Subject:
Mathematics, Algebra
Material Type:
Module
Provider:
Ohio Open Ed Collaborative
Tags:
College Algebra, Equations, Inequalities, Tmm0012
Language:
English
Media Formats:
eBook, Interactive, Video

# Equations and Inequalities ## Overview

This material covers Chapter 2: Equations and Inequalities chapter of the OpenStax College Algebra Text. This module contains an overview of learning objectives mapped to the OTM state standards, worksheets that correspond to chapter sections, interactive Desmos Activities that pair with the chapter, and a list of supplemental videos that correspond to the chapter content.

Image by Rudy and Peter Skitterians from Pixabay

# Equations and Inequalities

When two things are said to have an equal value to one another, the = symbol is used to represent that those two things have values that are equal to one another. This concept is used to draw relations between a large number of scenarios. If a homeowner is paying for lawn service by the hour, and they know the total cost, they could easily create an equation that represents how much they will pay for that service, depending on the number of hours worked. If a homeowner is looking at the rates for two different lawn care companies, they can discern which company is the better deal, given some information about the services they are looking for. These equations can also be graphed. Graphing equations provides a useful visual for analyzing key characteristics about an equation, and is used in a variety of contextual situations.

There are many types of equations, and various techniques for how to solve them. This chapter will present a variety of equations, and will present different strategies for how to solve. One of the key elements to focus on in this chapter is identifying the type of equation, and being able to apply the necessary method for solving that equation. Much like an excellent maintenance person can examine a large range of problems, and the skill to identify which tool and technique they must employ to solve the problem, effectively solving an equation involves a very similar technique.

# Chapter Sections:

• 2.1 The Rectangular Coordinate Systems and Graphs

• 2.2 Linear Equations in One Variable

• 2.3 Models and Applications

• 2.4 Complex Numbers

• 2.6 Other Types of Equations

• 2.7 Linear Inequalities and Absolute Value Inequalities

# OTM Outcomes

2.     Equations and Inequalities: Successful College Algebra students are proficient at solving a wide array of equations and inequalities involving linear, quadratic, higher-order polynomial, rational, exponential, logarithmic, radical, and piecewise-defined functions (including absolute value). The successful College Algebra student can:

2a. Recognize function families as they appear in equations and inequalities and choose an appropriate solution methodology for a particular equation or inequality and can communicate reasons for that choice.*

2b. Use correct, consistent, and coherent notation throughout the solution process to a given equation or inequality.*

2c. Distinguish between exact and approximate solutions and which solution methodologies result in which kind of solutions.*

2d. Demonstrate an understanding of the correspondence between the solution to an equation, the zero of a function, and the point of intersection of two curves.*

2e. Solve for one variable in terms of another.*

4b. Create linear models from data and interpret slope as a rate of change.*

5b. Use technology to verify solutions to equations and inequalities obtained algebraically.*

5d. Use technology and algebra in concert to locate and identify exact solutions.*

6a. Recognize when a result (theorem) is applicable and use the result to make sound logical conclusions and provide counter-examples to conjectures.*

# Chapter Section Objectives and related OTM standards:

2.1 The Rectangular Coordinate System and Graphs

• Plot ordered pairs in a Cartesian coordinate system. (Prerequisite)

• Graph equations by plotting points. (1b)

• Graph Equations with a graphing utility.  (1b)

• Find x-intercepts and y-intercepts. (2a)

• Use the distance formula. (NA)

• Use the midpoint formula. (NA)

2.2 Linear Equations in One Variable

• Solve equations in one variable algebraically. (2a, 2b)

• Solve a rational equation. (2a, 2b)

• Find a linear equation. (2e, 4b)

• Given the equations of two lines, determine whether their graphs are parallel or perpendicular. (4b)

• Write the equation of a line parallel or perpendicular to a given line. (4b or NA??)

2.3 Models and Applications

• Set up a linear equation to solve a real-world application (2a, 2b)

• Use a formula to solve a real world application (2a, 2b)

2.4 Complex Numbers

• Add and subtract complex numbers.(NA)

• Multiply and divide complex numbers. (NA)

• Solve quadratic equations with complex numbers.(Note: Although this is listed as a learning objective, it is never actually addressed in the text. For this reason, it is listed as NA, complex solutions are addressed in 2.5)

• Solve quadratic equations by factoring. (2a, 2b, 2d, 5b)

• Solve quadratic equations by the square root property. (2a, 2b)

• Solve quadratic equations by completing the square. (2a, 2b)

2.6 Other Types of Equations

• Solve equations involving rational exponents. (2a, 2b)

• Solve equations using factoring. (2a, 2b, 2d, 5b)

• Solve radical equations. (2a, 2b)

• Solve absolute value equations. (2a, 2b)

• Solve other types of equations (equations in quadratic form).(2a, 2b)

2.7 Linear Inequalities and Absolute Value Inequalities

• Use interval notation. (2a, 2b)

• Use properties of inequalities. (2a, 2b)

• Solve inequalities in one variable algebraically. (2a, 2b)

• Solve absolute value inequalities.(2a, 2b, 5b, 5d)

# Chapter 2 Worksheets

Please note that section 2.4 has been omitted as that content does not fall within the purview of the OTM standards for College Algebra.

The files below are worksheets created to support practice and learning for chapter 2 of the OpenStax College text.

# Chapter 2 Desmos Activities

The links below will direct to pre-built Desmos activites. These can be easily copied, remixed, or compiled into activites for teachers to customize to their class and objectives. For more on how educators can integrate the desmos activites into their class please click here to link to the desmos teacher support site:

Below is a short explanation of the Desmos activities linked.

Polygraph: Points
This Custom Polygraph is designed to spark vocabulary-rich conversations about points in the coordinate plane. Key vocabulary that may appear in student questions includes: right, left, above, below, quadrant, axis, positive, negative, coordinate, x-value (or abscissa), and y-value (or ordinate). In the early rounds of the game, students may notice graph features from the list above, even though they may not use those words to describe them. That’s where you can step in. After most students have played 2-3 games, consider taking a short break to discuss strategy, highlight effective questions, and encourage students in their use of increasingly precise academic language. Then ask them to play several more games, putting that precise language to work

Battle Boats
In this activity, students build coordinate plane proficiency through a guess-the-location-style game. This could be used as a remedial tool for those need more practice plotting points.

Polygraph: Lines
In this polygraph, students will work in pairs, one student will pick a line, it will be the job of the other person to eliminate lines that do not match that description. This will progress until the student has narrowed it down to one line. It requires the use of vocabulary and formulating a series of yes/no questions to correctly identify the line.

Exploring Length with Geoboards:
In this activity, students use Desmos-powered geoboards to explore length and to further develop their proficiency with the Pythagorean theorem. While distance formula is not covered in the OTM standards, we felt this was an excellent activity to familiarize students with the Pythagorean theorem.

# Linear Bundle:

The linear bundle contains a large collection of activities that are appropriate for this section. The activities that apply directly to this section are listed below:

Polygraph: Lines
In this polygraph, students will work in pairs, one student will pick a line, it will be the job of the other person to eliminate lines that do not match that description. This will progress until the student has narrowed it down to one line. It requires the use of vocabulary and formulating a series of yes/no questions to correctly identify the line.

Polygraph: Lines Part 2
This is a follow up that explores steepness, positive and negative slope, as well as the concept of increasing and decreasing

Put the point on the Line
This activity's focus is slope. The goal is to sharpen students’ focus on slope. In particular, the activity asks students to estimate first, then to calculate, then to notice proportionality as they place points on an imaginary line. Use student ideas here to define slope as a ratio of change in y-coordinates to change in x-coordinates. By the time students get to the end of the activity, they should have a number of ways of talking about this, but it’s unlikely they’ll write a fraction with ∆y in the numerator and ∆x in the denominator. They’ll be ready for you to introduce this idea.

Match My Line
In this activity, students work through a series of scaffolded linear graphing challenges to develop their proficiency with direct variation, slope-intercept, point-slope, and other linear function forms.

Land the Plane
In this activity, students practice finding equations of lines in order to land a plane on a runway. Most of the challenges are well-suited to slope-intercept form, but depending on the goals of an individual class or student they are easily adapted to other forms of linear equations.

Card Sort: Linear Functions
This activity asks students to notice and use properties of linear functions to make groups of three. Different properties will lead to different groupings by different students. Later we ask students to make conjectures about different groupings – why might another student have grouped the cards in a particular way?

Two Truths and a Lie: Lines
Students will practice their understanding of the features and vocabulary of linear equations by creating a line, writing two true and one false statement about it, and inviting their peers to separate truth from lies.

Marbleslides: Lines
In this delightful and challenging activity, students will transform lines so that the marbles go through the stars. Students will test their ideas by launching the marbles, and have a chance to revise before trying the next challenge.

Point Collector Activity:

A very good introductory activity to get students used to inequalities, and compound inequalities. Not really at the college level, but would be a great warmup or activity for a co-requisite College Algebra course.

# Chapter 2 Supplemental Videos

Below is a document that links to supplemental videos for this chapter. Note that these videos are not created by the publisher of the text, so some verbiage or problem solving strategies may vary from what is presented in the text.