# Evaluating Expressions Contributed by: 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
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
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
Module 2 ges must
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14/05/14 2:46 PM
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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
© 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
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
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