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This pdf includes the following topics:-

Common Core Math Standards

Solving Absolute Value Equations

Algebraically

Integrate Mathematical Practices

Absolute Value Equations with Fewer

than Two Solutions

Common Core Math Standards

Solving Absolute Value Equations

Algebraically

Integrate Mathematical Practices

Absolute Value Equations with Fewer

than Two Solutions

1.
2.2 Name Class Date

Solving Absolute Value 2.2 Solving Absolute Value

Equations Equations

Essential Question: How can you solve an absolute value equation?

Resource

Common Core Math Standards Locker

The student is expected to:

COMMON

Explore Solving Absolute Value Equations Graphically

CORE A-CED.A.1

Absolute value equations differ from linear equations in that they may have two solutions. This is indicated

Create equations and inequalities in one variable and use them to solve with a disjunction, a mathematical statement created by a connecting two other statements with the word

problems. Also A-REI.B.3, A-REI.D.11 “or.” To see why there can be two solutions, you can solve an absolute value equation using graphs.

Mathematical Practices A Solve the equation 2⎜x − 5⎟ − 4 = 2. y

8

Plot the function ƒ(x) = 2⎜x − 5⎟ − 4 on the grid. Then plot

COMMON

CORE MP.6 Precision 4

the function g(x) = 2 as a horizontal line on the same grid,

and mark the points where the graphs intersect. x

Language Objective 0

The points are (2, 2) and (8, 2). -8 -4 4 8

Explain to a partner why solutions to a variety of absolute value -4

equations make sense and contain more than one solution, one solution, B Write the solution to this equation as a disjunction:

x= 2 or x = 8 -8

or no solution.

Reflect

ENGAGE 1. Why might you expect most absolute value equations to have two solutions?

Why not three or four?

If the absolute value expression is not equal to zero, the expression inside an

Essential Question: How can you solve

absolute value can be either positive or negative. So, there can be at most two

an absolute value equation?

solutions. Looking at this graphically, an absolute value graph can intersect a

© Houghton Mifflin Harcourt Publishing Company

Possible answer: Isolate the absolute value

horizontal line at most two times.

expression, then write two related equations with a

2. Is it possible for an absolute value equation to have no solutions? one solution?

disjunction, also known as an “or” statement.

If so, what would each look like graphically?

Yes; yes; A graph with the horizontal line entirely below an upward-opening

absolute value function, or above a downward-opening absolute value function,

PREVIEW: LESSON will not have points of intersection and the equation will have no solutions. A graph

PERFORMANCE TASK

with the horizontal line passing through the vertex will have exactly 1 solution.

View the Engage section online. Discuss the photo

and why this situation can be represented by a

V-shaped path and an absolute value equation. Then

preview the Lesson Performance Task.

Module 2 ges must

be made throu

gh “File info” 77 Lesson 2

EDIT--Chan

DO NOT Key=NL-B;CA-B

Correction

Date

Class

lue

solute Va

Name

Solving Ab

2.2

Essential

Question:

How can

Equations

you solve

an absolu

ons and inequa

te value

lities in one

equation?

variable and

use them

to solve proble

ms.

phically

Resource

Locker

HARDCOVER PAGES 5762

A-CED.A.1

Create equati ations Gra

.11

Value Equ indicated

COMMON

Absolute

CORE

.3, A-REI.D ns. This is

Also A-REI.B two solutio

Solving in that they

may have ents with

the word

Explore equati ons cting two other statem graphs.

from linear created by a conne te value equation using

A2_MNLESE385894_U1M02L2 77 Absolute

value equati

ons differ ent

matical statem you can solve an

absolu

14/05/14 2:46 PM

Turn to these pages to

ction, a mathe solutions, y

with a disjun there can be two 8

why

“or.” To see

− 4 = 2.

2⎜x − 5⎟ plot 4

equation grid. Then x

Solve the − 4 on the

2⎜x − 5⎟ the same

grid,

on ƒ(x) = ntal line on 0 4 8

Plot the functi = 2 as a horizo ct. -4

find this lesson in the

on g(x) graphs interse -8

the functi where the -4

the points 2).

and mark 2) and (8,

s are (2, on as a disjun

ction: -8

The point this equati

solution to

Write the

2 or x =

8

hardcover student

x=

solutions?

to have two

Reflect

equations an

absolute value ssion inside

expect most the expre

might you l to zero, most two

1. Why three or four? is not equa can be at

Why not expression

edition.

ute value So, there ect a

If the absol or negative. can inters

r positive value graph

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value can absolute

absolute ically, an

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Look

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points of x will

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line

horizontal

© Houghto

with the

Lesson 2

77

5:07 AM

5/15/14

Module 2

L2 77

4_U1M02

SE38589

A2_MNLE

77 Lesson 2.2

Solving Absolute Value 2.2 Solving Absolute Value

Equations Equations

Essential Question: How can you solve an absolute value equation?

Resource

Common Core Math Standards Locker

The student is expected to:

COMMON

Explore Solving Absolute Value Equations Graphically

CORE A-CED.A.1

Absolute value equations differ from linear equations in that they may have two solutions. This is indicated

Create equations and inequalities in one variable and use them to solve with a disjunction, a mathematical statement created by a connecting two other statements with the word

problems. Also A-REI.B.3, A-REI.D.11 “or.” To see why there can be two solutions, you can solve an absolute value equation using graphs.

Mathematical Practices A Solve the equation 2⎜x − 5⎟ − 4 = 2. y

8

Plot the function ƒ(x) = 2⎜x − 5⎟ − 4 on the grid. Then plot

COMMON

CORE MP.6 Precision 4

the function g(x) = 2 as a horizontal line on the same grid,

and mark the points where the graphs intersect. x

Language Objective 0

The points are (2, 2) and (8, 2). -8 -4 4 8

Explain to a partner why solutions to a variety of absolute value -4

equations make sense and contain more than one solution, one solution, B Write the solution to this equation as a disjunction:

x= 2 or x = 8 -8

or no solution.

Reflect

ENGAGE 1. Why might you expect most absolute value equations to have two solutions?

Why not three or four?

If the absolute value expression is not equal to zero, the expression inside an

Essential Question: How can you solve

absolute value can be either positive or negative. So, there can be at most two

an absolute value equation?

solutions. Looking at this graphically, an absolute value graph can intersect a

© Houghton Mifflin Harcourt Publishing Company

Possible answer: Isolate the absolute value

horizontal line at most two times.

expression, then write two related equations with a

2. Is it possible for an absolute value equation to have no solutions? one solution?

disjunction, also known as an “or” statement.

If so, what would each look like graphically?

Yes; yes; A graph with the horizontal line entirely below an upward-opening

absolute value function, or above a downward-opening absolute value function,

PREVIEW: LESSON will not have points of intersection and the equation will have no solutions. A graph

PERFORMANCE TASK

with the horizontal line passing through the vertex will have exactly 1 solution.

View the Engage section online. Discuss the photo

and why this situation can be represented by a

V-shaped path and an absolute value equation. Then

preview the Lesson Performance Task.

Module 2 ges must

be made throu

gh “File info” 77 Lesson 2

EDIT--Chan

DO NOT Key=NL-B;CA-B

Correction

Date

Class

lue

solute Va

Name

Solving Ab

2.2

Essential

Question:

How can

Equations

you solve

an absolu

ons and inequa

te value

lities in one

equation?

variable and

use them

to solve proble

ms.

phically

Resource

Locker

HARDCOVER PAGES 5762

A-CED.A.1

Create equati ations Gra

.11

Value Equ indicated

COMMON

Absolute

CORE

.3, A-REI.D ns. This is

Also A-REI.B two solutio

Solving in that they

may have ents with

the word

Explore equati ons cting two other statem graphs.

from linear created by a conne te value equation using

A2_MNLESE385894_U1M02L2 77 Absolute

value equati

ons differ ent

matical statem you can solve an

absolu

14/05/14 2:46 PM

Turn to these pages to

ction, a mathe solutions, y

with a disjun there can be two 8

why

“or.” To see

− 4 = 2.

2⎜x − 5⎟ plot 4

equation grid. Then x

Solve the − 4 on the

2⎜x − 5⎟ the same

grid,

on ƒ(x) = ntal line on 0 4 8

Plot the functi = 2 as a horizo ct. -4

find this lesson in the

on g(x) graphs interse -8

the functi where the -4

the points 2).

and mark 2) and (8,

s are (2, on as a disjun

ction: -8

The point this equati

solution to

Write the

2 or x =

8

hardcover student

x=

solutions?

to have two

Reflect

equations an

absolute value ssion inside

expect most the expre

might you l to zero, most two

1. Why three or four? is not equa can be at

Why not expression

edition.

ute value So, there ect a

If the absol or negative. can inters

r positive value graph

be eithe

value can absolute

absolute ically, an

ing at this graph

Look

solutions. two times

. n?

line at most ns? one solutio

y

no solutio

g Compan

horizontal on to have

te value equati rd-opening

an absolu ically? an upwa

possible for each look like graph ly below function,

2. Is it line entire

Publishin

would horizontal ute value

If so, what with the ing absol

A graph e a down

ward-open ions. A graph

Yes; yes; no solut

Harcour t

ion, or abov equation

will have

value funct n and the ly 1 solut

ion.

absolute inters ectio have exact

points of x will

n Mifflin

the verte

will not have passing

through

line

horizontal

© Houghto

with the

Lesson 2

77

5:07 AM

5/15/14

Module 2

L2 77

4_U1M02

SE38589

A2_MNLE

77 Lesson 2.2

2.
Explain 1 Solving Absolute Value Equations

Algebraically

To solve absolute value equations algebraically, first isolate the absolute value expression on one side of the

EXPLORE

equation the same way you would isolate a variable. Then use the rule:

Solving Absolute Value Equations

If ⎜x⎟ = a (where a is a positive number), then x = a OR x = –a.

Graphically

Notice the use of a disjunction here in the rule for values of x. You cannot know from the original

equation whether the expression inside the absolute value bars is positive or negative, so you must work

through both possibilities to finish isolating x.

INTEGRATE TECHNOLOGY

]Example 1 Solve each absolute value equation algebraically. Graph the

solutions on a number line. Students have the option of completing the graphing

activity either in the book or online.

⎜3x⎟ + 2 = 8

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

Subtract 2 from both sides. ⎜3x⎟ = 6

Rewrite as two equations. 3x = 6 or 3x = −6 QUESTIONING STRATEGIES

Solve for x. x=2 or x = −2 How do you solve an absolute value equation

3⎜4x - 5⎟ - 2 = 19

graphically? Plot each side as if it were a

separate function of x, and find the x-coordinates

Add 2 to both sides. 3⎜4x - 5⎟ = 21

of the intersection points.

Divide both sides by 3. ⎜4x - 5⎟ = 7

Rewrite as two equations. 4x - 5 = 7 or 4x − 5 = -7

Why do you write the solutions to the absolute

value equation as a disjunction? If two values

Add 5 to all four sides. 4x = 12 or 4x = -2

of the variable both satisfy an equation, then one or

1

Solve for x. x = 3 or x = -_ the other can be correct.

2

-3 -2 -1 0 1 2 3

Your Turn

© Houghton Mifflin Harcourt Publishing Company

EXPLAIN 1

Solve each absolute value equation algebraically. Graph the solutions on a

number line. Solving Absolute Value Equations

3. 1 ⎜x + 2⎟ = 10

_

2

Algebraically

-24 -16 -8 0 8 16 24

⎜x + 2⎟ = 20

x + 2 = 20 or x + 2 = -20

x = 18 or x = -22 AVOID COMMON ERRORS

4. −2⎜3x − 6⎟ + 5 = 1 Some students may not isolate the absolute value

-2⎜3x - 6⎟ = -4 -4 -2 - 23 0 2

3

2 4 expression on one side of the equation as a first step

⎜3x - 6⎟ = 2

when solving the equation. Stress the importance of

3x - 6 = 2 3x - 6 = -2

or

this step so that the equation is in the form ⎜x⎟ = a,

x=_

8

3

or x=_

4

3 which has the solution x = a or x = –a.

Module 2 78 Lesson 2

PROFESSIONAL DEVELOPMENT QUESTIONING STRATEGIES

A2_MNLESE385894_U1M02L2 78 14/05/14 2:46 PM

Integrate Mathematical Practices How do you interpret the solutions to an

absolute value equation like ⎜x⎟ = a on a

This lesson provides an opportunity to address Mathematical Practice MP.6,

number line? Sample answer: The solutions are

which calls for students to “attend to precision” and communicate precisely.

the same distance from 0 on either side of the

Students find the solutions to absolute value equations both by graphing them,

number line.

with and without technology, and through algebra. Students learn that a

disjunction is often used to express the solutions to absolute value equations, and Why is it important to isolate the absolute

they use the properties of algebra to accurately and efficiently find the solutions to value expression when solving an absolute

various types of absolute value equations. value equation? So you can remove the absolute

value bars and rewrite the expression as a

disjunction.

Solving Absolute Value Equations 78

Algebraically

To solve absolute value equations algebraically, first isolate the absolute value expression on one side of the

EXPLORE

equation the same way you would isolate a variable. Then use the rule:

Solving Absolute Value Equations

If ⎜x⎟ = a (where a is a positive number), then x = a OR x = –a.

Graphically

Notice the use of a disjunction here in the rule for values of x. You cannot know from the original

equation whether the expression inside the absolute value bars is positive or negative, so you must work

through both possibilities to finish isolating x.

INTEGRATE TECHNOLOGY

]Example 1 Solve each absolute value equation algebraically. Graph the

solutions on a number line. Students have the option of completing the graphing

activity either in the book or online.

⎜3x⎟ + 2 = 8

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

Subtract 2 from both sides. ⎜3x⎟ = 6

Rewrite as two equations. 3x = 6 or 3x = −6 QUESTIONING STRATEGIES

Solve for x. x=2 or x = −2 How do you solve an absolute value equation

3⎜4x - 5⎟ - 2 = 19

graphically? Plot each side as if it were a

separate function of x, and find the x-coordinates

Add 2 to both sides. 3⎜4x - 5⎟ = 21

of the intersection points.

Divide both sides by 3. ⎜4x - 5⎟ = 7

Rewrite as two equations. 4x - 5 = 7 or 4x − 5 = -7

Why do you write the solutions to the absolute

value equation as a disjunction? If two values

Add 5 to all four sides. 4x = 12 or 4x = -2

of the variable both satisfy an equation, then one or

1

Solve for x. x = 3 or x = -_ the other can be correct.

2

-3 -2 -1 0 1 2 3

Your Turn

© Houghton Mifflin Harcourt Publishing Company

EXPLAIN 1

Solve each absolute value equation algebraically. Graph the solutions on a

number line. Solving Absolute Value Equations

3. 1 ⎜x + 2⎟ = 10

_

2

Algebraically

-24 -16 -8 0 8 16 24

⎜x + 2⎟ = 20

x + 2 = 20 or x + 2 = -20

x = 18 or x = -22 AVOID COMMON ERRORS

4. −2⎜3x − 6⎟ + 5 = 1 Some students may not isolate the absolute value

-2⎜3x - 6⎟ = -4 -4 -2 - 23 0 2

3

2 4 expression on one side of the equation as a first step

⎜3x - 6⎟ = 2

when solving the equation. Stress the importance of

3x - 6 = 2 3x - 6 = -2

or

this step so that the equation is in the form ⎜x⎟ = a,

x=_

8

3

or x=_

4

3 which has the solution x = a or x = –a.

Module 2 78 Lesson 2

PROFESSIONAL DEVELOPMENT QUESTIONING STRATEGIES

A2_MNLESE385894_U1M02L2 78 14/05/14 2:46 PM

Integrate Mathematical Practices How do you interpret the solutions to an

absolute value equation like ⎜x⎟ = a on a

This lesson provides an opportunity to address Mathematical Practice MP.6,

number line? Sample answer: The solutions are

which calls for students to “attend to precision” and communicate precisely.

the same distance from 0 on either side of the

Students find the solutions to absolute value equations both by graphing them,

number line.

with and without technology, and through algebra. Students learn that a

disjunction is often used to express the solutions to absolute value equations, and Why is it important to isolate the absolute

they use the properties of algebra to accurately and efficiently find the solutions to value expression when solving an absolute

various types of absolute value equations. value equation? So you can remove the absolute

value bars and rewrite the expression as a

disjunction.

Solving Absolute Value Equations 78

3.
Explain 2 Absolute Value Equations with Fewer

EXPLAIN 2 than Two Solutions

You have seen that absolute value equations have two solutions when the isolated absolute value expression

is equal to a positive number. When the absolute value is equal to zero, there is a single solution because

Absolute Value Equations with Fewer zero is its own opposite. When the absolute value expression is equal to a negative number, there is no

than Two Solutions solution because absolute value is never negative.

Example 2 Isolate the absolute value expression in each equation to determine

if the equation can be solved. If so, finish the solution. If not, write

QUESTIONING STRATEGIES “no solution.”

When does an absolute value equation have −5⎜x + 1⎟ + 2 = 12

fewer than two solutions? when the absolute Subtract 2 from both sides. − 5⎜x + 1⎟ = 10

value expression is equal to zero or equal to a Divide both sides by −5 . ⎜x + 1⎟ = −2

negative number Absolute values are never negative. No Solution

In the absolute value expression 3 ⎜2x − 4⎟ − 3 = −3

_

d⎜ax + b⎟ – c = –c for nonzero variables, how

5

Add 3 to both sides. 3 ⎜2x − 4⎟ =

_ 0

does d affect the solution? It does not affect it. 5

5.

Multiply both sides by _ ⎜2x − 4⎟ = 0

The first step is to add c to both sides to get 3

d⎜ax + b⎟ = 0. Because the product of a number Rewrite as one equation. 2x − 4 = 0

and 0 is 0, you can divide both sides by d to get Add 4 to both sides. 2x = 4

⎜ax + b⎟ = 0. Divide both sides by 2. x= 2

Your Turn

INTEGRATE TECHNOLOGY Isolate the absolute value expression in each equation to determine if the

equation can be solved. If so, finish the solution. If not, write “no solution.”

A graphing calculator can be used to check

⎜ ⎟

1 x+5 +7=5

3_ ⎜ ⎟

4x − 2 + 7 = 7

9_

© Houghton Mifflin Harcourt Publishing Company

5. 6.

the number of solutions to an absolute value 2 3

equation. Graph each side of the equation as a ⎜1

3 __

2 ⎟

x + 5 = -2 ⎜

9_

4

3 ⎟

x-2 =0

function and then count the number of intersection

⎜__2 x + 5⎟ = -_23

1

⎜_3x - 2⎟ = 0

4

points. 3

No solution x=_

2

AVOID COMMON ERRORS

Some students may think that if an absolute value

equation does not have two solutions, then there

must be no solution. Explain to students that when

the absolute value expression equals zero, there will

be one solution. For example, ⎜3x + 6⎟ = 0 has one

solution, x = –2, because 0 is neither positive nor Module 2 79 Lesson 2

COLLABORATIVE LEARNING

A2_MNLESE385894_U1M02L2 79 16/05/14 4:31 AM

Peer-to-Peer Activity

Have students work in pairs to brainstorm types of absolute value equations that

have two solutions, one solution, or no solution. For example, instruct one student

to write a conjecture about what type of absolute value equation has no solutions,

and give an example. Then have the other student solve the example and write an

explanation about whether the conjecture is correct or incorrect. Have students

switch roles and repeat the exercise using an equation that has a different number

of solutions.

79 Lesson 2.2

EXPLAIN 2 than Two Solutions

You have seen that absolute value equations have two solutions when the isolated absolute value expression

is equal to a positive number. When the absolute value is equal to zero, there is a single solution because

Absolute Value Equations with Fewer zero is its own opposite. When the absolute value expression is equal to a negative number, there is no

than Two Solutions solution because absolute value is never negative.

Example 2 Isolate the absolute value expression in each equation to determine

if the equation can be solved. If so, finish the solution. If not, write

QUESTIONING STRATEGIES “no solution.”

When does an absolute value equation have −5⎜x + 1⎟ + 2 = 12

fewer than two solutions? when the absolute Subtract 2 from both sides. − 5⎜x + 1⎟ = 10

value expression is equal to zero or equal to a Divide both sides by −5 . ⎜x + 1⎟ = −2

negative number Absolute values are never negative. No Solution

In the absolute value expression 3 ⎜2x − 4⎟ − 3 = −3

_

d⎜ax + b⎟ – c = –c for nonzero variables, how

5

Add 3 to both sides. 3 ⎜2x − 4⎟ =

_ 0

does d affect the solution? It does not affect it. 5

5.

Multiply both sides by _ ⎜2x − 4⎟ = 0

The first step is to add c to both sides to get 3

d⎜ax + b⎟ = 0. Because the product of a number Rewrite as one equation. 2x − 4 = 0

and 0 is 0, you can divide both sides by d to get Add 4 to both sides. 2x = 4

⎜ax + b⎟ = 0. Divide both sides by 2. x= 2

Your Turn

INTEGRATE TECHNOLOGY Isolate the absolute value expression in each equation to determine if the

equation can be solved. If so, finish the solution. If not, write “no solution.”

A graphing calculator can be used to check

⎜ ⎟

1 x+5 +7=5

3_ ⎜ ⎟

4x − 2 + 7 = 7

9_

© Houghton Mifflin Harcourt Publishing Company

5. 6.

the number of solutions to an absolute value 2 3

equation. Graph each side of the equation as a ⎜1

3 __

2 ⎟

x + 5 = -2 ⎜

9_

4

3 ⎟

x-2 =0

function and then count the number of intersection

⎜__2 x + 5⎟ = -_23

1

⎜_3x - 2⎟ = 0

4

points. 3

No solution x=_

2

AVOID COMMON ERRORS

Some students may think that if an absolute value

equation does not have two solutions, then there

must be no solution. Explain to students that when

the absolute value expression equals zero, there will

be one solution. For example, ⎜3x + 6⎟ = 0 has one

solution, x = –2, because 0 is neither positive nor Module 2 79 Lesson 2

COLLABORATIVE LEARNING

A2_MNLESE385894_U1M02L2 79 16/05/14 4:31 AM

Peer-to-Peer Activity

Have students work in pairs to brainstorm types of absolute value equations that

have two solutions, one solution, or no solution. For example, instruct one student

to write a conjecture about what type of absolute value equation has no solutions,

and give an example. Then have the other student solve the example and write an

explanation about whether the conjecture is correct or incorrect. Have students

switch roles and repeat the exercise using an equation that has a different number

of solutions.

79 Lesson 2.2

4.
Elaborate

7. Why is important to solve both equations in the disjunction arising from an absolute value

equation? Why not just pick one and solve it, knowing the solution for the variable will work

ELABORATE

when plugged backed into the equation?

The solution to a mathematical equation is not simply any value of the variable INTEGRATE MATHEMATICAL

that makes the equation true. Supplying only one value that works in the equation PRACTICES

implies that it is the only value that works, which is incorrect. Focus on Patterns

MP.8 Discuss with students how to solve an

absolute value equation of the form ⎜ax + b⎟ = c.

8. Discussion Discuss how the range of the absolute value function differs from the range of Students should routinely rewrite the next step as a

a linear function. Graphically, how does this explain why a linear equation always has exactly one disjunction, or a compound equation of the form

solution while an absolute value equation can have one, two, or no solutions?

The range of a non-constant linear function is all real numbers. The range of an ax + b = c or ax + b = –c and then solve each part

absolute value function is y ≥ k if the function opens upward and y ≤ k if the of the equation.

function opens downward. Because the graph of a linear function is a line, a

horizontal line will intersect it only once. Because the graph of an absolute value

QUESTIONING STRATEGIES

function is a V, a horizontal line can intersect it once, twice, or not at all.

How is the process of solving a linear absolute

value equation like the process of solving a

regular linear equation? Both processes are similar

9. Essential Question Check-In Describe, in your own words, the basic steps to solving absolute

value equations and how many solutions to expect.

initially, except that you isolate the absolute value

Isolate the absolute value expression. If the absolute value expression is equal to in one case, but isolate the variable in the case of

a positive number, solve for both the positive and negative case. If the absolute the linear equation. From there, the process is the

value expression is equal to zero, then remove the absolute value bars and solve same for each part of the disjunction of the two

the equation. There is one solution. If the absolute value expression is equal to a linear equations for the absolute value equation.

negative number, then there is no solution. © Houghton Mifflin Harcourt Publishing Company

PEER-TO-PEER ACTIVITY

Have students work in pairs. Have one student write

an absolute value equation and have the partner solve

it. The partner then explains why the solution(s)

makes sense. Students switch roles and repeat the

process. Encourage students to use the phrase

“distance from zero” and the statement “This

negative/positive integer makes the equation true.”

Module 2 80 Lesson 2 SUMMARIZE THE LESSON

How do you solve a linear absolute value

DIFFERENTIATE INSTRUCTION

equation? Isolate the absolute value

A2_MNLESE385894_U1M02L2 80 14/05/14 2:46 PM

Critical Thinking expression; write resulting equation as the

Some students may need help in deciding whether absolute value equations have disjunction of two linear equations; and solve

no solutions, one solution, or two solutions. You may want to suggest that they each equation.

always follow this solving plan: (1) Write the original equation; then (2) isolate

the absolute value expression on one side of the equal sign. It will have the form

⎜ax + b⎟ = c. (3) Rewrite the equation as two equations of the form ax + b = c

and ax + b = –c; and (4) solve each equation for x. There may be 0, 1, or 2

solutions. (5) If there are two solutions, write the answer using “or.” (6) Check the

solution(s) in the original problem.

Solving Absolute Value Equations 80

7. Why is important to solve both equations in the disjunction arising from an absolute value

equation? Why not just pick one and solve it, knowing the solution for the variable will work

ELABORATE

when plugged backed into the equation?

The solution to a mathematical equation is not simply any value of the variable INTEGRATE MATHEMATICAL

that makes the equation true. Supplying only one value that works in the equation PRACTICES

implies that it is the only value that works, which is incorrect. Focus on Patterns

MP.8 Discuss with students how to solve an

absolute value equation of the form ⎜ax + b⎟ = c.

8. Discussion Discuss how the range of the absolute value function differs from the range of Students should routinely rewrite the next step as a

a linear function. Graphically, how does this explain why a linear equation always has exactly one disjunction, or a compound equation of the form

solution while an absolute value equation can have one, two, or no solutions?

The range of a non-constant linear function is all real numbers. The range of an ax + b = c or ax + b = –c and then solve each part

absolute value function is y ≥ k if the function opens upward and y ≤ k if the of the equation.

function opens downward. Because the graph of a linear function is a line, a

horizontal line will intersect it only once. Because the graph of an absolute value

QUESTIONING STRATEGIES

function is a V, a horizontal line can intersect it once, twice, or not at all.

How is the process of solving a linear absolute

value equation like the process of solving a

regular linear equation? Both processes are similar

9. Essential Question Check-In Describe, in your own words, the basic steps to solving absolute

value equations and how many solutions to expect.

initially, except that you isolate the absolute value

Isolate the absolute value expression. If the absolute value expression is equal to in one case, but isolate the variable in the case of

a positive number, solve for both the positive and negative case. If the absolute the linear equation. From there, the process is the

value expression is equal to zero, then remove the absolute value bars and solve same for each part of the disjunction of the two

the equation. There is one solution. If the absolute value expression is equal to a linear equations for the absolute value equation.

negative number, then there is no solution. © Houghton Mifflin Harcourt Publishing Company

PEER-TO-PEER ACTIVITY

Have students work in pairs. Have one student write

an absolute value equation and have the partner solve

it. The partner then explains why the solution(s)

makes sense. Students switch roles and repeat the

process. Encourage students to use the phrase

“distance from zero” and the statement “This

negative/positive integer makes the equation true.”

Module 2 80 Lesson 2 SUMMARIZE THE LESSON

How do you solve a linear absolute value

DIFFERENTIATE INSTRUCTION

equation? Isolate the absolute value

A2_MNLESE385894_U1M02L2 80 14/05/14 2:46 PM

Critical Thinking expression; write resulting equation as the

Some students may need help in deciding whether absolute value equations have disjunction of two linear equations; and solve

no solutions, one solution, or two solutions. You may want to suggest that they each equation.

always follow this solving plan: (1) Write the original equation; then (2) isolate

the absolute value expression on one side of the equal sign. It will have the form

⎜ax + b⎟ = c. (3) Rewrite the equation as two equations of the form ax + b = c

and ax + b = –c; and (4) solve each equation for x. There may be 0, 1, or 2

solutions. (5) If there are two solutions, write the answer using “or.” (6) Check the

solution(s) in the original problem.

Solving Absolute Value Equations 80

5.
EVALUATE Evaluate: Homework and Practice

• Online Homework

Solve the following absolute value equations by graphing. • Hints and Help

• Extra Practice

1. ⎜x − 3⎟ + 2 = 5 2. 2⎜x + 1⎟ + 5 = 9

y y

8 12

4 8

ASSIGNMENT GUIDE x

4

-8 -4 0 4 8

Concepts and Skills Practice -4

x

-8 -4 0 4 8

Explore Exercise 1–4 -8 -4

Solving Absolute Value Equations

Graphically x=0 or x=6 x = −3 or x=1

Example 1

Solving Absolute Value Equations

Exercises 5–8 3. −2⎜x + 5⎟ + 4 = 2 4. ⎜_32 (x − 2)⎟ + 3 = 2

Algebraically y y

8 8

Example 2 Exercises 9–16

Absolute Value Equations with 4 4

Fewer than Two Solutions x x

-8 -4 0 4 -8 -4 0 4 8

-4 -4

INTEGRATE MATHEMATICAL -8 -8

x = −4 or x = −6 No solution

Focus on Reasoning

Solve each absolute value equation algebraically. Graph the solutions on a

© Houghton Mifflin Harcourt Publishing Company

MP.2 Remind students to check their solutions by number line.

substituting the values into the original equation and 5. ⎜2x⎟ = 3 6. ⎜_13 x + 4⎟ = 3

verifying that both solutions make the equation true.

When solving equations graphically, remind students -3 -2 -1 0 1 2 3 -24 -20 -16 -12 -8 -4 0

that the x-value of an intersection point is a solution

to the original equation.

2x = 3 or 2x = −3 (_31 ) x + 4 = 3 or (_31 ) x + 4 = -3

(_13 ) x = −1 (_13 )x = −7

3 3

x=_ or x = −_

2 2 or

x = −3 or x = − 21

Module 2 81 Lesson 2

COMMON

A2_MNLESE385894_U1M02L2 81

Exercise Depth of Knowledge (D.O.K.) CORE Mathematical Practices 14/05/14 3:10 PM

1–4 2 Skills/Concepts MP.5 Using Tools

5–16 2 Skills/Concepts MP.6 Precision

17 3 Strategic Thinking MP.4 Modeling

18, 21 3 Strategic Thinking MP.4 Modeling

19 3 Strategic Thinking MP.6 Precision

20 2 Skills/Concepts MP.6 Precision

22 3 Strategic Thinking MP.3 Logic

23–25 3 Strategic Thinking MP.6 Precision

81 Lesson 2.2

• Online Homework

Solve the following absolute value equations by graphing. • Hints and Help

• Extra Practice

1. ⎜x − 3⎟ + 2 = 5 2. 2⎜x + 1⎟ + 5 = 9

y y

8 12

4 8

ASSIGNMENT GUIDE x

4

-8 -4 0 4 8

Concepts and Skills Practice -4

x

-8 -4 0 4 8

Explore Exercise 1–4 -8 -4

Solving Absolute Value Equations

Graphically x=0 or x=6 x = −3 or x=1

Example 1

Solving Absolute Value Equations

Exercises 5–8 3. −2⎜x + 5⎟ + 4 = 2 4. ⎜_32 (x − 2)⎟ + 3 = 2

Algebraically y y

8 8

Example 2 Exercises 9–16

Absolute Value Equations with 4 4

Fewer than Two Solutions x x

-8 -4 0 4 -8 -4 0 4 8

-4 -4

INTEGRATE MATHEMATICAL -8 -8

x = −4 or x = −6 No solution

Focus on Reasoning

Solve each absolute value equation algebraically. Graph the solutions on a

© Houghton Mifflin Harcourt Publishing Company

MP.2 Remind students to check their solutions by number line.

substituting the values into the original equation and 5. ⎜2x⎟ = 3 6. ⎜_13 x + 4⎟ = 3

verifying that both solutions make the equation true.

When solving equations graphically, remind students -3 -2 -1 0 1 2 3 -24 -20 -16 -12 -8 -4 0

that the x-value of an intersection point is a solution

to the original equation.

2x = 3 or 2x = −3 (_31 ) x + 4 = 3 or (_31 ) x + 4 = -3

(_13 ) x = −1 (_13 )x = −7

3 3

x=_ or x = −_

2 2 or

x = −3 or x = − 21

Module 2 81 Lesson 2

COMMON

A2_MNLESE385894_U1M02L2 81

Exercise Depth of Knowledge (D.O.K.) CORE Mathematical Practices 14/05/14 3:10 PM

1–4 2 Skills/Concepts MP.5 Using Tools

5–16 2 Skills/Concepts MP.6 Precision

17 3 Strategic Thinking MP.4 Modeling

18, 21 3 Strategic Thinking MP.4 Modeling

19 3 Strategic Thinking MP.6 Precision

20 2 Skills/Concepts MP.6 Precision

22 3 Strategic Thinking MP.3 Logic

23–25 3 Strategic Thinking MP.6 Precision

81 Lesson 2.2