Systems of Inequalities
By the end of this lesson, you’ll be able to:
- Graph systems of inequalities in the coordinate plane.
- Identify the overlapping solution region.
- Use test points to confirm shading direction.
- Determine whether a boundary line is solid or dashed.
Key Ideas
A system of inequalities is simply two inequalities graphed together.
The solution is the area where both shadings overlap.
Boundary Lines
- \(\le\) or \(\ge\) → solid line
- \(<\) or \(>\) → dashed line
Shading Direction
- \(y > mx + b\) → shade above
- \(y < mx + b\) → shade below
Overlap
The intersection of shaded areas is the solution region.
Common Problem Types
1. Solid vs. Dashed Boundary Lines
Use a solid line for \(\le\) or \(\ge\) because points on the line are included.
Use a dashed line for \(<\) or \(>\) because boundary points are not included.
Example:
\(y < 3x - 1\) uses a dashed line.
2. Shading Direction
Shade the region representing all solutions to the inequality.
- Shade above the line for \(>\) or \(\ge\)
- Shade below the line for \(<\) or \(\le\)
Example:
\(y \ge -2x + 4\) → shade above the line.
3. Systems with Parallel Lines
If boundary lines are parallel:
- Their shading regions may overlap → solution region exists.
- Or they may never overlap → no solution.
Example:
If one inequality shades above a line and the other shades below a parallel line, the regions may not intersect.
4. Test-Point Verification
When unsure where to shade, plug a test point (often \((0,0)\) if available) into the inequality.
Example:
For \(y < x + 2\), test \((0,0)\):
\(0 < 2\) is true → shade the side containing \((0,0)\).
Worked Examples
Example 1
System: \[ \begin{cases} y \le x + 2 \\ y > -2x + 1 \end{cases} \]
- First → solid line, shade below
- Second → dashed line, shade above
Solution = overlapping region.
Example 2 — Test a Point
System: \[ \begin{cases} y < 3x + 5 \\ y \ge -x - 1 \end{cases} \]
Test \((0,0)\):
- \(0 < 5\) → ✓
- \(0 \ge -1\) → ✓
So \((0,0)\) is in the solution.
- Forgetting dashed vs solid lines.
- Shading the wrong direction.
- Not taking the overlap of regions.
Practice Problems
- Graph: \(y < 2x + 3\), \(y \ge -x + 1\)
- Does \((3,2)\) satisfy: \(y \ge x - 1\) and \(y < -2x + 5\)?
- Boundary type for \(y > -4x - 2\)?
1. Dashed line for \(<\), solid for \(\ge\). Shade below/above; overlap is solution.
2.
\(2 \ge 2\) → ✓
\(2 < -1\) → ✗
So not a solution.
3.
“>” → dashed line.
Summary
- Graph each inequality with correct line type.
- Shade for \(>\) above, for \(<\) below.
- The overlap is the solution region.
- Test points help confirm accuracy.
- \(>\) or \(<\) → dashed; \(\ge\) or \(\le\) → solid.
- Use \((0,0)\) as a quick shade-checking point.
- The solution area must satisfy all inequalities.