Which Point Is A Solution To The System Of Inequalities

Introduction

When students first encounter systems of inequalities, the question that often pops up is: which point is a solution to the system of inequalities? This query is more than a simple homework prompt—it is the gateway to understanding how multiple constraints interact on a coordinate plane. In this article we will unpack the concept, walk through a logical process for identifying valid points, and illustrate why mastering this skill matters in both academic and real‑world contexts. By the end, you will feel confident spotting the correct solution point and explaining the reasoning behind it.

Detailed Explanation

A system of inequalities consists of two or more inequality statements that involve the same set of variables. For example,

[ \begin{cases} y \ge 2x - 1 \ y < -x + 4 \end{cases} ]

represents a region where every point must satisfy both conditions simultaneously. The solution set is the intersection of the individual regions, typically shown as a shaded area on a graph.

Understanding which point is a solution to the system of inequalities requires recognizing three key ideas:

1. Inequality direction – “≥”, “≤”, “>”, “<” dictate whether the boundary line is included.
2. Region selection – Each inequality carves out a half‑plane; the solution is where these half‑planes overlap.
3. Verification – A candidate point must be tested against every inequality to confirm it belongs to the overlapping region.

For beginners, the safest approach is to treat the problem as a point‑checking exercise: plug the coordinates into each inequality and see if all statements hold true. If they do, the point is part of the solution set; if any fails, it is excluded.

Step‑by‑Step or Concept Breakdown

Below is a clear, logical flow that you can follow whenever you need to determine which point is a solution to the system of inequalities.

1. Write down each inequality clearly.

   • Example: (y \ge 2x - 1) and (y < -x + 4).

2. Identify the boundary lines.

   • Convert each inequality to an equation (replace “≥” or “<” with “=”).
   • For the example: (y = 2x - 1) and (y = -x + 4).

3. Graph the boundary lines (optional but helpful).

   • Use solid lines for “≥” or “≤” (boundary included).
   • Use dashed lines for “>” or “<” (boundary excluded).

4. Shade the appropriate side of each line.

   • For (y \ge 2x - 1), shade above the line.
   • For (y < -x + 4), shade below the line.

5. Locate the intersection of the shaded regions.

   • The overlapping area is the solution set.

6. Pick a candidate point (often an integer coordinate for simplicity).

7. Test the point against every original inequality.

   • Substitute the (x) and (y) values into each inequality.
   • If all inequalities are satisfied, the point is a solution.

8. Conclude which point is a solution to the system of inequalities based on the test results.

Bullet‑point summary:

• Write the system.
• Convert to equations for boundaries.
• Graph (solid/ dashed).
• Shade each half‑plane.
• Find the overlapping region.
• Test candidate points.
• Confirm the point satisfies every inequality.

Real Examples

Let’s apply the step‑by‑step method to a concrete example and see which point is a solution to the system of inequalities.

Example 1

Consider the system

[ \begin{cases} y \ge x + 2 \ y \le 5 - x \end{cases} ]

We want to know which point is a solution to the system of inequalities among the following candidates: ((1,3)), ((2,4)), ((3,6)).

Testing ((1,3)):

• (3 \ge 1 + 2 \Rightarrow 3 \ge 3) ✔️
• (3 \le 5 - 1 \Rightarrow 3 \le 4) ✔️

Both conditions hold, so ((1,3)) is a solution.

Testing ((2,4)):

• (4 \ge 2 + 2 \Rightarrow 4 \ge 4) ✔️
• (4 \le 5 - 2 \Rightarrow 4 \le 3) ❌

Fails the second inequality; therefore, ((2,4)) is not a solution.

Testing ((3,6)):

• (6 \ge 3 + 2 \Rightarrow 6 \ge 5) ✔️
• (6 \le 5 - 3 \Rightarrow 6 \le 2) ❌

Again, the second inequality fails, so ((3,6)) is excluded.

Thus, the only point that satisfies both inequalities is ((1,3)). This illustrates how systematic testing reveals which point is a solution to the system of inequalities.

Example 2 – Non‑integer Point

Sometimes the solution set contains points with fractional coordinates. Suppose we have

[ \begin{cases} y > 2x - 1 \ y < -x + 5 \end{cases} ]

Pick the point ((\frac{3}{2}, 2)). Substituting:

• (2 > 2(\frac{3}{2}) - 1 \Rightarrow 2 > 3 - 1 \Rightarrow 2 > 2) ❌ (strict inequality fails).

Now try ((\frac{3}{2}, 2.1)):

• (2.1 > 2(\frac{3}{2}) - 1 \Rightarrow 2.1 > 2) ✔️
• (2.1 < -\frac{3}{2} + 5 \Rightarrow 2.1 < 3.5) ✔️

Both hold, confirming that ((\frac{3}{2}, 2.1)) is a solution even though it is not an integer. This demonstrates that which point is a solution to the system of inequalities can be any coordinate that meets every condition, not just whole numbers.

Scientific or Theoretical Perspective

From a theoretical standpoint, a system of inequalities defines a convex polyhedron (in two dimensions, a convex polygon or unbounded region) in the variable space. The set of all solutions is the intersection of half‑spaces, each half‑space being the region that satisfies a single inequality.

This approach to solving systems of inequalities is a fundamental tool in mathematics, with applications extending far beyond simple exercises. It provides a clear, step-by-step methodology for analyzing and determining the feasible region. This region, defined by the intersection of all the half-spaces, represents the set of all possible solutions to the system. Understanding this intersection is crucial in various fields, including economics, engineering, and computer science. For example, in economics, resource allocation problems often involve maximizing profit subject to constraints defined by inequalities representing available resources. In engineering, designing structures requires satisfying constraints related to material strength and stability. The ability to systematically identify valid solutions is paramount in these applications.

Furthermore, the concept of convex polyhedra has significant implications in optimization. Many optimization problems aim to find the maximum or minimum value of a function within a defined region. The feasible region, determined by the system of inequalities, becomes the domain of the optimization problem. Algorithms used to solve these optimization problems often rely on the properties of convex sets, making this mathematical framework particularly powerful.

In conclusion, understanding how to solve systems of inequalities through testing and graphical representation is not merely a computational skill. It's a foundational concept that illuminates the relationship between mathematical inequalities and real-world constraints. By systematically identifying the feasible region, we gain insights into the possible solutions and can apply this knowledge to solve a wide range of problems across diverse disciplines. The ability to confidently test points and interpret graphical representations empowers individuals to analyze and model complex scenarios, ultimately leading to more informed and effective decision-making.