Graphing Inequalities Note Sheet
Student Notes & Practice Algebra Foundations
Graphing Inequalities
Name:
Date: Period:
Core Concepts & Graphing Rules
1. What is an Inequality?
A mathematical sentence comparing values using inequality symbols rather than an equals sign.
The Graph: Displays all solutions that make the statement true (infinite numbers).
2. Circle vs. Solid Dot
○
Open Circle: \(<\) or \(>\)
Boundary is NOT included.
●
Solid Dot: \(\le\) or \(\ge\)
Boundary IS included ("or equal to").
3. Arrow Direction
Variable on Left (e.g. \(x > a\)):
Symbol points the way:
\(>\) or \(\ge\) : Shade Right → \(<\) or \(\le\) : Shade ← Left
Variable on Right: Read from the variable or flip: \(3 < x \iff x > 3\).
Part 1: Graphing with Variable on the Left Plot the boundary point (circle or dot) and draw the solution ray
- \(x > 3\)
0 1 2 3 4 5 6
- \(x \le -1\)
-4 -3 -2 -1 0 1 2
- \(x < 5\)
2 3 4 5 6 7 8
- \(x \ge 0\)
-3 -2 -1 0 1 2 3
- \(x \le 2\)
-1 0 1 2 3 4 5
- \(x > -4\)
-7 -6 -5 -4 -3 -2 -1
- \(x \ge 6\)
3 4 5 6 7 8 9
- \(x < -2\)
-5 -4 -3 -2 -1 0 1
- \(x > 1\)
-2 -1 0 1 2 3 4
- \(x \le -3\)
-6 -5 -4 -3 -2 -1 0
- \(x \ge 4\)
1 2 3 4 5 6 7
- \(x < 7\)
4 5 6 7 8 9 10
Part 2: Variable on the Right Remember: read from the variable's perspective
- \(2 < x\)
-1 0 1 2 3 4 5
- \(-1 \ge x\)
-4 -3 -2 -1 0 1 2
- \(5 \le x\)
2 3 4 5 6 7 8
- \(0 > x\)
-3 -2 -1 0 1 2 3
Self-Check Verification:
1. Boundary: Open circle for \(<, >\) vs. Solid dot for \(\le, \ge\) | 2. Test Point: Pick a shaded value to confirm it makes the inequality true
