# 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)

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