Which Inequality Is Shown In The Graph Below

The graph illustrates a linear inequality, and understanding how to read it helps answer the question which inequality is shown in the graph below. By examining the boundary line, the direction of the shading, and the type of line used, you can pinpoint the exact inequality represented. This article walks you through the key visual cues, explains the underlying mathematics, and provides a clear, step‑by‑step method for identifying the correct inequality from any plotted graph.

Understanding Graphs of Inequalities

When a linear inequality is graphed, the solution set is depicted as a region on the coordinate plane. The boundary of this region is a straight line that corresponds to the related equation. The line may be solid or dashed, and the shading indicates which side of the line satisfies the inequality. Recognizing these elements is essential for answering questions like which inequality is shown in the graph below.

Key Visual Elements

• Solid line – indicates that the boundary points are included in the solution (≤ or ≥).
• Dashed line – indicates that the boundary points are not included ( < or >).
• Shading direction – points toward the half‑plane that contains all solutions.
• Intercept form – the slope and y‑intercept of the line reveal the equation’s coefficients.

Identifying the Boundary Line

To determine the inequality, start by writing the equation of the boundary line in slope‑intercept form: y = mx + b.

1. Find the slope (m) by counting the rise over run between two clear points on the line.
2. Determine the y‑intercept (b) where the line crosses the y‑axis.
3. Write the equation using the identified slope and intercept.

Example: If the line passes through (0, 2) and (2, 4), the slope is 1 and the y‑intercept is 2, giving the equation y = x + 2.

Interpreting the Shading Region

The shading tells you whether the inequality is “greater than” or “less than” the boundary line.

• Above the line (y‑values larger than the line’s y‑value) → y > mx + b or y ≥ mx + b.
• Below the line (y‑values smaller than the line’s y‑value) → y < mx + b or y ≤ mx + b.

The type of line (solid vs. dashed) decides whether the equality sign is included.

Common Forms of Linear Inequalities

Understanding the standard forms helps you match the graph to the correct inequality quickly.

Step‑by‑Step Guide to Determine the Inequality

1. Locate two distinct points on the boundary line.
2. Calculate the slope (rise/run) and write the equation y = mx + b.
3. Check the line style: solid → include equality; dashed → exclude equality.
4. Pick a test point not on the line (commonly the origin (0, 0) if it isn’t on the line).
5. Substitute the test point into the equation to see whether it satisfies the inequality.
6. Determine the direction of shading based on whether the test point’s coordinates make the inequality true.
7. Write the final inequality using the appropriate symbols (≤, ≥, <, >) and the correct variable side.

Quick Checklist

• Solid vs. dashed? → decides ≤/≥ vs. </.
• Shading above or below? → decides >/≥ vs. </≤.
• Is the test point inside the shaded region? → confirms the sign.

Example Analysis

Suppose the graph shows a line passing through (0, ‑1) and (3, 2) with a solid line, and the region below the line is shaded.

1. Slope: (2 ‑ (‑1)) / (3 ‑ 0) = 3/3 = 1 → m = 1.
2. Y‑intercept: b = ‑1 → equation y = x – 1.
3. Line style: solid → includes equality (≥ or ≤).
4. Test point: (0, 0) lies in the shaded region. Substitute: 0 ? (‑1) → 0 > ‑1, so the inequality is y > x – 1.
5. Because the line is solid, the correct form is y ≥ x – 1 if the boundary is included; however, the shading is strictly below, so the inequality is y ≤ x – 1? Wait, we need consistency. Actually, if the shading is below and the test point (0,0) satisfies 0 ≤ (0) – 1? That's false. So the correct inequality is y < x – 1? Let's re‑evaluate: The line is solid, shading below, test point (0,0) is above the line? Actually line at x=0 gives y = -1, so point (0,0) is above the line, not below. If shading is below, then (0,0) is not in the shaded region; a point like (0,‑2) would be. So the inequality is y ≤ x – 1 (solid line, includes equality). This example illustrates the importance of careful testing.

Frequently Asked Questions

Q: How do I know whether to use ≤ or <?
A: Look at the line style. A solid line means the boundary is part

Q: How do I know whether to use ≤ or <?
A: Look at the line style. A solid line means the boundary is part of the solution set, so the inequality includes equality (≤ or ≥). A dashed line indicates the boundary itself is not satisfied, so the inequality is strict (< or >).

Q: What if the shaded region is to the left of a vertical line?
A: For a vertical boundary x = c, shading to the left corresponds to x < c (dashed) or x ≤ c (solid). Shading to the right yields x > c or x ≥ c, respectively. The same test‑point method works: pick a point with an x‑coordinate clearly on one side (e.g., the origin if c ≠ 0) and substitute it into the tentative inequality.

Q: Can I use a test point that lies on the axis but not on the line? A: Yes, any point not on the boundary line is valid. The origin (0, 0) is convenient unless the line passes through it; in that case choose another simple point such as (1, 0) or (0, 1) and verify that it is not on the line before substituting.

Q: What should I do if the line is horizontal?
A: A horizontal line has the form y = k. If the shading is above the line, the inequality is y > k (dashed) or y ≥ k (solid). If the shading is below, it is y < k or y ≤ k. Again, the line style dictates whether the equality symbol is included.

Q: How can I avoid mixing up the direction of the inequality when the slope is negative?
A: Focus on the shading relative to the line, not the sign of the slope. After you have the equation y = mx + b, ask: “Does increasing y move you into the shaded region?” If yes, use > or ≥; if no, use < or ≤. The test‑point step removes any ambiguity caused by a negative slope.

Q: Is there a shortcut for graphs that already show the inequality in words?
A: Some textbooks label the shaded region directly (e.g., “y ≥ 2x − 3”). In such cases, simply verify that the line style matches the symbol (solid for ≥/≤, dashed for >/<) and that the shading direction agrees with the inequality. If anything mismatches, re‑examine the graph for possible mis‑labeling.

Common Pitfalls to Watch For

1. Assuming the test point must be the origin.
   If the boundary passes through (0, 0), the origin gives no information; choose another point.

2. Confusing “above” with “greater than” when the line is not vertical.
   “Above” always corresponds to larger y‑values, regardless of slope.

3. Overlooking the line style when writing the final answer.
   A dashed line never yields ≤ or ≥; a solid line never yields < or >.

4. Misreading the scale on the axes.
   Ensure you use the same unit length for both axes when computing slope; otherwise the slope will be incorrect.

5. Forgetting to simplify the inequality. After substituting the test point, reduce any fractions or combine like terms before deciding on the final symbol.

Conclusion

Translating a graphed region into an algebraic inequality is a systematic process: identify the boundary line, determine its equation, note whether the line is solid or dashed, select a reliable test point, and use the test point’s outcome to choose the correct inequality symbol. By consistently applying the checklist—solid/dashed, shading direction, test‑point verification—and staying alert to common errors, you can confidently convert any linear‑inequality graph into its precise mathematical form. This skill not only aids in solving systems of inequalities but also strengthens the link between visual intuition and algebraic reasoning, a cornerstone of successful problem‑solving in mathematics.