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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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.

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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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.

Linear Inequalities in One Variable Linear Inequality An inequality says that two expressions are not equal. Linear Inequality

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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.

3.1 Linear Inequalities in One Variable Using the Addition Property of Inequality Solve and graph the solution:

Linear Inequalities in One Variable Using the Addition Property of Inequality Solve and graph the solution:

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

Linear Inequalities in One Variable Using the Multiplication Property of Inequality Solve and graph the solution:

Linear Inequalities in One Variable Using the Multiplication Property of Inequality Solve and graph the solution:

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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.

Linear Inequalities in One Variable Solving a Linear Inequality Solve and graph the solution:

Linear Inequalities in One Variable Solving a Linear Inequality with Fractions Solve and graph the solution:

Linear Inequalities in One Variable Solving a Three-Part Inequality Solve and graph the solution:

Linear Inequalities in One Variable Solving a Three-Part Inequality Solve and graph the solution:

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Linear Inequalities in One Variable Solving Applied Problems Using Linear Inequalities In addition to the familiar phrases “less than” and “greater than”, it is important to accurately interpret the meaning of the following: Word ExpressionInterpretation a is at least b a is no less than b a is at most b a is no more than b

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Linear Inequalities in One Variable Solving Applied Problems Using Linear Inequalities A rectangle must have an area of at least 15 cm 2 and no more than 60 cm 2. If the width of the rectangle is 3 cm, what is the range of values for the length? Step 1 Read the problem. Step 2 Assign a variable. Let L = the length of the rectangle. Step 3Write an inequality. Area equals width times length, so area is 3L; and this amount must be at least 15 and no more than 60.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Linear Inequalities in One Variable Solving Applied Problems Using Linear Inequalities A rectangle must have an area of at least 15 cm 2 and no more than 60 cm 2. If the width of the rectangle is 3 cm, what is the range of values for the length? Step 4Solve. Step 5State the answer. In order for the rectangle to have an area of at least 15 cm 2 and no more than 60 cm 2 when the width is 3 cm, the length must be at least 5 cm and no more than 20 cm.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Linear Inequalities in One Variable Solving Applied Problems Using Linear Inequalities A rectangle must have an area of at least 15 cm 2 and no more than 60 cm 2. If the width of the rectangle is 3 cm, what is the range of values for the length? Step 6 Check. If the length is 5 cm, the area will be 3 5 = 15 cm 2 ; if the length is 20 cm, the area will be 3 20 = 60 cm 2. Any length between 5 and 20 cm will produce an area between 15 and 60 cm 2.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Linear Inequalities in One Variable Solving Applied Problems Using Linear Inequalities You have just purchased a new cell phone. According to the terms of your agreement, you pay a flat fee of $6 per month, plus 4 cents per minute for calls. If you want your total bill to be no more than $10 for the month, how many minutes can you use? Step 1Read the problem. Step 2 Assign a variable. Let x = the number of minutes used during the month.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Linear Inequalities in One Variable Solving Applied Problems Using Linear Inequalities You have just purchased a new cell phone. According to the terms of your agreement, you pay a flat fee of $6 per month, plus 4 cents per minute for calls. If you want your total bill to be no more than $10 for the month, how many minutes can you use? Step 3Write an inequality. You must pay a total of $6, plus 4 cents per minute. This total must be less than or equal to $10.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Linear Inequalities in One Variable Solving Applied Problems Using Linear Inequalities Step 4 Solve.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Linear Inequalities in One Variable Solving Applied Problems Using Linear Inequalities Step 5 State the answer. If you use no more than 100 minutes of cell phone time, your bill will be less than or equal to $10. Step 6Check. If you use 100 minutes, you will have a total bill of $10, or $6 + $0.04(100).