Lesson 4.4 Solving Inequalities in One Variable

CURRIKI ALGEBRA UNIT 4

Expressions and Equations

Lesson 4.4 Solving Inequalities in One Variable

Unit 4: Expressions and Equations

The five lessons (4.1-4.5) provide the instruction and practice that supports the culminating activity in the final unit project.

Lesson 4.4 Solving Inequalities in one variable

Students should learn of the existence of the complex number system, but will not solve quadratics with complex solutions until Algebra II.

Common Core State Standardsby Cluster

Lesson Preparation and Resources for Teachers

Graph the solution set of an inequality

Graphing equations and inequalities

Inequalities (Learn Without Limits)

Linear Inequalities (Sal Kahn video – set)

Introduction to Linear Equations

Guided Practice (with Answer Key)

Graphing Inequalities (Sal Kahn video)

Graphing Inequalities (with absolute values – Extension activity)

Graphing Inequalities in One Variable (video)

Writing and Solving Inequalities (assessment)

Compound Inequalities (Homework or Assessment)

Solving Soiree (Extension Activity)

Problem Solving Math Stars (Extension Activity)

Graphing Inequalities (Sal Kahn video) – if students have a solid understanding of solving and graphing quadratics (usually by 10th or 11th grade)

Graphing Inequalities Worksheet

TE_Quadratic Inequalities Worksheet

Instructional Materials for Students (print one copy for each student)

Guided Practice (Practice writing inequalities)

Donut Shop Inequalities (Individual Activity)

Writing and Solving Inequalities (assessment)

Blank Number Lines

Graphing Inequalities (blank coordinate grids)

SE_Quadratic Inequalities Worksheet

Time: 50-minute lesson

Lesson Objectives:

Students will:

• Solve inequalities in one variable
• Create inequalities that describe numbers or relationships

Lesson Content

1.Background building Activity for Students (5 minutes)

1. Vocabulary Building

Provide students with a copy of the Vocabulary Page. Students should create their own definition of the terms, which includes a picture or example. Ask students to keep the math-specific vocabulary words in mind while they work through the warm-up problem.

1. Warm-up Problem

Play the video Introduction to Linear Equations. Students visually witness a real-life example of an inequality. This video can launch a discussion about the terms less than and greater than and the idea of inequality.

1. Discuss the video as a class. Have students share thoughts and ideas about inequalities. Ask for examples of real-life situations where inequities are used.

Some possible ideas: comparing speeds to fastest animals to a car; comparing planets’ temperatures to sizes; determining the maximum number of players to dollars.

d.Revisit the vocabulary page. Ask students to write in other math-specific words that they heard during the Warm-up problem. Review those words as a whole class and ask students to write the meanings in their Vocabulary Page.

2.Focus Question based on Today’s lesson (25 minutes)

Today’s focus question (write it on the board): How can inequalities be written and graphed to show comparisons?

1. Whole group activity: Show the “Question” (in the green boxes) from Inequalities from Learn Without Limits. If individual white boards are available, ask students to create a graphical representation of each inequality; display the solution and allow students to self-check. Discuss any difficulties students had in graphing. If white boards are not available, print Blank Number Lines and ask students to create graphical representations of the inequalities.

1. Small group Activity (teacher observes students while they work to check understanding.) Show the video Graphing Inequalities in One Variable. After showing the video, provide students with the worksheet Guided Practice: Equal or Not. Students first write and then graph the inequalities. Students should discuss how their graphs are alike and how they are different. Be sure to discuss when to use a dotted line and when to use a solid line. Also ask: How do you determine which side of the line represents the solution set?

1. If students have learned how to solve quadratic equations, challenge students to solve a quadratic inequality. Show the video Graphing Inequalities. Watch the first 6 minutes and then ask students to solve: -x(2x – 14) ≥24. After solving, show the remaining 3:48 minutes of the video and ask students to self-check. Remind students that when multiplying or dividing an inequality by a negative number, the inequality direction is reversed.

1. Individual Activity: Provide each student with the Donut Shop Inequalities. Students work on the problem individually to assess their understanding.

Students share their solutions to the problems. Discuss as a class methods used to solve the inequalities. Allow students the chance to share their solutions with the class to help solidify their understanding of the topic. Students should ask questions to prepare them for the upcoming assessment.

3.Whole Class Discussion

Ask students to write a simple algorithm for solving inequalities. After they have finished, present the following algorithm. Discuss what it means and how to apply it to algebra problems that involve inequalities.

Algorithm:

1. When solving inequalities, the rules for solving equations apply with only one exception. When solving the equation requires multiplying or dividing both sides by a negative number, then the direction of the inequality symbol must be reversed. Otherwise all rules for solving equations (i.e., adding a constant to both sides, subtracting a constant from both sides, multiplying both sides by a constant, and squaring both sides) will still hold. (See: Multiplying Inequalities and Dividing Inequalities Problem four of six in Inequalities for examples of the reversing the direction of the inequality rule.)
2. When solving quadratic inequalities, begin by rewriting the inequality so that it is compared to zero. Next, factor the quadratic, use the quadratic equation or complete the square to find the values for x. Using logical arguments, determine possible values for x as it compares to zero (remember, if the product of the roots is less than zero, one root is positive and the other must be negative; if the product of the roots is greater than zero, then both roots are positive or both roots are negative).

1. The solution set of an inequality is represented as a graph

• If there is one variable, a single number line will be used. The graph has a point representing the value of the variable – the dot is solid when graphing less than or equal to and greater than or equal to; the dot is empty when the graph is less than or greater than. The direction of the arrow depends on if graphing less than (arrow to the left) or greater than (arrow to the right)
• When there are two variables, begin by graphing the related equation; this is done by replacing the inequality symbol with an equal sign and then graphing the resulting equations. Secondly, determine an ordered pair (point) that is a solution to equation and shade the side of its graphed line it falls on; this represents the solution set for the equation.
• If there is a system of inequalities; follow step (2); the solution set is the intersection of the shaded parts of the graph.
• When graphing a quadratic inequality, use a number line to indicate the roots and highlight solutions. Remember, the graph of a quadratic equation is a parabola, so when graphing a quadratic inequality the solution will be bound by the parabola.

4.Assessment Activity (5 minutes)

Provide each student with a copy of Writing and Solving Inequalities and Graphing Inequalities Sheets. Allow students time to complete. Collect and assess student understanding of solving inequalities.

5.Extension Activities

Graphing Inequalities Learn how to solve and graph inequalities with absolute values.

Solving Soiree Card Game: This card game allows students to solve equations and inequalities in a fun and interactive way. There are a variety of links for this game, including:

Solving Soiree Game Cards; Solving Soiree Activity Sheet; Solving Soiree Answer Sheet.

Solving and Graphing Inequalities and Mixed Reviews This lesson plan may be conducted as a review; this lesson includes great “tickets out the door” which can be used as additional assessment items.

Problem Solving Math Stars This is a multi-problem type game set up as a PowerPoint. This is a great way to review a variety of Algebra problems including inequalities.

6. Homework assignment for additional independent practice

Compound Inequalities This worksheet can be assigned as homework or as an assessment.

Inequalities Using the Learn Without Limits problems, ask students to write a word problem that would be represented by one of the problems from each section. The sections are: Graphs of Inequalities; Adding in inequalities; Subtracting in inequalities; Multiplying in inequalities; Dividing in inequalities; Solving inequalities with combined steps; and Compound inequalities.

Quadratic Inequalities Worksheet: This worksheet can be assigned as an in-class assignment of homework. Instruct students to first graph the quadratic function (they may use a graphing calculator); then solve the inequality as shown in the video, then shade the graph accordingly.

1