1. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 3.1 - 1 3.1 Linear Inequalities in One Variable

2. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 2 3.1 Linear Inequalities in One Variable Objectives 1.Graph intervals on a number line. 2.Solve linear inequalities using the addition property. 3.Solve linear inequalities using the multiplication property. 4.Solve linear inequalities with three parts.

3. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 3 3.1 Linear Inequalities in One Variable Graphing intervals on a number line Solving inequalities is closely related to solving equations. Inequalities are algebraic expressions related by We solve an inequality by finding all real numbers solutions for it.

4. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 4 3.1 Linear Inequalities in One Variable Graphing intervals on a number line –5–4–3–2–1012345 One way to describe the solution set of an inequality is by graphing. We graph all the numbers satisfying x < –1 by placing a right parenthesis at –1 on the number line and drawing an arrow extending from the parenthesis to the left. This arrow represents the fact that all numbers less than –1 are part of the graph.

5. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 5 3.1 Linear Inequalities in One Variable Interval Notation and the Infinity Symbol The set of numbers less than –1 is an example of an interval. We can write the solution set of this inequality using interval notation. The symbol does not actually represent a number. A parenthesis is always used next to the infinity symbol. The set of real numbers is written as in interval notation.

6. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 6 3.1 Linear Inequalities in One Variable EXAMPLE 1 Graphing Intervals Written In Interval Notation on Number Lines Write the inequality in interval notation and graph it. –5–4–3–2–1012345 This statement says that x can be any number greater than or equal to 1. This interval is written. We show this on the number line by using a left bracket at 1 and drawing an arrow to the right. The bracket indicates that the number 1 is included in the interval.

7. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 7 3.1 Linear Inequalities in One Variable EXAMPLE 2 Graphing Intervals Written In Interval Notation on Number Lines Write the inequality in interval notation and graph it. –5–4–3–2–1012345 This statement says that x can be any number greater than –2 and less than or equal to 3. This interval is written. We show this on the number line by using a left parenthesis at – 2 and a right bracket at 3 and drawing a line between. The parenthesis indicates that the number –2 is not included in the interval and the bracket indicates that the 3 is included in the interval.

8. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 8 3.1 Linear Inequalities in One Variable Types of Intervals Summarized Open Intervals Set Interval Notation Graph a b a b

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 9 3.1 Linear Inequalities in One Variable Types of Intervals Summarized Half Open Intervals b a b a b a Set Interval Notation Graph

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 10 3.1 Linear Inequalities in One Variable Types of Intervals Summarized Closed Interval ba Set Interval Notation Graph

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 11 3.1 Linear Inequalities in One Variable Linear Inequality An inequality says that two expressions are not equal. Linear Inequality Examples:

12. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 12 3.1 Linear Inequalities in One Variable Solving Linear Inequalities Using the Addition Property Solving an inequality means to find all the numbers that make the inequality true. Usually an inequality has a infinite number of solutions. Solutions are found by producing a series of simpler equivalent equations, each having the same solution set. We use the addition and multiplication properties of inequality to produce equivalent inequalities.

13. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 13 3.1 Linear Inequalities in One Variable Addition Property of Inequality

14. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 14 3.1 Linear Inequalities in One Variable Using the Addition Property of Inequality Solve and graph the solution: Check: Substitute –4 for x in the equation x – 5 = 9. The result should be a true statement. This shows –4 is a boundary point.

15. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 15 3.1 Linear Inequalities in One Variable Using the Addition Property of Inequality Solve and graph the solution: Now we have to test a number on each side of –4 to verify that numbers greater than –4 make the inequality true. We choose –3 and –5. –5–4–3–2–1012345

16. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 16 3.1 Linear Inequalities in One Variable Using the Addition Property of Inequality Solve and graph the solution: Check: Substitute 3 for m in the equation 3 + 7m = 8m. The result should be a true statement. This shows 3 is a boundary point.

17. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 17 3.1 Linear Inequalities in One Variable Using the Addition Property of Inequality Solve and graph the solution: Now we have to test a number on each side of 3 to verify that numbers less than or equal to 3 make the inequality true. We choose 2 and 4. –5–4–3–2–1012345

18. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 18 3.1 Linear Inequalities in One Variable Multiplication Property of Inequality

19. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 19 3.1 Linear Inequalities in One Variable Using the Multiplication Property of Inequality Solve and graph the solution: Check: Substitute –8 for m in the equation 3m = –24. The result should be a true statement. This shows –8 is a boundary point.

20. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 20 3.1 Linear Inequalities in One Variable Using the Multiplication Property of Inequality Solve and graph the solution: Now we have to test a number on each side of –8 to verify that numbers greater than or equal to –8 make the inequality true. We choose –9 and –7. –16–14–12–10–8–6–4– 2024

21. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 21 3.1 Linear Inequalities in One Variable Using the Multiplication Property of Inequality Solve and graph the solution: Check: Substitute – 5 for k in the equation –7k = 35. The result should be a true statement. This shows –5 is a boundary point.

22. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 22 3.1 Linear Inequalities in One Variable Using the Multiplication Property of Inequality Solve and graph the solution: Now we have to test a number on each side of –5 to verify that numbers less than or equal to –5 make the inequality true. We choose –6 and –4. –16–14–12–10–8–6–4– 2024

23. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 23 3.1 Linear Inequalities in One Variable Solving a Linear Inequality Steps used in solving a linear inequality are: Step 1Simplify each side separately. Clear parentheses, fractions, and decimals using the distributive property as needed, and combine like terms. Step 2Isolate the variable terms on one side. Use the additive property of inequality to get all terms with variables on one side of the inequality and all numbers on the other side. Step 3Isolate the variable. Use the multiplication property of inequality to change the inequality to the form x k.

24. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 24 3.1 Linear Inequalities in One Variable Solving a Linear Inequality Solve and graph the solution: Step 1 Step 2

25. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 25 3.1 Linear Inequalities in One Variable Solving a Linear Inequality Solve and graph the solution: Step 3 –10–9–8–7–6–5–4– 3–2–10

26. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 26 3.1 Linear Inequalities in One Variable Solving a Linear Inequality with Fractions Solve and graph the solution: First Clear Fractions: Multiply each side by the least common denominator, 15.

27. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 27 3.1 Linear Inequalities in One Variable Solving a Linear Inequality with Fractions Solve and graph the solution: Step 1 Step 2

28. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 28 3.1 Linear Inequalities in One Variable Solving a Linear Inequality with Fractions Solve and graph the solution: Step 3 –16–14–12–10–8–6–4– 2024

29. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 29 3.1 Linear Inequalities in One Variable Solving Linear Inequalities with Three Parts In some applications, linear inequalities have three parts. When linear inequalities have three parts, it is important to write the inequalities so that: 1.The inequality symbols point in the same direction. 2.Both inequality symbols point toward the lesser numbers.

30. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 30 3.1 Linear Inequalities in One Variable Solving a Three-Part Inequality Solve and graph the solution: This statement says that x – 2 is greater than or equal to 3 and less than or equal to 7. To solve this inequality, we need to isolate the variable x. To do this, we must add 2 to the expression, x – 2. To produce an equivalent statement, we must also add 2 to the other two parts of the inequality as well.

31. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 31 3.1 Linear Inequalities in One Variable Solving a Three-Part Inequality Solve and graph the solution: 345678910111213

32. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 32 3.1 Linear Inequalities in One Variable Solving a Three-Part Inequality Solve and graph the solution: 0 –1 –2 1 2