Representing Inequalities as Regions

Strict Inequalities ($<$ or $>$ ): These are represented using a dashed (or dotted) line. This indicates that points exactly on the line are NOT part of the solution set.

Non-Strict Inequalities ($\le$ or $\ge$ ): These are represented using a solid line. This indicates that the boundary itself is included in the solution set.

Vertical and Horizontal Boundaries: Inequalities involving only $x$ (e.g., $x < k$) result in vertical boundaries, while those involving only $y$ (e.g., $y \ge k$) result in horizontal boundaries.

Step 1: Graph the Boundary: Treat the inequality as an equation and plot the line. Ensure the line style (solid or dashed) matches the inequality symbol.

Step 2: Select a Test Point: Choose a coordinate $(x, y)$ that is clearly not on the boundary line. The origin $(0, 0)$ is the most common choice because it simplifies calculations.

Step 3: Evaluate the Inequality: Substitute the test point coordinates into the original inequality. If the resulting statement is true, the region containing that point is the solution; if false, the region on the opposite side of the line is the solution.

Step 4: Shade the Region: Apply shading to the identified solution area to visually represent all possible $(x, y)$ pairs that satisfy the constraint.

Check the Inequality Sign: Always double-check if the sign requires a solid or dashed line; using the wrong line type is a frequent source of lost marks.

The Origin Test: If the boundary line passes through $(0,0)$, you cannot use it as a test point. Instead, pick a simple point like $(1,0)$ or $(0,1)$.

Negative Coefficients: Remember that if you rearrange an inequality and multiply or divide by a negative number, the inequality sign must be flipped (e.g., $-2y < 4$ becomes $y > -2$).

Verification: After shading, pick a second point deep within the shaded region and plug it into the original inequality to ensure it holds true.