2. Linear Equations in One Variable
Type Example Solution
Linear equations x – 8 = 3(x + 5) A number
in one variable x = -23
Graph

3. Linear Inequalities in One Variables
Type Example Solution
Linear inequalities –3x + 5 > A set of numbers;
in one variable x < an interval
Graph

4. Linear Equations in Two Variables
Type Example Solution
Linear equations x + y = A set of ordered
in two variables pairs; a line
Graph

5. An ordered pair is a solution of the linear inequality if it makes the inequality a true statement.

6. Example
Determine whether (1, 5) and (6, –2) are solutions of the inequality 3x – y < 5.
Solution
3(1) –
–2 < 5
TRUE
3x – y < 5
3(6) – (–2) 5
20 < 5
FALSE
3x – y < 5
The pair (1, 5) is a solution of the inequality, but (6, –2) is not.

7. Example: Graph 7x + y > –14 y First : Graph the boundary line
7x + y = –14 or y = -7x - 14
(0, 0)
Second: Pick a test point not on the boundary: (0,0)
Test it in the original inequality.
7(0) + 0 > –14
0 > –14
True, so shade the side containing (0,0).

8. To Graph a Linear Inequality in Two Variables
1. Graph the boundary line found by replacing the inequality sign with an equal sign. If the inequality sign is > or <, graph a dashed boundary line (indicating that the points on the line are not solutions of the inequality). If the inequality sign is  or , graph a solid boundary line (indicating that the points on the line are solutions of the inequality).
2. Choose a point, not on the boundary line, as a test point. Substitute into the original inequality. (0,0) or (1,0) or (0,1)
3. If a true statement is obtained in Step 2, shade the half-plane that contains the test point. If a false statement is obtained, shade the half-plane that does not contain the test point.

9. y = x Example Graph Solution First: Graph the boundary -3 2 -2 3 -1 1 6 5 4
y = x -4 -5
Example
Graph
Solution
First: Graph the boundary
y = x ( Solid boundary line, > )
(0, 1)
Second: We choose a test point on one side of the boundary, say (0, 1).
Substituting into the inequality we get
We finish drawing the solution set by shading the half-plane that includes (0, 1).

10. a) b) c) d)