2-7 Two-Variable Inequalities

M11.D.2.1.2: Identify or graph functions, linear equations, or linear inequalities on a coordinate

plane

Objectives

Graphing Linear Inequalities

Graphing Two-Variable Absolute Value Inequalties

Vocabulary

A linear inequality is an inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line.

A solid boundary line means the line is part of the solution.

ex. ≥ or ≤

A dashed boundary line means the line is not part of the solution.

ex. > or <

Vocabulary

To figure out whether to shade above or below the line, Solve first for y.

If y is greater or greater than or equal to the line, you shade above.

ex. y > 2x + 1

If y is less than or less than or equal to the line, you shade below.

ex. y ≤ 2x + 1

Step 1: Graph the boundary line y = x + 1. Since the inequality is greater than, not greater than or equal to, use a dashed boundary line.

32

Graph y > x + 1.32

Step 2: Since the inequality is greater than, y-values must be more than those on the boundary line. Shade the region above the boundary line.

Graphing a Linear Inequality

A restaurant has only 15 eggs until more are delivered. An

order of scrambled eggs requires 2 eggs. An omelet requires 3

eggs. Write an inequality to model all possible combinations of

orders of scrambled eggs and omelets the restaurant can fill till

more eggs arrive. Graph the inequality.

Relate:plus

is less than or equal to

number of eggs needed for x orders of scrambled eggs

number of eggs needed for y orders

of omelets15

Define: Let x = the number of orders for scrambled eggs.

Let y = the number of orders for omelets.

Write: 2 x + 3 y 1515<–

Real World Example

(continued)

Step 1: Find two points on the boundary line. Use the points to graph the boundary line.

when x = 0, 2(0) + 3y = 15

3y = 15

y = 5

when y = 0, 2x + 3(0) = 15

2x = 15

x = 152

Graph the points ( , 0) and (0, 5). Since the

inequality is less than or equal to, use a solid

boundary line.

152

Continued

(continued)

Step 2: Since the inequality is less than, y-values must be less than those on the boundary line. Shade the region below the boundary line.

All ordered pairs with whole-number coordinates in the shaded area and on the boundary line represent a combination of x orders of scrambled eggs and y orders of omelets that the restaurant could fill.

Continued

Graph y |2x| – 3.

Since the inequality is greater than or equal to, the boundary is solid and the shaded region is above the boundary.

>–

Graphing Absolute Value Inequalities

Write an inequality for each graph. The boundary line is given.

a. Boundary: y = | x – 2| – 1

The boundary line is dashed. The shaded region is above the boundary. This is the graph of y > |x – 2| – 1.

b. Boundary: y = – x + 312

The boundary is solid. The shaded region is below

the boundary. This is the graph of y – x + 3.12

<–

Writing Inequalities

Homework

Pg 104 & 105 # 1,2,10,11,12,20,21