Graph Linear Inequalities

1. Graph Linear Inequalities Chapter 2.8

2. Quick Review of Inequalities • The symbol “<”is an arrowhead pointing left • When we write it means, both, that a is to the left ofb on the (horizontal) number line, and that a is less than b • The symbol “>” is an arrowhead pointing right • When we write it means, both, that a is to the right ofb on the (horizontal) number line, and that a is greater than b • The number line need not be oriented horizontally • The y-axis in the coordinate plane is a number line that is oriented vertically

3. Quick Review of Inequalities • On a vertical number line the symbol “<” means below and it also means less than • When we write it means, both, that a is belowb on the (vertical) number line, and that a is less than b • On a vertical number line the symbol “>” means above and it also means greater than • When we write it means, both, that a is aboveb on the (vertical) number line, and that a is greater than b

4. Quick Review of Inequalities

5. Quick Review of Inequalities

6. Graphing Inequalities • In the inequality , we choose x to represent every value that makes the inequality true • Some of the values that would make this statement true (and could thus replace x) are 3.5, 5, 5.1, 10, 1000000000, and so on • All of these values are to the left of 3

7. Graphing Inequalities • In the inequality , y represents every number value that is greater than 3 • Some of these values include 3.75, 4.25, , and

8. Graphing Linear Inequalities • In this section you will graph linear inequalities • The form of a linear inequality is one of the following • We can use a modified table of values to get an idea of what the graphs of the above look like • Consider the linear inequality

9. Graphing Linear Inequalities The graph of these points is shown on the next slide

10. Graphing Linear Inequalities

11. Graphing Linear Inequalities

12. Graphing Linear Inequalities • Recall the difference between, for example, and • In the first case, x includes all values greater than 3, but x cannot be 3 itself • In the second case, x includes all values greater than 3, and x can also be 3 • We represented the difference on a graph with an open circle at 3 for the first case and a closed (filled) circle at 3 for the second case • We need an analogous way to represent the difference between and

13. Graphing Linear Inequalities • For the case of , the points that represent solutions are all points above the line , but no points on the line • To show that these points are not included, we draw the line using a dashed line • For the case of , the points that represent solutions are all points above the line , and all points on the line • To show that these points are included, we draw the line with a solid line

14. Graphing Linear Inequalities : use dashed line : use solid line

15. Graphing Linear Inequalities • Below are points to remember when graphing linear inequalities • For , draw the dashed line and shade above the line • For , draw the solid line and shade above the line • For , draw the dashed line and shade below the line • For , draw the solid line and shade below the line • Note that we have not yet considered inequalities for horizontal or vertical lines!

16. Guided Practice Graph the inequality in the coordinate plane. • (HINT: you will have to divide both sides by a negative number!)

17. Guided Practice

18. Guided Practice

19. Guided Practice

20. Guided Practice

21. Inequalities for Vertical/Horizontal Lines • Recall that a horizontal line has the form and a vertical line has the form • The corresponding inequalities are

22. Inequalities for Vertical/Horizontal Lines • Graph horizontal inequalities as follows • : draw a dashed line through b on the y-axis and shade above the line • : draw a solid line through b on the y-axis and shade above the line • : draw a dashed line through b on the y-axis and shade below the line • : draw a solid line through b on the y-axis and shade below the line

23. Inequalities for Vertical/Horizontal Lines • Graph vertical inequalities as follows • : draw a dashed line through aon the x-axis and shade to the right of the line • : draw a solid line through aon the x-axis and shade to the right of the line • : draw a dashed line through aon the x-axis and shade to the left of the line • : draw a solid line through aon the x-axis and shade to the left of the line

24. Inequalities for Vertical/Horizontal Lines

25. Solutions to Linear Inequalities • For a linear equality (or ), a point is a solution to the equation if, by substituting the values of x and y into the equation, the result is a true statement • This is the same as saying that the point is a point on the graph of • For example, the point is a solution to sinceis a true statement • The point lies on the graph of

26. Solutions to Linear Inequalities • On the other hand, the point is not a solution to sinceis not a true statement • So the point does not lie on the graph of

27. Solutions to Linear Inequalities

28. Solutions to Linear Inequalities • Similarly, any point that makes the linear inequality (for example) true is a solution to the inequality • This means that the point is in the shaded area • For example, is a solution to sinceis true • In this case, is a solution sinceand is true

29. Solutions to Linear Inequalities

30. Guided Practice Tell whether the given ordered pair is a solution of .

31. Guided Practice Tell whether the given ordered pair is a solution of . This is a false statement, so this point is not a solution

32. Guided Practice Tell whether the given ordered pair is a solution of . This is a true statement, so the point is a solution

33. Guided Practice Tell whether the given ordered pair is a solution of . This is a true statement, so the point is a solution

34. Guided Practice Tell whether the given ordered pair is a solution of . This statement is false, so the point is not a solution.

35. Exercise 2.8 • Page 135, #3-18