Graph The Following System of Inequalities Calculator

Graphing systems of inequalities helps visualize the solution regions for multiple linear inequalities. This calculator allows you to input up to three inequalities and see their graphical representation on a coordinate plane.

How to use this calculator

To graph a system of inequalities:

Enter each inequality in the format "y ≥ mx + b" or "y ≤ mx + b" in the input fields
Click "Calculate" to generate the graph
View the solution region where all inequalities are satisfied

The calculator will display the graph with the solution region shaded in green. The boundary lines will be shown in blue, and the intersection points will be marked with red dots.

How it works

This calculator solves systems of linear inequalities by:

Finding the intersection points of the boundary lines
Testing points in each region to determine which satisfy all inequalities
Shading the region that satisfies all conditions simultaneously

Key formulas

For two inequalities y ≥ m₁x + b₁ and y ≤ m₂x + b₂:

Intersection point: x = (b₂ - b₁)/(m₁ - m₂)
Solution region: The area where both inequalities are true

Note: The calculator currently supports up to three inequalities. For more complex systems, consider using graphing software.

Worked example

Let's graph the system:

y ≥ 2x + 1
y ≤ -x + 4

Step 1: Find the intersection point

Set 2x + 1 = -x + 4 → 3x = 3 → x = 1 → y = 3

Intersection at (1, 3)

Step 2: Test a point (0,0)

0 ≥ 2(0) + 1? No
0 ≤ -0 + 4? Yes

This point doesn't satisfy both inequalities

Step 3: Shade the region where both inequalities are true

The solution is the area above the first line and below the second line, bounded by their intersection.

FAQ

How many inequalities can I graph at once?

This calculator supports up to three inequalities simultaneously. For more complex systems, consider using specialized graphing software.

What if the inequalities don't intersect?

If the boundary lines are parallel, the system will either have no solution (if they're the same line) or an infinite solution region (if they're different lines).

Can I graph inequalities with absolute value?

Currently, this calculator only supports linear inequalities in the format y ≥ mx + b. For absolute value inequalities, you may need to break them into cases.