1. Review Graphing Linear Equations

2. We have used 3 different methods for graphing equations.
1) using a t-table
2) using slope-intercept form
3) using x- and y-intercepts
The goal is to determine which method is the easiest to use for each problem!

3. Here’s your cheat sheet!
If the equation is in STANDARD FORM (Ax + By = C), graph using the intercepts.
If the equation is in SLOPE-INTERCEPT FORM (y = mx + b), graph using slope and intercept or a t-table (whichever is easier for you).
If the equation is in neither form, rewrite the equation in the form you like the best!

4. Graph Which graphing method is easiest?
Using slope and y-intercept (or t-table)!
These notes will graph using m and b
m = , b = 2

5. Review: Graphing with slope-intercept
Start by graphing the y-intercept (b = 2).
From the y-intercept, apply “rise over run” using your slope. rise = 1, run = -3
Repeat this again from your new point.
Draw a line through your points. -3 1 -3 1
Start here

6. Graph -2x + 3y = 12 Which graphing method is easiest?
Using x- and y-intercepts! (The equation is in standard form)
Remember, plug in 0 to find the intercepts.

7. Review: Graphing with intercepts: -2x + 3y = 12
Find your x-intercept:
Let y = 0
-2x + 3(0) = 12
x = -6; (-6, 0)
Find your y-intercept:
Let x = 0
-2(0) + 3y = 12
y = 4; (0, 4)
3. Graph both points and draw a line through them.

8. Graphing Linear Inequalities in Two Variables

9. Solution of Linear Inequalities
Expressions of the type x + 2y ≤ 8 and 3x – y > 6 are called linear inequalities in two variables.
A solution of a linear inequality in two variables is an ordered pair (x, y) which makes the inequality true.
Example: (1, 3) is a solution to x + 2y ≤ 8
since (1) + 2(3) = 7 ≤ 8.
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Solution of Linear Inequalities

10. Example: The solution set for x + 2y ≤ 8 is the shaded region.
The solution set, or feasible set, of a linear inequality in two variables is the set of all solutions.
Example: The solution set for x + 2y ≤ 8 is the shaded region. x y 2
The solution set is a half-plane. It consists of the line x + 2y ≤ 8 and all the points below and to its left.
The line is called the boundary line of the half-plane.
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Feasible Set

11. Example: The boundary line of the solution set of 3x – y ≥ 2 is solid.
If the inequality is ≤ or ≥ , the boundary line is solid; its points are solutions.
3x – y = 2 x y
3x – y < 2
Example: The boundary line of the solution set of 3x – y ≥ 2 is solid.
3x – y > 2
If the inequality is < or >, the boundary line is dotted; its points are not solutions. x y
Example: The boundary line of the solution set of x + y < 2 is dotted.
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Boundary lines

12. Example: For 2x – 3y ≤ 18 graph the boundary line.
A test point can be selected to determine which side of the half-plane to shade. x y
Example: For 2x – 3y ≤ 18 graph the boundary line.
(0, 0) 2 -2
The solution set is a half-plane.
Use (0, 0) as a test point.
(0, 0) is a solution. So all points on the (0, 0) side of the boundary line are also solutions.
Shade above and to the left of the line.
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Test Point

13. Graphing an Inequality
To graph the solution set for a linear inequality:
1. Graph the boundary line.
2. Select a test point, not on the boundary line, and determine if it is a solution.
3. Shade a half-plane.
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Graphing an Inequality

14. Example: Graph an Inequality
Example: Graph the solution set for x – y > 2.
1. Graph the boundary line x – y = 2 as a dotted line. x y
(0, 0)
2. Select a test point not on the line, say (0, 0).
(2, 0)
(0, -2)
(0) – 0 = 0 > 2 is false.
3. Since this is a not a solution, shade in the half-plane not containing (0, 0).
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Example: Graph an Inequality

15. Inequalities in One Variable
Solution sets for inequalities with only one variable can be graphed in the same way. x y 4
- 4
Example: Graph the solution set for x < - 2. x y 4
- 4
Example: Graph the solution set for x ≥ 4.
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Inequalities in One Variable