This document provides instructions for graphing inequalities on a coordinate plane. It begins by graphing examples of inequalities on a number line and plane. It then explains how to graph linear inequalities by writing them as equations, making a table of x-y values, plotting the points, and drawing the boundary line. Examples are provided of graphing inequalities in slope-intercept form such as y ≥ -3x + 2. The document also covers testing points, determining whether to shade above or below the line, and applying the process to graph more complex inequalities like 3x - 4y > 12. It concludes with an example of using inequalities to model real-world constraints and find maximum/minimum values through optimization

1. Solving
&
Graphing
Inequalities
Half - Plane

2. Objective- To graph inequalities
on the coordinate plane.
Recall…
Graph n < 3 on a number line.
-3 -2 -1 0 1 2 3 4

3. Graph y > 3 on the coordinate plane.
x
y

4. Graph x -2 on the coordinate plane.≤
x
y

5. How to graph a line
• Write the inequality as an equation
• E.g. y - 3x + 2≥  y = -3x + 2
• Make a table of x,y values
• x y = -3x+2
• 0 -3(0) + 2 = 2  (0,2)
• 1 -3(1) +2 = -1  (1, -1)
• 2 -3(2) +2 = -4  (2, -4)
• Graph these points and draw a line.

6. Graph y -3x + 2 on the coordinate plane.
x
y
≥
Boundary Line
y = -3x + 2
a = -3
b = 2
=
−3
1
Test a point not on the line
test (0,0)
0 -3(0) + 2≥
Not true!

7. Graph y -3x + 2 on the coordinate plane.
x
y
≥
Instead of testing a point
If in y = ax + b form...
Shade
up
Shade
down
Solid
line
Dashed
line
≥ ≤
> <

8. Graph on the coordinate plane.
3x - 4y > 12
-3x -3x
-4y > -3x + 12
-4 -4
y < x - 3
3
4
a =
b = -3
3
4
Boundary Line
x
y

9. Problem
If you have less than $5.00 in nickels and dimes,
find an inequality and sketch a graph to describe
how many of each coin you have.
Let n = # of nickels
Let d = # of dimes
0.05 n + 0.10 d < 5.00
or
5 n + 10 d < 500

10. 5n + 10d < 500
n d
0 50
100 0
0 10 20 30 40 50 60 70 80 90 100
n
d
60
50
40
30
20
10
0

11. Find the Vertex(es) or Vertices
• What were the (x,y) coordinates for the
triangle of constraints we just made?

12. Optimization
• Optimization is when we use a polygon of
constraints (the shape we made with the
inequalities) to find out the maximum or
minimum value of another equation.
• We need the vertices: (0,50) (100,0) (0,0)
• We substitute them into another equation to
and calculate the value.

13. Optimization
x y Optimization
Equation
P = 3x – 2 y
0 0 3(0) -2(0)
= 0
0 50 3(0) -2(50)
= -100 Min
100 0 3(100) -2(0)
= 300 Max

14. Check-up
Assignment
Page 43, Q. 3,4,6
**Will need graph
paper