2-2
Solving Inequalities by Adding
or Subtracting
Going Deeper
Essential question: How can you use properties to justify solutions to inequalities
that involve addition and subtraction?
Standards for
Mathematical Content
• State the Addition and Subtraction Properties of
Inequality in words. You can add the same number
to both sides of a true inequality and the result
will be a true inequality. You can subtract the
same number from both sides of a true inequality
and the result will be a true inequality.
A-REI.2.3 Solve linear…inequalities…in one
variable…
Prerequisites
Solving Equations by Adding or Subtracting
Graphing and Writing Inequalities
Differentiated Instruction
Visual learners may find it helpful to see the
addition and subtraction properties of inequality
modeled with weights on a balance scale. An
unbalanced scale with a weight on each side
will remain unbalanced when the same weight
is placed on each side. Likewise, an unbalanced
scale with several weights on each side will remain
unbalanced when the same weight is removed from
each side.
Math Background
Just as addition and subtraction equations can
be solved solved by using inverse operations to
isolate the variable, so can inequalities that involve
addition and subtraction. The justification for
this is the Addition and Subtraction Properties of
Inequality. Although these properties are stated
using > and <, they also hold true for ≥ and ≤.
Because the solutions to inequalities are often
infinite, solutions are often represented in set
notation or graphed on a number line.
2
EXAMPLE
Questioning Strategies
• Why is the Addition Property of Inequality used
to solve the inequalities? The inequalities involve
INTR O D U C E
• How do you know what type of circle to draw on
the number line when graphing the solution? Use an empty circle for < (or >) because there
is no equal part. Use a solid circle for ≥ (or ≤)
because there is an equal part.
Extra Example
Solve. Write the solution using set notation. Graph
your solution.
A.x - 4 > -1 {x | x > 3}; the graph has an empty
circle on 3, and the line to the right of 3 is shaded.
TEAC H
1
B. x - 2 ≤ 3 {x | x ≤ 5}; the graph has a solid circle
on 5, and the line to the left of 5 is shaded.
engage
Questioning Strategies
• What are the Addition and Subtraction Properties
of Equality? If a = b, then a + c = b + c and
a – c = b - c.
Chapter 2 79
Lesson 2
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subtraction. Because addition is the inverse of
subtraction, addition can be used to isolate the
variable.
Review solving an addition or subtraction equation,
such as x + 3 = -5, reminding students of the
properties of equality that justify the solution steps.
Represent the solution x = -8 with a point on
a number line at -8. Then review graphing and
writing inequalities. On the number line, extend an
arrow from the point for -8 to the right and have
students tell what inequality this represents. Repeat
for an arrow from the point for -8 to the left. Tell
students that they can solve inequalities involving
addition and subtraction in the same way as they
solved addition and subtraction equations and
graph the solutions on a number line.
Name
Class
Notes
2-2
Date
Solving Inequalities by Adding or
Subtracting
Going Deeper
Essential question: How can you use properties to justify solutions to inequalities
that involve addition and subtraction?
A-REI.2.3
1
ENGAGE
Properties of Inequality
You have solved addition and subtraction equations by performing inverse operations
that isolate the variable on one side. The value on the other side is the solution.
Inequalities involving addition and subtraction can be solved similarly using the following
inequality properties. These properties are also true for ≥ and ≤.
If a > b, then a + c > b + c.
If a < b, then a + c < b + c.
If a > b, then a - c > b - c.
If a < b, then a - c < b - c.
Addition Property of Inequality
Subtraction Property of Inequality
REFLECT
1a. How do the Addition and Subtraction Properties of Inequality compare to the
Addition and Subtraction Properties of Equality?
The Addition and Subtraction Properties of Inequality are similar to the Addition
and Subtraction Properties of Equality, except that they contain inequality symbols
instead of equal signs. Because of the inequality symbols, each property is stated
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twice, once for < and once for >.
Most linear inequalities have infinitely many solutions. When using set notation, it is not
possible to list all the solutions in braces. The solution x ≤ 1 in set notation is {x | x ≤ 1}.
Read this as “the set of all x such that x is less than or equal to 1.”
{ x | x ≤ 1}
the set of
all x
such that
x is less than or
equal to 1
A number line graph can be used to represent the solution set of a linear inequality.
x>1
• To represent < or >, mark the endpoint with an empty circle.
1
• To represent ≤ or ≥, mark the endpoint with a solid circle.
• Shade the part of the line that contains the solution set.
x<1
x≤1
x≥1
1
1
1
Chapter 2
79
Lesson 2
A-REI.2.3
2
EXAMPLE
Adding to Find the Solution Set
Solve. Write the solution using set notation. Graph your solution.
x-3<2
A
<2+
3
Addition
3
3
add
x<
5
Property of Inequality;
to both sides.
Simplify.
Write the solution set using set notation.
{x | x < 5}
Graph the solution set on a number line.
–4 –3 –2 –1
0
1
2
3
4
5
6
7
8
9
x - 5 ≥ -3
B
x-5+
≥ -3 +
5
Addition
5
5
add
x≥
2
Property of Inequality;
to both sides.
Simplify.
Write the solution set using set notation.
{x | x ≥ 2}
Graph the solution set on a number line.
-4 -3 -2 -1
0
1
2
3
4
5
6
7
8
9
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© Houghton Mifflin Harcourt Publishing Company
x-3+
REFLECT
2a. Is 5 in the solution set of the inequality in Part A? Explain.
No; the inequality symbol < means that only values less than 5 are in the
solution set.
2b. Suppose the inequality symbol in Part A had been >. Describe the solution set.
The solution set would have been all values greater than 5.
2c. Suppose the inequality symbol in Part B had been ≤. Describe the solution set.
The solution set would have been 2 and all values less than 2.
Chapter 2
Chapter 2
80
Lesson 2
80
Lesson 2
3
CLOS E
EXAMPLE
Essential Question
How can you use properties to justify solutions to
inequalities that involve addition and subtraction?
Questioning Strategies
• How do the inequalities in this example
differ from the inequalities in 2 EXAMPLE ?
Use the Addition Property of Inequality to justify
adding the same number to both sides of an
inequality that involves subtraction to isolate the
variable. Use the Subtraction Property of Inequality
to justify subtracting the same number from both
sides of an inequality that involves addition to
isolate the variable.
These inequalities involve addition instead of
subtraction.
• How is solving these inequalities similar to
solving the inequalities in 2 EXAMPLE ?
The inverse of the operation involved in the
inequalities is used to isolate the variable.
• How is solving these inequalities different from
solving the inequalities in 2 EXAMPLE ? The
Summarize
Have students write a journal entry in which they
describe how to solve an inequality involving
addition and an inequality involving subtraction.
They should include the properties that justify the
steps in their description. Encourage students to
use a variety of inequality signs. Have them write
their solutions in set notation and represent them
with graphs on a number line. Then have them
explain the decisions they had to make to graph the
solutions.
inverse operation used is subtraction instead of
addition.
EXTRA EXAMPLE
Solve. Write the solution using set notation. Graph
your solution.
A. x + 3 < -2 {x | x < -5}; the graph has an empty
circle on -5, and the line to the left of -5 is
shaded.
B. x + 5 ≥ 1 {x | x ≥ -4}; the graph has a solid
circle on -4, and the line to the right of -4 is
shaded.
PR ACTICE
Where skills are
taught
Highlighting
the Standards
2 EXAMPLE
EXS. 2, 5
3 EXAMPLE
EXS. 1, 3, 4
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This lesson provides numerous opportunities
to address Mathematical Practices Standard
6 (Attend to precision). Emphasize the need
for accuracy when writing inequality symbols
in the solution steps of an inequality and in
set notation and in graphing solutions of
inequalities on number lines.
Where skills are
practiced
Avoid Common Errors
Students may use the same operation instead of the
inverse operation to isolate the variable. Encourage
these students to write the application of the
appropriate property of inequality instead of simply
adding or subtracting in their head.
Chapter 2
81
Lesson 2
Notes
A-REI.2.3
3
EXAMPLE
Subtracting to Find the Solution Set
Solve. Write the solution using set notation. Graph your solution.
x+4>3
A
x+4-
>3-
4
Subtraction
4
x > -1
Property of Inequality
Simplify.
Write the solution set using set notation.
{x | x > -1}
Graph the solution set on a number line.
-4 -3 -2 -1
0
1
2
3
4
5
6
7
8
9
x + 2 ≤ -1
B
x+2-
≤ -1 -
2
Subtraction
2
x ≤ -3
Property of Inequality
Simplify.
Write the solution set using set notation.
{x | x ≤ -3}
Graph the solution set on a number line.
–5 –4 –3 –2 –1
0
1
2
3
4
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REFLECT
3a. Is -3 in the solution set of the inequality in Part B? Explain.
Yes; the inequality symbol ≤ means that the solutions are less than or equal to -3,
and -3 = -3.
3b. Suppose the inequality symbol in Part A had been ≥. Describe the solution set.
The solution set would have been all values greater than or equal to -1.
3c. Suppose the inequality symbol in Part B had been <. Describe the solution set.
The solution set would have been all values less than -3.
Chapter 2
81
Lesson 2
PRACTICE
Solve. Justify your steps. Write the solution in set notation. Graph your solution.
1. x + 1 ≤ -2
x ≤ -3
0
1
2
3
4
5
6
7
8
9
7
8
9
7
8
9
8
9
8
9
Subtraction Property of Inequality
Simplify.
{x | x ≤ -3}
2. x - 2 > 1
-4 -3 -2 -1
x - 2 + 2 > 1 +2
x>3
0
1
2
3
4
5
6
Addition Property of Inequality
Simplify.
{x | x > 3}
3. x + 6 < 6
–4 –3 –2 –1
x + 6 - 6 < 6 -6
x<0
0
1
2
3
4
5
6
Subtraction Property of Inequality
Simplify.
{x | x < 0}
4. x + 3 < 2
–4 –3 –2 –1
x + 3 - 3 < 2 -3
x < -1
0
1
2
3
4
5
6
7
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
–4 –3 –2 –1
x + 1 - 1 ≤ -2 -1
Subtraction Property of Inequality
Simplify.
{x | x < -1}
5. x - 4 ≥ -4
x - 4 + 4 ≥ -4 + 4
x≥0
-4 -3 -2 -1
0
1
2
3
4
5
6
7
Addition Property of Inequality
Simplify.
{x | x ≥ 0}
Chapter 2
Chapter 2
82
Lesson 2
82
Lesson 2
ADD I T I O NA L P R AC TI C E
AND PRO BL E M S O LV I N G
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
Answers
Additional Practice
1. b > 7
2. t ≥ 3
3. x ≥ 5
4. g < -6
5. m ≤ 0
6. d < -4
7. 29 + h > 40; h > 11
© Houghton Mifflin Harcourt Publishing Company
8. 287 + m ≤ 512; m ≤ 225
9. 34 + p ≥ 97; p ≥ 63
Problem Solving
1. 4 + h ≤ 10; h ≤ 6
2. m + 255 > 400; m > 145
3. q + 9 ≥ 20; q ≥ 11
4. 40 + e ≥ 60; e ≥ 20
5. A
6. J
7. C
Chapter 2
83
Lesson 2
Name
Class
Notes
2-2
Date
Additional Practice
© Houghton Mifflin Harcourt Publishing Company
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