2.8 Two Variable Inequalities
Vocabulary:
Linear Inequality: An inequality in two variables whose graph is a region of the
coordinate plane that is bounded by a line. To graph a linear inequality, first graph the
boundary line, then which side needs to be shaded and whether the boundary line is
included.
Helpful Tips: Boundary Line
 When the inequality is expressed with < or >, the boundary line is dashed. (- - -);
indicating that the line is not part of the solution.
 When the inequality is expressed with ≤ or ≥, the boundary line is solid;
indicating that the line is part of the solution.
Helpful Tips: Shading
 For an inequality with y < or y ≤, shade below the line.
 For an inequality with y > or y ≥, shade above the line.
 Choose a test point above or below the boundary line. If the test point makes the
inequality true, shade the region containing that point.
Graphing a Linear Inequality:
Example 1: y < 1/2x -3
Step 1: make sure the inequality looks like y = mx + b; then graph the inequality.
Step 2: Determine if the boundary line is dashed or solid. (In this example it will be
dashed.)
Step 3: Determine the shaded region. (Look at the Helpful Tips).
Example 2: 3x - 2y ≤ 9 (H.L. Question #8)
Step 1: make sure the inequality looks like y = mx + b; then graph the inequality. (Watch
the dividing of a negative number!! Remember the inequality flips!!!)
Step 2: Determine if the boundary line is dashed or solid. (In this example it will be
solid.)
Step 3: Determine the shaded region. (Look at the Helpful Tips).
Real World Connection:
Example 3: At least 35 performers of the Big Tent Circus are in the grand finale. Some
pile into cars, while other balance on bicycles. Seven performers are in each car, and five
performers are on each bicycle. Draw a graph showing all the combinations of cars and
bicycles possible for the finale.
Relate: the # of performers in each car plus the # of performers in each bicycle
Define: x = # of cars
y = # of bicycles.
Expression: 7x + 5y ≥ 35.
Step 1: Find the intercepts of the boundary line.
When y = 0, 7x + 5(0) = 35
7x = 35
X=5
Determine if the boundary line is shaded or dashed.
Step 2: Choose a test point and shade the region.
When x = 0, 7(0) + 5y = 35
5y = 35
y=7
Graphing Absolute Value Inequalities:
Example 4: y ≤ │x – 4 │ + 5
1: Find the Vertex. (- b/m; C)
2: Graph the absolute value. Also to graph make the inequality into y =│x – 4 │+ 5
(Remember the sign next to the absolute value, tells if the V is facing up or down).
3: Determine if the boundary line is dashed or solid. (In this example it will be solid.)
4: Determine the shaded region. (Look at the Helpful Tips).
Example 5: -y + 3 > │x + 1 │
1: Make the inequality into y = -│x – 1 │+ 3
2: Find the Vertex. (- b/m; C)
3: Graph the absolute value. (Remember the sign next to the absolute value, tells if the V
is facing up or down).
4: Determine if the boundary line is dashed or solid. (In this example it will be dashed.)
5: Determine the shaded region.