Visualizing Solutions: Plotting Inequalities and Finding Regions

 

The Art of Plotting Inequalities

Inequalities can be represented on a graph as shaded regions that satisfy the inequality equation. These regions are bounded by lines or curves, the nature of which depends on whether the inequality is linear, quadratic, or of another form.

Steps to Graphing Inequalities

1. Graph the Boundary: Start by graphing the boundary line or curve as if the inequality were an equation (e.g., $y = mx + c$ for a linear inequality). If the inequality is strict ($<$ or $>$), use a dashed line. If it's inclusive ($\leq$ or $\geq$), use a solid line.
2. Shade the Region: Determine which side of the boundary to shade by choosing a test point not on the boundary (often, the origin $(0,0)$ is a convenient choice) and seeing if it satisfies the inequality. Shade the region that contains the solutions.

 

Plotting Inequalities

Example

Show, graphically, the region that is satisfied by all three inequalities below: $$4x + 3y \ge 15, \ y \lt 3x, \ x \lt 4$$

 

Step 1: Graph the boundaries

First draw each of the straight lines of the boundaries. This is found by converting all the inequalities to equals, sketching the line and making the line either solid (If the inequality is $\ge$ or $\le$) or dotted (If the inequality is $\gt$ or $\lt$ )

 

 

Step 2: Shade the Regions

Now using the inequalities shade the not wanted regions so all that is left is the region that satisties all three inequalities.