Introduction
Understanding which system of inequalities is represented by the graph is a fundamental skill in algebra and coordinate geometry. When a graph displays one or more shaded regions bounded by lines, each line corresponds to an inequality, and the overlapping (or separate) regions describe the solution set of the system. This article walks you through a step‑by‑step process to decode any such graph, explains the underlying concepts, and provides a concrete example to solidify your comprehension. By the end, you will be able to look at a visual representation and confidently write the corresponding system of inequalities.
Steps to Identify the System of Inequalities
-
Observe the Boundary Lines
- Solid lines indicate that the boundary is included in the solution (i.e., “≤” or “≥”).
- Dashed (or broken) lines mean the boundary is excluded (“<” or “>”).
Italic tip: the style of the line tells you whether the inequality is strict or non‑strict.
-
Determine the Equation of Each Line
- Choose two points on the line (or use the slope‑intercept form).
- Calculate the slope m = (y₂‑y₁)/(x₂‑x₁).
- Use the point‑slope formula y‑y₁ = m(x‑x₁) to rewrite the equation in standard form Ax + By = C or slope‑intercept form y = mx + b.
-
Convert the Equation to an Inequality
- If the shaded region is above the line, the inequality is y ≥ mx + b (or y > mx + b for a dashed line).
- If the shaded region is below the line, the inequality is y ≤ mx + b (or y < mx + b).
- For horizontal lines (y = k), the inequality is y ≥ k (above) or y ≤ k (below).
- For vertical lines (x = k), the inequality is x ≥ k (right) or x ≤ k (left).
-
Check a Test Point (Optional but Helpful)
- Pick a point that is clearly inside the shaded region (often the origin (0,0) if it lies there).
- Substitute the coordinates into the derived inequality; if the statement holds true, the inequality direction is correct.
-
Write the Complete System
- List all inequalities together, preserving the correct symbols (≤, ≥, <, >).
- confirm that each inequality corresponds to a distinct boundary line in the graph.
Analyzing the Graph
- Intersection Points: The point(s) where two lines cross are solutions that satisfy both equations as equalities. These points are useful for verifying the system.
- Feasible Region: The area that satisfies all inequalities simultaneously is called the feasible region. Its shape (triangle, quadrilateral, unbounded strip, etc.) gives clues about the nature of the system.
- Consistency: If the shaded regions do not overlap, the system has no solution (inconsistent). If they overlap, the system is consistent, and the solution set may be a line segment, a point, or an area.
Common Scenarios
| Scenario | Description | Typical Graph Features |
|---|---|---|
| Two intersecting lines | Each line represents a different inequality; the feasible region is often a convex polygon. | One solid line, one dashed line; shading on opposite sides. On the flip side, |
| Parallel lines | Inequalities may be y ≤ mx + b and y ≥ mx + c (or the reverse). | Parallel solid or dashed lines; feasible region is a strip. And |
| Identical lines | The same line appears twice with different inequality symbols. | Overlapping solid/dashed lines; feasible region may be the line itself or the area on one side. |
| Non‑linear boundaries | Curved lines (e.g., circles, parabolas) represent non‑linear inequalities. | Shading inside/outside the curve; test points help determine direction. |
This is the bit that actually matters in practice.
Example Problem
Graph Description: The graph shows two lines.
- Line A: passes through (0, 2) and (2, 0); it is drawn solid.
- Line B: passes through (0, ‑1) and (1, 1); it is drawn dashed.
- The region below Line A and above Line B is shaded.
Step‑by‑Step Solution
-
Line A Equation:
- Slope m = (0‑2)/(2‑0) = –1.
- Using point (0, 2): y – 2 = –1(x – 0) ⇒ y = –x + 2.
- Since the shaded area is below this line and the line is solid, the inequality is y ≤ –x + 2.
-
Line B Equation:
- Slope m = (1‑(‑1))/(1‑0) = 2.
- Using point (0, ‑1): y + 1 = 2(x – 0) ⇒ y = 2x – 1.
- The shaded region is above this line and the line is dashed, so the inequality is y > 2x – 1.
-
System of Inequalities:
[ \begin{cases} y \leq -x + 2 \ y > 2x - 1 \end{cases} ]
-
Verification: Choose a point in the shaded region, e.g., (0, 0).
- 0 ≤ –0 + 2 ✔️
- 0 > 2·0 – 1 ⇒ 0 > –1 ✔️
The point satisfies both inequalities, confirming the system.
Frequently Asked Questions (FAQ)
-
Q1: What if the graph shows a curved boundary?
A: Treat the curve as the boundary of a non‑linear inequality. Determine whether the feasible region lies inside or outside the curve, then write the appropriate symbol (≤, ≥, <, >). -
Q2: How do I know if the system has infinitely many solutions?
*
