LEI - Linear Equations and Inequalities (Overview)

Linear Equations and Inequalities

Introduction

You pay $40 a month to get 5GB of data to use for your phone. Last month, you went over by 35 MB and you are charged $0.50 per MB. How much do you owe? If you can figure this problem out, then you can work with linear equations. The study of linear equations is one of the most important in all of Algebra and used to model many situations in the real world. In this unit, you will practice writing, solving, and graphing linear equations and inequalities. We will explore these functions in three different forms: as a graph, as an equation, and in a table. Connecting each of these representations will be essential for you as we work on problem solving!

Essential Questions

• How do I solve and justify my answer to an equation or inequality in one variable?
• How do I graph a linear equation or inequality in two variables?
• How do I use graphs to represent and solve real-world equations and inequalities?
• What is a function and how do I use it to model real-world situations?
• How do I interpret the parts of a function in the context of the problem?
• How do I interpret key features of graphs in context?
• Why are sequences functions?
• How do I write recursive and explicit formulas for arithmetic sequences?

Key Terms

The following key terms will help you understand the content in this module.

Algebra - The branch of mathematics that deals with relationships between numbers, utilizing letters and other symbols to represent specific sets of numbers, or to describe a pattern of relationships between numbers.

Arithmetic Sequence - A sequence of numbers in which the difference between any two consecutive terms is the same.

Average Rate of Change - The change in the value of a quantity by the elapsed time. For a function, this is the change in the y-value divided by the change in the x-value for two distinct points on the graph.

Coefficient - A number multiplied by a variable in an algebraic expression.

Constant Rate of Change - With respect to the variable x of a linear function y = f(x), the constant rate of change is the slope of its graph.

Continuous - Describes a connected set of numbers, such as an interval.

Discrete - A set with elements that are disconnected.

Domain - The set of x-coordinates of the set of points on a graph or in a given set of ordered pairs. The value that is the input in a function or relation.

End Behaviors - The appearance of a graph as it is followed farther and farther in either direction.

Equation - A number sentence that contains an equals symbol.

Explicit Formula - A formula that allows direct computation of any term for a sequence a(1), a(2), a(3), . . . , a(n), . . . .

Expression - A mathematical phrase involving at least one variable and sometimes numbers and operation symbols. Any mathematical calculation or formula combining numbers and/or variables using sums, differences, products, quotients including fractions, exponents, roots, logarithms, functions, or other mathematical operations.

Factor - For any number x, the numbers that can be evenly divided into x are called factors of x. For example, the number 20 has the factors 1, 2, 4, 5, 10, and 20.

Inequality - Any mathematical sentence that contains the symbols > (greater than), < (less than), ≤ (less than or equal to), or ≥ (greater than or equal to).

Interval Notation - A notation representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Parentheses and/or brackets are used to show whether the endpoints are excluded or included.

Linear Function - A function with a constant rate of change and a straight line graph.

Linear Model - A linear function representing real-world phenomena. The model also represents patterns found in graphs and/or data.

Ordered Pair - A pair of numbers, (x, y), that indicates the position of a point on a Cartesian plane.

Parameter - The independent variable or variables in a system of equations with more than one dependent variable.

Range - The set of all possible outputs of a function.

Recursive Formula -A formula that requires the computation of all previous terms to find the value of $a_n$ .

Slope - The ratio of the vertical and horizontal changes between two points on a surface or a line.

Substitution - To replace one element of a mathematical equation or expression with another.

Term - A value in a sequence--the first value in a sequence is the 1st term, the second value is the 2nd term, and so on. A term is also any of the monomials that make up a polynomial.

Variable - A letter or symbol used to represent a number.

X-Intercept - The point where a line meets or crosses the x-axis.

Y-Intercept - The point where a line meets or crosses the y-axis.

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