6.5 Graphing

Linear Inequalities

in Two Variables

Wow, graphing really is fun!

4 3 2 1 0

In addition to level 3.0 and above and beyond what was taught in class,  the student may:· Make connection with other concepts in math· Make connection with other content areas.

The student will understand that linear relationships can be described using multiple representations. - Represent and solve equations and inequalities graphically. - Write equations in slope-intercept form, point-slope form, and standard form. - Graph linear equations and inequalities in two variables. - Find x- and y-intercepts.

The student will be able to: - Calculate slope. - Determine if a point is a solution to an equation. - Graph an equation using a table and slope-intercept form. 

With help from theteacher, the student haspartial success with calculating slope, writing an equation in slope-intercept form, and graphing an equation.

Even with help, the student has no success understanding the concept of a linear relationships.

Learning Goal #1 for Focus 4 (HS.A-CED.A.2, HS.REI.ID.10 & 12, HS.F-IF.B.6,

HS.F-IF.C.7, HS.F-LE.A.2): The student will understand that linear relationships can be described using multiple representations.

What is a linear inequality?

• A linear inequality in x and y is an inequality that can be written in one of the following forms.

• ax + by < c• ax + by ≤ c• ax + by > c• ax + by ≥ c

• An ordered pair (a, b) is a solution of a linear equation in x and y if the inequality is TRUE when a and b are substituted for x and y, respectively.

• For example: is (1, 3) a solution of 4x – y < 2?

• 4(1) – 3 < 2• 1 < 2 This is a true statement so (1,

3) is a solution.

Check whether the ordered pairs are solutions of 2x - 3y ≥ -2.

a. (0, 0) b. (0, 1) c. (2, -1)

(x, y) Substitute ConclusionA (0,0) 2(0) – 3(0) = 0 ≥ -2 (0,0) is a

solution.

B (0,1) 2(0) – 3(1) = -3 ≥-2 (0, 1) is NOT a solution.

C (2,-1) 2(2) – 3(-1) = 7 ≥ -2 (2, -1) is a solution.

Graph the inequality 2x – 3y ≥ -2

Every point in the shaded region is a solution of the inequality and every other point is not a solution.

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2

1

-1

-2

-3

1 2 3 4 -3 -2 -1

Steps to graphing a linear inequality:

1. Sketch the graph of the corresponding linear equation.

1. Use a dashed line for inequalities with < or >.

2. Use a solid line for inequalities with ≤ or ≥.

3. This separates the coordinate plane into two half planes.

2. Test a point in one of the half planes to find whether it is a solution of the inequality.

3. If the test point is a solution, shade its half plane. If not shade the other half plane.

Sketch the graph of 6x + 5y ≥ 30

1. Use x- and y-intercepts:(0, 6) & (5, 0)This will be a solid line.

2. Test a point. (0,0)6(0) + 5(0) ≥ 300 ≥ 30 Not a solution.

3. Shade the side that doesn’t include (0,0).

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2

-2

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-6

2 4 6 8 -6 -4 -2

Sketch the graph y < 6.

1. This will be a dashed line at y = 6.

2. Test a point. (0,0)0 < 6 This is a solution.

3. Shade the side that includes (0,0).

6

4

2

-2

-4

-6

2 4 6 8 -6 -4 -2

Sketch the graph of 2x – y ≥ 1

1. Use x- and y-intercepts:(0, -1) & (1/2, 0)

This will be a solid line.

2. Test a point. (0,0)2(0) - 0 ≥ 10 ≥ 1 Not a solution.

3. Shade the side that doesn’t include (0,0).

3

2

1

-1

-2

-3

1 2 3 4 -3 -2 -1