copyright © 2010 pearson education, inc. all rights reserved sec 3.1 - 1 3.1 linear inequalities in...

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 3.1 - 1

3.1

Linear Inequalities in One Variable

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.1 - 2


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.


EXAMPLE 1 Graphing Intervals Written In Interval

Notation on Number Lines

Write the inequality in interval notation and graph it.



EXAMPLE 2 Graphing Intervals Written In Interval

Notation on Number Lines

Write the inequality in interval notation and graph it.


Linear Inequality

An inequality says that two expressions are not equal.

Linear Inequality



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:


Using the Addition Property of Inequality




Multiplication Property of Inequality

Multiplication Property of Inequality


Using the Multiplication Property of Inequality



3.1 Linear Inequalities in One Variable

Solving a Linear Inequality

Steps used in solving a linear inequality are:

Step 1 Simplify each side separately. Clear parentheses, fractions, and decimals using the distributive property as needed, and combine like terms.

Step 2 Isolate 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 3 Isolate the variable. Use the multiplication property of inequality to change the inequality to the form x < k or x > k.


Solving a Linear Inequality



Solving a Linear Inequality with Fractions



Solving a Three-Part Inequality




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 Expression Interpretation

a is at least b

a is no less than b

a is at most b

a is no more than b




A rectangle must have an area of at least 15 cm2 and no more than 60 cm2. 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 3 Write an inequality. Area equals width times length, so area is 3L; and this amount must be at least 15 and no more than 60.





Step 4 Solve.

Step 5 State the answer. In order for the rectangle to have an area of at least 15 cm2 and no more than 60 cm2 when the width is 3 cm, the length must be at least 5 cm and no more than 20 cm.





Step 6 Check. If the length is 5 cm, the area will be 3 • 5 = 15 cm2; if the length is 20 cm, the area will be 3 • 20 = 60 cm2. Any length between 5 and 20 cm will produce an area between 15 and 60 cm2.




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 1 Read the problem.

Step 2 Assign a variable. Let x = the number of minutes used during the month.




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 3 Write an inequality. You must pay a total of $6, plus 4 cents per minute. This total must be less than or equal to $10.




Step 4

Solve.




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 6 Check. If you use 100 minutes, you will have a total bill of $10, or $6 + $0.04(100).

copyright © 2010 pearson education, inc. all rights reserved sec 3.1 - 1 3.1 linear inequalities in...

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