# A.6 and *A.5bc - Linear Inequalities

A.6    The student will graph linear equations and linear inequalities in two variables, including

a)  determining the slope of a line when given an equation of the line, the graph of the line, or two points on the line.  Slope will be described as rate of change and will be positive, negative, zero, or undefined; and

b)  writing the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line.

A.5  The student will

b)  represent the solution of linear inequalities in two variables graphically;

c)  solve practical problems involving inequalities;

Bloom's Level: Apply, Analyze

### BIG IDEAS

• I can master the challenge of a ski slope for snowboarding, determine the gradient of a road, figure the pitch of a roof, and build a handicap accessible ramp to a door.
• I will understand slope is rate of change where one value changing proportionately effects the other value, and the pattern of this relationship can be represented by a line that facilitates analysis and prediction.

### UNDERSTANDING THE STANDARD

• Changes in slope may be described by dilations or reflections or both.
• Changes in the y-intercept may be described by translations.
• Linear equations can be graphed using slope, x- and y-intercepts, and/or transformations of the parent function.
• The slope of a line represents a constant rate of change in the dependent variable when the independent variable changes by a  constant amount.
• The equation of a line defines the relationship between two variables.
• The graph of a line represents the set of points that satisfies the equation of a line.
• A line can be represented by its graph or by an equation.
• The graph of the solutions of a linear inequality is a half-plane  bounded by the graph of its related linear equation.  Points on the boundary are included unless it is a strict inequality.
• Parallel lines have equal slopes.
• The product of the slopes of perpendicular lines is -1 unless one of the lines has an undefined slope.

·  A solution to an inequality is the value or set of values that can be substituted to make the inequality true.

·  The graph of the solutions of a linear inequality is a half-plane bounded by the graph of its related linear equation.  Points on the boundary are included unless the inequality contains only < or > (no equality condition).

·  Practical problems may be modeled and solved using linear inequalities.

·  Solutions and intervals may be expressed in different formats, including set notation or using equations and inequalities.

-  Examples may include:

 Equation/ Inequality Set Notation x = 3 {3} x = 3 or x = 5 {3, 5} y≥ 3 {y: y ≥ 3} Empty (null) set ∅ { }

·  Properties of Real Numbers and Properties of Inequality are applied to solve inequalities.

### ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

·  A.5bRepresent the solution of a linear inequality in two variables graphically.

·  A.5cSolve practical problems involving linear inequalities.

·  A.5cDetermine whether a coordinate pair is a solution of a linear inequality or a system of linear inequalities.

·  A.5bdDetermine and verify algebraic solutions using a graphing utility.

### KEY VOCABULARY

parent function, transformations, slope, intercepts, coordinates, graph, positive slope, negative slope, zero slope, undefined slope, equation of a line, vertical, horizontal, dilations, reflections, translation, rate of change, dependent variable, independent variable, boundary, parallel, perpendicular

Updated: Oct 27, 2017