§ 2.8

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§ 2.8 Solving Linear Inequalities

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Solving Linear Inequalities. § 2.8. Linear Inequalities in One Variable. A linear inequality in one variable is an inequality that can be written in the form ax + b < c where a , b , and c are real numbers and a is not 0. - PowerPoint PPT Presentation

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Page 1: § 2.8

§ 2.8

Solving Linear Inequalities

Page 2: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 22

Linear Inequalities in One Variable

A linear inequality in one variable is an inequality that can be written in the form

ax + b < c

where a, b, and c are real numbers and a is not 0.

This definition and all other definitions, properties and steps in this section also hold true for the inequality symbols >, , or .

Page 3: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 33

Graphing Solutions to Linear Inequalities• Use a number line.• Use a closed circle at the endpoint of an interval if

you want to include the point.• Use an open circle at the endpoint if you DO NOT

want to include the point.

7Represents the set {xx 7}.

-4Represents the set {xx > – 4}.

Solutions to Linear Inequalities

Interval notation, is used to write solution sets of inequalities.• Use a parenthesis if you want to include the number• Use a bracket if you DO NOT want to include the number..

Page 4: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 44

Solutions to Linear Inequalitiesx < 3

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

–1.5 x 3

–2 < x < 0-5 -4 -3 -2 -1 0 1 2 3 4 5

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

Interval notation: (–∞, 3)

Interval notation: (–2, 0)

Interval notation: [–1.5, 3)

NOT included

Included

Page 5: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 55

Addition Property of InequalityIf a, b, and c are real numbers, then

a < b and a + c < b + c are equivalent inequalities.

Multiplication Property of Inequality1. If a, b, and c are real numbers, and c is positive, then

a < b and ac < bc are equivalent inequalities.

2. If a, b, and c are real numbers, and c is negative, thena < b and ac > bc are equivalent inequalities.

Properties of Inequality

Page 6: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 66

Solving Linear Inequalities in One Variable1) Clear the inequality of fractions by multiplying both sides

by the LCD of all fractions of the inequality.2) Remove grouping symbols by using the distributive

property.3) Simplify each side of equation by combining like terms.4) Write the inequality with variable terms on one side and

numbers on the other side by using the addition property of equality.

5) Get the variable alone by using the multiplication property of equality.

Solving Linear Inequalities

Page 7: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 77

Solving Linear Inequalities

Example: Solve the inequality. Graph the solution and give your answer in interval notation.

2(x – 3) < 4x + 10

2x – 6 < 4x + 10 Distribute.

– 6 < 2x + 10 Subtract 2x from both sides.

– 16 < 2x Subtract 10 from both sides.

– 8 < x or x > – 8 Divide both sides by 2.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 (–8, ∞)

Page 8: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 88

Since 0 is always greater than –7, the solution is all real numbers. (Any value we put in for x in the original statement will give us a true inequality.)

Solving Linear InequalitiesExample: Solve the inequality. Graph the solution

and give your answer in interval notation.x + 5 x – 2

x x – 7 Subtract 5 from both sides.

0 – 7 Subtract x from both sides.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4

(Always true!)

(–∞ ∞)

Page 9: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 99

Solving Linear InequalitiesExample: Solve the inequality. Give your answer in

interval notation.a.) 9 < z + 5 < 13

b.) –7 < 2p – 3 ≤ 5a.) 9 < z + 5 < 13 4 < z < 8 Subtract 5 from all three parts.

(4, 8)

b.) –7 < 2p – 3 ≤ 5 –4 < 2p ≤ 8 Add 3 to all three parts. –2 < p ≤ 4 Divide all three parts by 2.

(–2, 4]

Page 10: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 1010

3x + 9 5(x – 1)

3x + 9 5x – 5 Use distributive property on right side.

3x – 3x + 9 5x – 3x – 5 Subtract 3x from both sides.

9 2x – 5 Simplify both sides.

14 2x Simplify both sides.

7 x Divide both sides by 2.

9 + 5 2x – 5 + 5 Add 5 to both sides.

Solution:7]

Solving Linear Inequalities

Example: Solve the inequality. Give your answer in graph form.

Page 11: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 1111

A compound inequality contains two inequality symbols.

To solve the compound inequality, perform operations simultaneously to all three parts of the inequality (left, middle and right).

Compound Inequalities

0 4(5 – x) < 8

This means 0 4(5 – x) and 4(5 – x) < 8.

Page 12: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 1212

The solution is (3,5].

– 20 – 4x < – 12 Simplify each part. 5 x > 3 Divide each part by –4.Remember that the sign changes direction when you divide by a negative number.

0 – 20 20 – 20 – 4x < 8 – 20 Subtract 20 from each part.

Solving Compound Inequalities

Example: Solve the inequality. Give your answer in interval notation.

0 20 – 4x < 8 Use the distributive property.

0 20 – 4x < 8

Page 13: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 1313

Inequality Applications

Example: Six times a number, decreased by 2, is at least 10. Find the number.

1.) UNDERSTAND

Let x = the unknown number.

“Six times a number” translates to 6x,

“decreased by 2” translates to 6x – 2,

“is at least 10” translates ≥ 10.

Continued

Page 14: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 1414

decreased

Six times a number

6x

by 2

2

is at least

10

10

Finding an Unknown Number

Example continued:

2.) TRANSLATE

Continued

Page 15: § 2.8

Martin-Gay, Beginning and Intermediate Algebra, 4ed 1515

Finding an Unknown Number

Example continued:3.) SOLVE

6x – 2 ≥ 106x ≥ 12 Add 2 to both sides.x ≥ 2 Divide both sides by 6.

4.) INTERPRETCheck: Replace “number” in the original statement of the problem with a number that is 2 or greater. Six times 2, decreased by 2, is at least 10

6(2) – 2 ≥ 10 10 ≥ 10 State: The number is 2.