State the Addition and Subtraction Properties of ... · and Subtraction Properties of Equality, except that they contain inequality symbols instead of equal signs. Because of the

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2-2

Standards for Mathematical Content

A-REI.2.3 Solve linear…inequalities…in one variable…

PrerequisitesSolving Equations by Adding or Subtracting

Graphing and Writing Inequalities

Math BackgroundJust 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.

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.

Questioning Strategies• What are the Addition and Subtraction Properties

of Equality? If a = b, then a + c = b + c and a – c = b - c.

• 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.

Differentiated InstructionVisual 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.

Questioning Strategies• Why is the Addition Property of Inequality used

to solve the inequalities? The inequalities involve subtraction. Because addition is the inverse of subtraction, addition can be used to isolate the variable.

• 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 ExamplESolve. 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.

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.

INTRODUCE

TEACH

EngagE1

ExamplE2

Solving Inequalities by Adding or SubtractingGoing DeeperEssentialquestion: How can you use properties to justify solutions to inequalities that involve addition and subtraction?

Chapter 2 79 Lesson 2

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Adding to Find the Solution Set

Solve. Write the solution using set notation. Graph your solution.

A x - 3 < 2

x - 3 + < 2 + Property of Inequality;

add to both sides.

x < Simplify.

Write the solution set using set notation.

Graph the solution set on a number line.

B x - 5 ≥ -3

x - 5 + ≥ -3 + Property of Inequality;

add to both sides.

x ≥ Simplify.



REFLECT

2a. Is 5 in the solution set of the inequality in Part A? Explain.

2b. Suppose the inequality symbol in Part A had been >. Describe the solution set.

2c. Suppose the inequality symbol in Part B had been ≤. Describe the solution set.

E X A M P L E2A-REI.2.3

–1–2–3–4 0 21 3 4 5 876 9

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

3

5

2

5

3

5

3

5

Addition

Addition

{x | x < 5}

{x | x ≥ 2}

No; the inequality symbol < means that only values less than 5 are in the

solution set.

The solution set would have been all values greater than 5.

The solution set would have been 2 and all values less than 2.


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Name Class Date 2-2

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 ≤.

REFLECT

1a. How do the Addition and Subtraction Properties of Inequality compare to the

Addition and Subtraction Properties of Equality?

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.”

A number line graph can be used to represent the solution set of a linear inequality.

• To represent < or >, mark the endpoint with an empty circle.

• To represent ≤ or ≥, mark the endpoint with a solid circle.

• Shade the part of the line that contains the solution set.

E N G A G E1

Addition Property of Inequality If a > b, then a + c > b + c.If a < b, then a + c < b + c.

Subtraction Property of Inequality If a > b, then a - c > b - c.If a < b, then a - c < b - c.

Solving Inequalities by Adding or SubtractingGoing DeeperEssential question: How can you use properties to justify solutions to inequalities that involve addition and subtraction?

A-REI.2.3

x > 1

1

1

1

1

x < 1

x ≤ 1

x ≥ 1

the set ofall x such that

x is less than orequal to 1

{ x | x ≤ 1}

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

twice, once for < and once for >.


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Notes


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Questioning Strategies• How do the inequalities in this example

differ from the inequalities in EXAMPLE2 ?

These inequalities involve addition instead of subtraction.

• How is solving these inequalities similar to

solving the inequalities in EXAMPLE2 ?

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 EXAMPLE2 ? The inverse operation used is subtraction instead of addition.

EXTRA EXAMPLESolve. 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.

Avoid Common ErrorsStudents 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.

Essential QuestionHow can you use properties to justify solutions to inequalities that involve addition and subtraction?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.

SummarizeHave 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.

Where skills are taught

Where skills are practiced

EXAMPLE2 EXS. 2, 5

EXAMPLE3 EXS. 1, 3, 4

EXAMPLE3 CLOSE

PRACTICE

Highlighting the Standards

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.


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P R A C T I C E

Solve. Justify your steps. Write the solution in set notation. Graph your solution.

1. x + 1 ≤ -2

2. x - 2 > 1

3. x + 6 < 6

4. x + 3 < 2

5. x - 4 ≥ -4

–1–2–3–4 0 21 3 4 5 876 9

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

–1–2–3–4 0 21 3 4 5 876 9

–1–2–3–4 0 21 3 4 5 876 9

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

x + 1 - 1 ≤ -2 -1 Subtraction Property of Inequality

x ≤ -3 Simplify.

{x | x ≤ -3}

x - 2 + 2 > 1 + 2 Addition Property of Inequality

x > 3 Simplify.

{x | x > 3}

x + 6 - 6 < 6 - 6 Subtraction Property of Inequality

x < 0 Simplify.

{x | x < 0}

x + 3 - 3 < 2 - 3 Subtraction Property of Inequality

x < -1 Simplify.

{x | x < -1}

x - 4 + 4 ≥ -4 + 4 Addition Property of Inequality

x ≥ 0 Simplify.

{x | x ≥ 0}


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Subtracting to Find the Solution Set

Solve. Write the solution using set notation. Graph your solution.

A x + 4 > 3

x + 4 - > 3 - Property of Inequality

x > Simplify.



B x + 2 ≤ -1

x + 2 - ≤ -1 - Property of Inequality

x ≤ Simplify.



REFLECT

3a. Is -3 in the solution set of the inequality in Part B? Explain.

3b. Suppose the inequality symbol in Part A had been ≥. Describe the solution set.

3c. Suppose the inequality symbol in Part B had been <. Describe the solution set.

E X A M P L E3A-REI.2.3

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

–2 –1–3–4–5 210 3 4

Subtraction

Subtraction

{x | x > -1}

{x | x ≤ -3}

Yes; the inequality symbol ≤ means that the solutions are less than or equal to -3,

and -3 = -3.

The solution set would have been all values greater than or equal to -1.

The solution set would have been all values less than -3.

4

2

-3

4

2

-1


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Notes


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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

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

ADDITIONAL PRACTICE AND PROBLEM SOLVING


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Notes

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Problem Solving


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2-2Name Class Date

Additional Practice



Documents

State the Addition and Subtraction Properties of ... · and Subtraction Properties of Equality, except that they contain inequality symbols instead of equal signs. Because of the