Linear functions form the basis of linear inequalities. A linear inequality in two variables relates two variables using an inequality symbol, such as y > 2x – 4. Its graph is a region of the coordinate plane bounded by a line. The line is a boundary line, which divides the coordinate plane into two regions.

To graph y ≥ 2x – 4, make the boundary line solid, and shade the region above the line. To graph y > 2x – 4, make the boundary line dashed because y-values equal to 2x – 4 are not included.

Think of the underlines in the symbols ≤ and ≥ as representing solid lines on the graph.

Helpful Hint

Ex 1A: Graph the inequality .

The boundary line is which has a y-intercept of 2 and a slope of .Draw the boundary line dashed because it is not part of the solution. Then shade the region above the boundary line to show .

Check Choose a point in the solution region, such as (3, 2) and test it in the inequality.

The test point satisfies the inequality, so the solution region appears to be correct.

?

2 > 1 ?

Graph the inequality y ≤ –1.

Recall that y= –1 is a horizontal line.

Step 1 Draw a solid line for y=–1 because the boundary line is part of the graph.

Step 2 Shade the region below the boundary line to show where y < –1.

Ex 1B:

Check The point (0, –2) is a solution because –2 ≤ –1. Note that any point on or below y = –1 is a solution, regardless of the value of x.

Graph 3x + 4y ≤ 12 using intercepts.Ex 2:

Substitute x = 0 and y = 0 into 3x + 4y = 12 to find the intercepts of the boundary line.

y-intercept

x-intercept

3x + 4y = 123(0) + 4y = 12

3x + 4(0) = 12

4y = 12

3x + 4y = 12

y = 3

3x = 12x = 4 (4, 0)

(0, 3)

(0, 3)

(4, 0)

Ex 3: A school carnival charges $4.50 for adults and $3.00 for children. The school needs to make at least $135 to cover expenses.A. Using x as the adult ticket price and y as the child ticket price, write and graph an inequality for the amount the school makes on ticket sales.B. If 25 child tickets are sold, how many adult tickets must be sold to cover expenses?

An inequality that models the problem is 4.5x + 3y ≥ 135.

Find the intercepts of the boundary line.

4.5(0) + 3y = 135

4.5x + 3(0) = 135

y = 45

x = 30 (30, 0)

(0, 45)

If 25 child tickets are sold,

Substitute 25 for y in 4.5x + 3y ≥ 135.

Multiply 3 by 25.

A whole number of tickets must be sold.

At least 14 adult tickets must be sold.

4.5x + 3(25) ≥ 135

4.5x + 75 ≥ 135

4.5x ≥ 60, so x ≥ 13.3_

You can graph a linear inequality that is solved for y with a graphing calculator. Press and use the left arrow key to move to the left side.

Each time you press you will see one of the graph styles shown here. You are already familiar with the line style.

Solve for y. Graph the solution.

8x – 2y > 8

–2y > –8x + 8

Ex 4:

y < 4x – 4

Note that the graph is shown in the STANDARD SQUARE window. ( 6:ZStandard followed by 5:ZSquare).

Use the calculator option to shade below the line y < 4x – 4.