State the Addition and Subtraction Properties of ...

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

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

Prerequisites Solving Equations by Adding or Subtracting Graphing and Writing Inequalities

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 ) 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.

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.

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Name

Class

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?

2-2

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 .

Addition Property of Inequality Subtraction Property of Inequality

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.

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 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

x is less than or such that 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.

1

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

1 x < 1

1 x 1

1

x > 1 x 1

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A-REI.2.3

2 E X A M P L E Adding to Find the Solution Set

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

A

x - 3 < 2

x-3+ 3 . 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.

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Notes

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3 EXAMPLE

Questioning Strategies ? How do the inequalities in this example

differ from the inequalities in 2 EXAMPLE ? 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 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.

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.

CLOSE

Essential Question How 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.

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.

PRACTICE

Where skills are taught

2 EXAMPLE

3 EXAMPLE

Where skills are practiced EXS. 2, 5

EXS. 1, 3, 4

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.

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A-REI.2.3

3 EXAMPLE

Subtracting to Find the Solution Set

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

A

x + 4 > 3

x+4- 4 >3- 4

Subtraction

x > -1

Simplify.

Write the solution set using set notation. {x | x > -1}

Property of Inequality

Graph the solution set on a number line.

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

B

x + 2 -1

x + 2 - 2 -1 - 2

x -3

Subtraction Simplify.

Write the solution set using set notation. {x | x -3}

Property of Inequality

Graph the solution set on a number line.

?5 ?4 ?3 ?2 ?1 0 1 2 3 4

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 1

x - 2 + 2 > 1 +2 x > 3

{x | x > 3}

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

Addition Property of Inequality Simplify.

3. x + 6 < 6

x + 6 - 6 < 6 -6 x < 0

{x | x < 0}

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

Subtraction Property of Inequality Simplify.

4. x + 3 < 2

x + 3 - 3 < 2 -3 x < -1

{x | x < -1}

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

Subtraction Property of Inequality Simplify.

5. x - 4 -4

x - 4 + 4 -4 + 4 x 0

{x | x 0}

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

Addition Property of Inequality Simplify.

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ADDITIONAL PRACTICE AND PROBLEM SOLVING

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

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