Analyzing Linear Inequalities

Introduction

Linear inequalities open up an opportunity for real-world problems to have multiple correct answers! Let's go back to our phone bill problem to understand this further. You are paying $40 a month to get 5GB of data to use for your phone. As long as you stay under 5GB in the month, you won't be charged a overage fee. This could be represented by an inequality because any value under 5GB would be okay in this scenario. There are limits such as this everywhere - speed limits, age restrictions, monthly budgets, etc. In all of these scenarios, there isn't just one correct answer, but multiple values that would work!

Essential Questions

How do I solve and justify my answer to a linear inequality in one variable?
How do I graph a linear inequality in two variables?
How do I use graphs to represent and solve real-world inequalities?
How can I solve a system of two inequalities using a graph?

Key Terms

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

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.

Constraints - restrictions (limitations, boundaries) that need to be placed upon variables used in equations and inequalities that model real-world situations.

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.

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.

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.

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.

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

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.

IMAGES CREATED BY GAVS