What is a 'Weak Inequality'?

A weak inequality uses $\le$ or $\ge$ and includes the boundary line in its solution set. It is represented by a solid line on a graph.

What is the visual difference between the boundary of $y > x + 2$ and $y \ge x + 2$?

The boundary for $y > x + 2$ is a dotted line because it is a strict inequality, while the boundary for $y \ge x + 2$ is a solid line because it includes the points on the line.

When should you use the 'shading the unwanted area' technique?

This technique is best used when there are multiple inequalities (a system). By shading the areas that do not satisfy the conditions, the valid solution region $R$ remains clear and easy to identify.

How does the region for $y x^2$?

The region $y x^2$ consists of the area above/inside the 'U' shape of the parabola.

What is a common mistake when testing the point $(0,0)$ for a boundary line like $y = 2x$?

The mistake is attempting to use a test point that lies exactly on the boundary line. If the line passes through the origin, you must choose a different point, such as $(1,0)$ or $(0,1)$, to determine the shading side.

What happens to the inequality sign if you divide both sides by a negative number while rearranging for $y$?

The inequality sign must be flipped (e.g., $ $). Failing to flip the sign will result in shading the completely wrong side of the boundary line.

Why might a student lose marks even if they shade the correct area on a graph?

They may have used the wrong line type (solid instead of dotted) or failed to clearly label the required region as 'R' or 'unshaded' as per the question's specific instructions.

Define a 'Strict Inequality' in the context of graphing.

A strict inequality uses $ $ and defines a region that does not include its boundary. Visually, this is represented by a dotted line.

What does the region 'R' represent on a graph of multiple inequalities?

Region 'R' represents the feasible region, which is the set of all points that satisfy every inequality in the system simultaneously.

Why is coordinate testing necessary after drawing a boundary line?

The boundary line only shows where the equality happens; testing a point determines which of the two divided sections of the plane contains the values that satisfy the inequality.

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Inequalities on Graphs

Summary

Inequalities on graphs represent sets of points $(x, y)$ that satisfy one or more mathematical constraints. Unlike equations that produce lines or curves, inequalities define entire regions of the Cartesian plane, using specific visual conventions to distinguish between included and excluded boundaries.

1. Definition & Core Concepts

Graphical Inequalities: These are mathematical statements involving two variables, typically $x$ and $y$ , that describe a specific area or region on a coordinate grid.
The Solution Region: The set of all coordinates $(x, y)$ that make the inequality true is represented visually as a shaded (or unshaded) area, often labeled with the letter $R$ .
Boundary Lines: Every inequality is bounded by a line or curve derived from its corresponding equation (e.g., the boundary for $y < 2x$ is the line $y = 2x$ ).

A Cartesian graph showing a linear boundary line and a shaded region R representing the solution set of an inequality.

2. Boundary Line Conventions

3. The Testing Method

4. Shading Strategies

5. Key Distinctions

6. Exam Strategy & Tips