Linear Inequalities

1. Linear Inequalities • By • Dr. Julia Arnold • Math 04 • Intermediate Algebra

2. This module may be used as a supplement to aid the Academic Systems lesson. As so, your test for this lesson will still be on Academic Systems lesson 5.3’s Evaluate. So, let’s begin…………………………………………………...

3. The graph of a linear inequality is always a half-plane where the equality represents the boundary of the half-plane. This is the half-plane y < 2x+4 The line is the boundary and is y=2x+4

4. When you are given an inequality to graph there are two ways to discover which side of the line to shade. In the last example, y < 2x + 4 , we have the equation given, solved for y, and y is to the left of the inequality symbol. For example, 2x + 3y > 7 is not written as above. When your inequality is solved for y and y is on the left then “y <“ means shade below the line and “y >” means shade above the line.

5. Thus y > -2x +3 would be shaded above the line. The dotted line represents the boundary y = -2x + 3 and is dotted instead of solid because the inequality above does not have the = sign with the > sign.

6. The second way to know which way to shade is by using a test point. Suppose we have 2x - 3y < 6 As long as you can graph the line you don’t have to solve for y. There are infinitely many points above and below the line. Fill in the chart below with points above the line. x y 0 -5 7

7. The second way to know which way to shade is by using a test point. 2x - 3y < 6 You could choose many different numbers for y. Some are: x y 0 -5 7 0 0 4

8. The second way to know which way to shade is by using a test point. Do these points satisfy the inequality? 2x - 3y < 6 x y 0 -5 7 Let’s check (0,0) 2(0) - 3(0) < 6 yes, 0 < 6 is true. 0 0 4 Let’s check (7,4) 2(7) - 3(4) < 6 14 -12 < 6 2 < 6 again true. Let’s check (-5,0) 2(-5) - 3(0) < 6 -10 < 6 also true.

9. The second way to know which way to shade is by using a test point. Since all of these points satisfy the inequality, we should shade on the same side of the line in which the points are located. Thus…... 2x - 3y < 6 x y 0 -5 7 0 0 4 .

10. The second way to know which way to shade is by using a test point. Since all of these points satisfy the inequality, we should shade on the same side of the line in which the points are located. 2x - 3y < 6 x y 0 -5 7 0 0 4 .

11. What if the points do not satisfy the inequality? Shade the opposite side of the line. For example: x + y < 5 Pick a point above the line.

12. What if the points do not satisfy the inequality? Shade the opposite side of the line. x + y < 5 I picked (5,4) Let’s check it in the inequality: 5 + 4 < 5 9 < 5 which is false. So,Shade opposite side of line from the point

13. What if the points do not satisfy the inequality? Shade the opposite side of the line. x + y < 5 I picked (5,4) . Let’s check it in the inequality: 5 + 4 < 5 9 < 5 which is false. So,Shade opposite side of line from the point

14. How do we put two graphs together and solve a system? When solving a system of equalities you are looking for a common point. When solving a system of inequalities you are looking for a common region as illustrated by x - 4y > 2 2x - y > -3 The brown color shows the over-lapping of the red and green and the solution of the system.

15. Let’s solve the system: y < 6 x > -7/2 The boundary lines are respectively horizontal, and vertical lines.

16. Let’s solve the system: y < 6 x > -7/2 The boundary lines are respectively horizontal, and vertical lines. The over-lapping color is where the solution lies.

17. You can use colored pencils and produce the same results or shading with your pencil.