Graphing and Solving Inequalities

Finding an Inequality Boundary

Boundary Point: A solution(s) that makes the inequality true (equal). It could be the smallest number(s) that make it true. Or it is the largest number(s) that makes it NOT true.

EX: Find the boundary point of 2 5 1x

To find a boundary replace the inequality

symbol with an equality symbol.

2 5 1x 2 6x

3x

Solving an Inequality

In order to find the points that satisfy an inequality statement:

1. Find the boundary

2. Test every region to find which one(s) satisfies the original statement

Reminder: Compound Inequalities

The following are examples to algebraically write the following graphs:

0 4

0≤x<4

-1 2

x<-1 or x>2

Solving a 1 Variable Inequality

2 22 5 3 4 3x x x x

2 22 5 3 4 3x x x x 2 6 0x x

3 2 0x x

3 or 2x x

0

x

x = -4 x = 0 x = 3

9 ≤ 3 -3 ≤ 3 30 ≤ 24False True False

Find the Boundary Test Every Region

3 2x

Represent the solutions to the following inequality algebraically and on a number line.

2 22 4 5 4 3 4 4 4 3 2 2

2 3 5 3 3 3 4 3 3 2 2

2 0 5 0 3 0 4 0 3

Change inequality to equality

Solve

Plot Boundary Point(s)

Pick a point in each region

Substitute into Original

Shade True Region(s) Algebraic

Solution

Closed or Open Dot(s)?

Graphical Solution

Solving a 2 Variable Inequality

(0,0)

Graphically represent the solutions to the following inequality.

Find the Boundary

32 3y x

32 3y x

Plot points for the equality

Test Every Region

(3,0)

0 > -3 0 > 1.5

True False

320 0 3 3

20 3 3

Solid or Dashed?

Pick a point in each region

Substitute into Original

Shade True Region(s)

Reminder: Cover-Up Method

Plot : -2x + 5y = -10Find the

intercepts

X Y

00 -25

If the graph is in Ax+By=C

form.

Solution to a System of Inequalities

A solution to a System of Inequalities is the coordinate(s) that makes ALL of the inequalities true. The graph of all the points that make the system true is called the Feasible Region.

EX: Prove (-4,5) is a solution to the system below2 6

3 7

x y

x y

It must make

EVERY inequality

true. 2 4 6 5

8 6 5 14 5

3 4 7 5 12 7 5

19 5True True

Solving a System of Inequalities

2

2

2 5 3

4 3

y x x

y x x

(0 ,0)

0 ≥ -3True

(0 ,0)

0 < 3True

Test Every Region

Graphically represent the solutions to the following system of inequalities:

Find the Boundaries

2 4 3y x x Plot points for the equalities one at a time

22 5 3y x x

Solid or Dashed?

0 1 3x x 0 2 1 3x x

Find which side to shade for each inequality

20 0 5 0 3 2

0 0 4 0 3

Shade the Feasible Region