Algebra
Graphing Linear Inequalities – Shading a Region
While a linear equation graphs as a single line, a linear inequality like y > mx + c graphs as an entire shaded region – every point in that region makes the inequality true. The boundary line is drawn first (using the equation y = mx + c), and then one side of it is shaded to represent all the solutions.
The boundary line itself is drawn dashed for a strict inequality (> or <), since points exactly on the line don't satisfy the inequality, or solid for a “greater than or equal to”/“less than or equal to” inequality (≥ or ≤), since the line's points are included in the solution.
Inequalities as Regions
A simple inequality like x > 3 lives on a number line: it's every point to the right of 3, not including 3 itself (shown with an open circle, since 3 is not part of the solution).
On a coordinate plane, the same idea works with two variables. y > 2x + 1 is every point above the dashed boundary line y = 2x + 1 (dashed, again because the line itself isn't included).
The dashed line splits the plane into two halves. The point (0, 5) sits inside the shaded half, confirming it's a solution — matching the worked check below.
Graphing an Inequality
Draw the boundary line y = mx + c (dashed for strict inequalities, solid for ≥/≤). Test a point not on the line to decide which side to shade.
At x = 0, the boundary line gives y = 1. Since 5 > 1, the point is above the line, which matches y > 2x + 1: yes, it's in the region.
Real-Life Application
- Budget planning: shading the region of affordable combinations of two purchases.
- Manufacturing constraints: shading the feasible region for production given resource limits.
- Scheduling: shading the valid combinations of time spent on two tasks.
Key Takeaways
- A linear inequality graphs as a shaded region, bounded by a line.
- Strict inequalities (>, <) use a dashed line; ≥/≤ use a solid line.
- Test a point to determine which side of the line to shade.