 2 inches wide and ��

27 z 2 inches tall. What is its perimeter?

14z ��

3 in.


Name Date Class


You have solved inequalities by adding or subtracting. Now you will solve inequalities by multiplying or dividing.

To solve an inequality involving multiplication or division, you can multiply or divide each side by the same number to isolate the variable. When the number you multiply or divide by is positive, the direction of the inequality symbol does not change.

Solve and graph the inequality.

1 __ 4 x � 3

(4) 1 __ 4 x � 3(4) Multiply both sides by 4.

x � 12 Simplify.

To graph the solution, draw a number line. Use a closed dot at 12 and shade to the right.

11 12 13 14 15 1610


5x � 10

5x ___ 5 � 10 ___

5 Divide both sides by 5.

x � 2 Simplify.

To graph the solution, draw a number line. Use an open dot at 2 and shade to the left.

1 2 3 4 50

PracticeComplete the steps to solve and graph the inequality.

1. 2 � x __ 3 2. 4x � �20

2( 3 ) � x __ 3 ( 3 ) 4x ____

4

� �20 ____ 4

6 � x x � �5

6 7 8 9 10 115 -9 -8 -7 -6 -5 -4-10

Solve and graph each inequality.

3. 1 __ 2 x � 7 4. �5 � x __

8

x � 14 �40 � x

11 12 13 14 15 1610 -44-43-42-41-40-39-45

5. 1 ___ 10

x � �6 6. 7x � 21

x � �60 x � 3

-64-63-62-61-60-59-65 1 2 3 4 5 60

ReteachingSolving Inequalities by Multiplying or Dividing 70


When you solve an inequality, if the number you multiply or divide by is negative, you need to switch the direction of the inequality symbol.


� x ___ 12

� 4

(�12) �x ___ 12

� 4(�12) Multiply by �12 and switch the symbol direction.

x � �48 Simplify.

To graph the solution, draw a number line. Use an open dot at �48 and shade to the left.

-49-48-47-46-45-44-50


18 � �6x

18 ___ �6

� �6x ____ �6

Divide by �6 and switch the symbol direction.

�3 � x Simplify.

To graph the solution, draw a number line. Use a closed dot at –3 and shade to the right.

-4 -3 -2 -1 - 0 1-5

PracticeComplete the steps to solve and graph the inequality.

7. � x __ 9

� 2 8. �5x � �25

( �9 )� x __ 9

� 2( �9 ) �5x _____ �5

� �25 _____ �5

x � �18 x � 5

-19-18-17-16-15-14-20 2 3 4 5 61 7

Solve and graph each inequality.

9. � 1 __ 5

x � 2 10. � 1 __ 7 x � �5

x � �10 x � 35

-14 -13 -12 -11 -10 -9-15 31 32 33 34 35 3630

11. �8x � �24 12. �3x � �12

x � 3 x � 4

1 2 3 4 50 6 1 2 3 4 50 6

13. Marvin earns $15 for each lawn he mows. He is trying to earn at least $120 for a trip. Write and solve an inequality to find the number of lawns he needs to mow to earn enough money. Let x equal the number of lawns.

15x � 120; at least 8 lawns


reteaching transforming linear functions 6 - … · 2015-06-08 · you have graphed linear...

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