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3.5 Graphing Linear
Inequalities in Two
Variables
What is a Linear Inequality?
• An inequality containing x and y
whose boundary is a straight line.
Checking Solutions
• Solutions of linear inequalities are
ordered pairs (x, y) that make the
inequality true.
• To check if a point is a solution:
▫ Plug it in for x and y
▫ Simplify
▫ Does it make a true statement?
Example
• Is (-1, 9) a solution of 2x + y < -3 ?
Example
• Is (2, -2) a solution of x – 3y ≥ 8 ?
Example
• Is (-3, 4) a solution of y > -1 ?
Example
• Is (0, 0) a solution of y > x ?
You Try!
• Is (-4, -1) a solution of 5x – 2y ≤ -1?
Graphs of Linear
Inequalities
• Linear inequalities, like linear
equations, cannot be “solved” to get a
number answer.
• Instead, we use a graph to show all
solutions!
• Graphs will contain a “boundary
line” and a shaded “half plane.”
Dashed or Solid
• A dashed line means points on
the line are not solutions.
▫ Use for < and >.

• A solid line means points on the
line are solutions.
▫ Use for ≤ and ≥.
Where to Shade?

• The shaded area shows all possible
points that make the inequality true.
• Test a point.
▫ We usually use (0,0).
▫ If it is on the line, choose a different
point.
• If it is a solution, shade that side.
• If it is not a solution, shade the other
side.
Graphing Simple Linear
Inequalities
• Linear inequalities with only one
variable have horizontal or vertical
boundary lines.
• x: vertical (up and down)
• y: horizontal (side to side)
Example:
• Graph y ≤ 2.
Example:
• Graph x > 1.
Example:
• Graph y ≥ -5.
You Try!
Graph y > -1

Graph 3.5 > x
Slope-Intercept Form
• Remember: y = mx + b
• m = slope
▫ rise over run!

• (0, b) = y-intercept
Example
• Graph y ≤ 3x – 1
Example
• Graph y > ½ x + 3
Example
• Graph y ≥ - x – 2
You Try!
• Graph y < x + 2
Standard Form
• Remember: ax + by = c
• Two choices:
▫ Solve for y. Rewrite as y = mx + b.
- or ▫ Find intercepts: (0, y) and (x, 0)
Graphing Practice/Review
• Graph 3x + 2y = -6 both ways.
Example
• Graph –x + 2y > 2
Example
• Graph 3x – 2y < 6
Example
• Graph 3x – 2y ≥ 0
You Try!
• Graph –x + 4y > – 2

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