Which Inequality Is Represented By This Graph

Which Inequality Is Represented by This Graph?

The question which inequality is represented by this graph often appears in algebra classrooms when students encounter shaded regions on a coordinate plane. Recognizing the correct inequality from a visual cue requires understanding how boundary lines, shading direction, and line style interact to convey mathematical relationships. This article walks you through the essential concepts, step‑by‑step strategies, and common pitfalls so you can confidently decode any graph that poses the same query.

Introduction to Graphical Inequalities

When an inequality involves two variables, its solution set is not a single number but a region in the Cartesian plane. Instead of a point, the graph shades an area that satisfies the condition. The boundary of that area is usually a straight line, and the way the line is drawn—solid or dashed—indicates whether points on the line are included in the solution.

Solid line → the boundary points are part of the solution (≤ or ≥). Dashed line → the boundary points are not part of the solution (< or >).

The shading side tells you whether the inequality is satisfied for points above, below, left, or right of the line. Mastering these visual cues answers the core query which inequality is represented by this graph.

How to Read a Graphical Inequality

Identify the Boundary Line 1. Write the equation of the line in slope‑intercept form (y = mx + b) or standard form (Ax + By = C).

Plot key points such as the y‑intercept and another point using the slope.

Determine Line Style

If the line is solid, the inequality uses ≤ or ≥.
If the line is dashed, the inequality uses < or >.

Test a Point to Find the Shading Direction

Choose a simple test point not on the line (commonly (0,0) if it isn’t on the line). 2. Substitute the coordinates into the original inequality.
If the statement is true, shade the side containing that point; otherwise, shade the opposite side.

Match the Graph to an Inequality

After completing the steps above, you can write the inequality that exactly matches the observed graph. This process directly addresses the question which inequality is represented by this graph.

Common Types of Inequalities and Their Graphs

Inequality Type	Boundary Line	Shading Region	Typical Notation
y ≤ mx + b	Solid line	Below the line	“y is less than or equal to …”
y ≥ mx + b	Solid line	Above the line	“y is greater than or equal to …”
y < mx + b	Dashed line	Below the line	“y is less than …”
y > mx + b	Dashed line	Above the line	“y is greater than …”
x ≤ c	Vertical line	Left of the line	“x is less than or equal to …”
x ≥ c	Vertical line	Right of the line	“x is greater than or equal to …”
x < c	Dashed vertical	Left of the line	“x is less than …”
x > c	Dashed vertical	Right of the line	“x is greater than …”

Understanding these patterns helps you quickly answer which inequality is represented by this graph without re‑deriving the entire solution set each time.

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Step‑by‑Step Example

Suppose you are presented with the following graph:

A solid line passes through the points (0, 2) and (4, 0).
The region below the line is shaded.

Step 1 – Find the equation.
The slope (m = \frac{0-2}{4-0} = -\frac{1}{2}).
Using the y‑intercept (0, 2), the equation is (y = -\frac{1}{2}x + 2).

Step 2 – Determine line style.
The line is solid, so the inequality includes the boundary: ≤ or ≥.

Step 3 – Choose a test point.
The origin (0, 0) is not on the line. Substitute into (y \le -\frac{1}{2}x + 2): (0 \le -\frac{1}{2}(0) + 2 \Rightarrow 0 \le 2) (true). Since the test point satisfies the inequality and the shading is below the line, the correct inequality is (y \le -\frac{1}{2}x + 2).

This concrete walkthrough illustrates how to answer which inequality is represented by this graph in a systematic way.

Frequently Asked Questions

Q1: What does a dashed line imply?
A dashed line means the boundary points are excluded; the inequality uses strict symbols (< or >).

Q2: Can the shading be on both sides of a line?
No. A single inequality produces shading on one side of the boundary line. If both sides are shaded, the graph represents a system of inequalities, not a single one.

Q3: How do I handle vertical lines?
Vertical boundaries are expressed as (x = c). The inequality will be (x \le c) (solid) or (x < c) (dashed) for shading to the left, and (x \ge c) or (x > c) for shading to the right.

Q4: Why is the test point (0,0) sometimes invalid?
If the line passes through the origin, substituting (0,0) yields a true statement that does not help determine the shading side. In such cases, pick another point like (1,0) or (0,1) that is not on the line.

Q5: Does the slope affect the inequality direction?
The slope determines the orientation of the line but does not change the inequality symbols. Only the shading side relative to the line matters.

Conclusion

Decoding a graph to answer which inequality is represented by this graph hinges on three visual clues: line style, shading direction, and the underlying equation. By systematically determining the boundary line, interpreting the line’s solid or dashed nature, and testing a