Graph Linear Inequalities

Graph Linear Inequalities

Chapter 2.8

Quick Review of Inequalities

• The symbol “<”is an arrowhead pointing left

• When we write it means, both, that a is to the left of b 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 of b 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


• On a vertical number line the symbol “<” means below and it also means less than

• When we write it means, both, that a is below b 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 above b on the (vertical) number line, and that a is greater than b



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

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

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


The graph of these points is shown on the next slide

Values below

Values below

Values below

Values below

Values below

Values below

Values below




• 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


• 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


: use dashed line : use solid line


• 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!

Guided Practice

Graph the inequality in the coordinate plane.

a)

b)

c)

d) (HINT: you will have to divide both sides by a negative number!)

Guided Practice

Guided Practice

Guided Practice

Guided Practice

Inequalities for Vertical/Horizontal Lines

• Recall that a horizontal line has the form and a vertical line has the form

• The corresponding inequalities are


• 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


• Graph vertical inequalities as follows• : draw a dashed line through a on the x-axis and shade to the right of the line• : draw a solid line through a on the x-axis and shade to the right of the line• : draw a dashed line through a on the x-axis and shade to the left of the line• : draw a solid line through a on the x-axis and shade to the left of the line


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 since

is a true statement

• The point lies on the graph of


• On the other hand, the point is not a solution to since

is not a true statement

• So the point does not lie on the graph of



• 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 since

is true

• In this case, is a solution since

and is true


Guided Practice

Tell whether the given ordered pair is a solution of .

a)

b)

c)

d)

Guided Practice


a)

This is a false statement, so this point is not a solution

Guided Practice


b)

This is a true statement, so the point is a solution

Guided Practice


c)

This is a true statement, so the point is a solution

Guided Practice


d)

This statement is false, so the point is not a solution.

Exercise 2.8

• Page 135, #3-18

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Graph Linear Inequalities