Writing and Solving Inequalities How can you represent relationships using inequalities?

Vocabulary An inequality is a statement that compares two expressions that are not strictly equal by using one of the following inequality signs. SymbolMeaning is less than is less than or equal to is greater than is greater than or equal to is not equal to

Vocabulary A solution of an inequality is any value of the variable that makes the inequality true. One way you can find solutions is by using a table.

Writing and Solving Inequalities Example 1 Mr. Bowker, being a teacher, can afford to spend at most $50 for a birthday dinner at a restaurant, including a 15% tip. Describe some costs that are within his budget. A.Which inequality symbol can be used to represent “at most”? _________ B.Complete the verbal model for the situation. Cost before tip (dollars) 15% Cost before tip (dollars) Budget limit (dollars)

Writing and Solving Inequalities Example 1 Mr. Bowker, being a teacher, can afford to spend at most $50 for a birthday dinner at a restaurant, including a 15% tip. Describe some costs that are within his budget. C.Write and simplify an inequality for the model. _________________________

Writing and Solving Inequalities Example 1 D.Complete the table to find some costs that are within Mr. Bowker’s budget. CostSubstituteCompareSolution? No

Writing and Solving Inequalities Example 1 1.Can Mr. Bowker spend $40 on the meal before the tip? Explain. _____________________________________ _____________________________________ 2.What whole dollar amount is the most Mr. Bowker can spend before the tip? Explain. _____________________________________ _____________________________________

Writing and Solving Inequalities Example 1 3.The solution set of an equation or inequality consists of all values that make the statement true. Describe the whole dollar amounts that are in the solution set for this situation. _____________________________________ _____________________________________

Writing and Solving Inequalities Example 1 4.Suppose Mr. Bowker also has to pay a 6% meal tax. Write an inequality to represent the new situation. Then identify two solutions. _____________________________________ _____________________________________ _____________________________________

Summary In your own words, describe what an inequality is and what it means to have a solution to an inequality. What do inequalities have in common with equations? How are they different? Next lesson, we will learn how to solve inequalities in a systematic way.

Solving Inequalities by Adding or Subtracting How can you use properties to justify solutions to inequalities that involve addition and subtraction?

Introduction

What inequality does the graph represent? What about this one? Introduction

You can solve inequalities involving addition and subtraction in the same way as you solved addition and subtraction equations. You can also graph the solutions on a number line. Introduction

Properties of Inequality

Addition Property of Inequality Subtraction Property of Inequality How do the Addition and Subtraction Properties of Inequality compare to the Addition and Subtraction Properties of Equality? ________________________________________________ ________________________________________________

Set Notation for Solution Sets

Graphing the Solution Set of a Linear Inequality A number line graph can be used to represent the solution set of a linear inequality.

Adding to Find the Solution Set Example 1 Solve. Write the solution using set notation. Graph your solution. A. Write the solution set using set notation. __________________ Graph the solution set on a number line. ______________ Property of Inequality; add______ to both sides. Simplify.

Adding to Find the Solution Set Example 1 Solve. Write the solution using set notation. Graph your solution. B. Write the solution set using set notation. __________________ Graph the solution set on a number line. ______________ Property of Inequality; add______ to both sides. Simplify.

Adding to Find the Solution Set Example 1

Subtracting to Find the Solution Set Example 2 Solve. Write the solution using set notation. Graph your solution. A. Write the solution set using set notation. __________________ Graph the solution set on a number line. ______________ Property of Inequality Simplify.

Subtracting to Find the Solution Set Example 2 Solve. Write the solution using set notation. Graph your solution. B. Write the solution set using set notation. __________________ Graph the solution set on a number line. ______________ Property of Inequality Simplify.

Adding to Find the Solution Set Example 2

Summary 1.How can you use properties to justify solutions to inequalities that involve addition and subtraction? 2.Describe how to solve an inequality involving addition and an inequality involving subtraction. Include the properties that justify the steps in your description. Write your solutions in set notation and represent them with graphs on a number line.

Solving Inequalities by Multiplying or Dividing How can you use properties to justify solutions to inequalities that involve multiplication and division?

Introduction

Multiplying or Dividing by a Negative Number

1.When solving inequalities, if you multiply by a negative number, you must _____________________________________ 2.When solving inequalities, if you divide by a negative number, you must _____________________________________

More Properties of Inequality Multiplication Property of Inequality Division Property of Inequality

Multiplying to Find the Solution Set Example 1 Solve. Write the solution using set notation. Graph your solution. A. Solution set:____________________ ____________ Property of Inequality Simplify.

Solve. Write the solution using set notation. Graph your solution. B. Solution set:____________________ Multiplying to Find the Solution Set Example 1 Simplify.

Multiplying to Find the Solution Set Example 1

Multiplying to Find the Solution Set Example 2 Solve. Write the solution using set notation. Graph your solution. A. Solution set:____________________ ____________ Property of Inequality Simplify.

Multiplying to Find the Solution Set Example 2 Solve. Write the solution using set notation. Graph your solution. B. Solution set:____________________ Simplify.

Dividing to Find the Solution Set Example 2

Summary Describe how to solve an inequality involving multiplication and an inequality involving division. Include the properties that justify your steps. Write your solutions in set notation and represent them with graphs on a number line.

Solving Two-Step and Multi-Step Inequalities How can you use properties to justify solutions to multi-step inequalities?

Introduction

Solving Inequalities With More Than One Step Example 1 Find the solution set. Justify each step and graph the solution set. Solution set: __________________ _____________ Property of Addition Combine like terms _____________ Property of Inequality Simplify. _____________ Property of Inequality Simplify.

Solving Inequalities With More Than One Step Example 1

The Distributive Property

Using the Distributive Property Example 2 Find the solution set. Justify each step and graph the solution set. ______________ Property ______________ Property of Inequality Simplify. ______________ Property of Inequality ______________ the inequality symbol; simplify. Solution set:______________

Using the Distributive Property Example 2

Summary Create a table that shows the properties of inequality. The table should give each rule in words and an example of the rule.

Solving Inequalities with Variables on Both Sides How can you use properties to justify solutions of inequalities with variables on both sides?

Introduction

Using Properties to Justify Solutions Example 1 Find the solution set. Justify each step and graph your solution. _________________ Property _________________________________ Simplify. _________________________________ Simplify. _________________________________ Simplify. Solution set:____________________

Using Properties to Justify Solutions Example 1 1.Why is the Distributive Property applied first in the solution? _______________________________________ 2.Could the properties have been applied in a different order than shown in Example 1? If so, would this make finding the solution easier or more difficult? Explain. _______________________________________ _______________________________________ _______________________________________

Using Properties to Justify Solutions Example 1

Using Properties to Justify Solutions Example 2

Summary Write a step-by-step method for solving inequalities with variables on both sides. One method should involve the distributive property.

Solving Compound Inequalities How can you solve special compound inequalities?

Introduction

Compound Inequalities

Solving Compound Inequalities Example 1

Solving Compound Inequalities Example 2

Solving Compound Inequalities Example 3

Solving Compound Inequalities Example 4

Solving Compound Inequalities

Summary In your own words, describe how to solve compound inequalities.

Solving Absolute Value Equations How can you use graphing to solve equations involving absolute value?

Introduction

Solving an Absolute Value Equation by Graphing Example 1

Solving an Absolute Value Equation Using Algebra Example 2 Write the equation. Subtract 1 from both sides. Simplify. Divide both sides by 2. Simplify

Solving an Absolute Value Equation Using Algebra Example 2 or Add 3 to both sides of both equations. orSimplify.

Solving an Absolute Value Equation Using Algebra Example 2

Solving a Real-World Problem Example 3

1.Show how to use algebra to find the time(s) when Sal is 1 mile from home.

Summary

Solving Absolute Value Inequalities How does solving absolute value inequalities relate to solving compound inequalities?

Introduction

Summary Describe how to solve an absolute value inequality. Address how you know whether the solution of the inequality will result in an AND or an OR compound inequality.