Finding Ordered Pair Solutions For Linear Inequality Systems

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In mathematics, a system of linear inequalities is a set of two or more linear inequalities that involve the same variables. The solution set of a system of linear inequalities is the region in the coordinate plane that satisfies all the inequalities in the system simultaneously. This article delves into the process of identifying ordered pairs that belong to the solution set of a given system of linear inequalities, offering a comprehensive guide for students and enthusiasts alike.

Understanding Linear Inequalities

Before we dive into solving systems, let's first grasp the concept of a single linear inequality. A linear inequality is similar to a linear equation, but instead of an equals sign (=), it uses an inequality symbol such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). Graphically, a linear inequality represents a region in the coordinate plane, bounded by a line. The line itself is included in the solution if the inequality includes "equal to" (≥ or ≤), and it is excluded if the inequality is strict (> or <).

The line that bounds the region is determined by the equation formed when the inequality symbol is replaced with an equals sign. For instance, the inequality y ≥ -1/2x corresponds to the line y = -1/2x. This line acts as a boundary, and the solution to the inequality lies on one side of this line. To determine which side, we can test a point not on the line. If the point satisfies the inequality, then the region containing that point is the solution region. If it doesn't, the other side is the solution.

For example, consider the inequality y ≥ -1/2x. Let's test the point (0,1). Substituting these values into the inequality, we get 1 ≥ -1/2(0), which simplifies to 1 ≥ 0. This is true, so the region above the line y = -1/2x (including the line itself) represents the solution to the inequality.

Similarly, for the inequality y < 1/2x + 1, the boundary line is y = 1/2x + 1. This time, let's test the point (0,0). Substituting, we get 0 < 1/2(0) + 1, which simplifies to 0 < 1. This is also true, so the region below the line y = 1/2x + 1 (excluding the line itself) is the solution region.

Solving Systems of Linear Inequalities

A system of linear inequalities consists of two or more linear inequalities considered together. The solution set of a system is the region where the solutions of all the individual inequalities overlap. This overlapping region represents all the points (ordered pairs) that satisfy every inequality in the system.

To find the solution set graphically, you would graph each inequality on the same coordinate plane. The region where the shaded areas (representing the solutions of each inequality) overlap is the solution set of the system. This region may be bounded (a closed shape) or unbounded (extending infinitely in one or more directions).

However, without graphing, we can determine if a given ordered pair is a solution to the system by substituting the coordinates of the point into each inequality. If the point satisfies all inequalities, then it is a solution to the system. If it fails to satisfy even one inequality, it is not a solution.

Determining Ordered Pairs in the Solution Set: A Step-by-Step Approach

Let's apply this knowledge to the given system of linear inequalities:

  1. y ≥ -1/2x
  2. y < 1/2x + 1

We are given a set of ordered pairs and need to determine which ones are in the solution set. We will test each ordered pair in both inequalities.

Testing the Ordered Pair (5, -2)

  • For inequality 1: Substitute x = 5 and y = -2 into y ≥ -1/2x.

    -2 ≥ -1/2(5)

    -2 ≥ -2.5

    This is true, so (5, -2) satisfies the first inequality.

  • For inequality 2: Substitute x = 5 and y = -2 into y < 1/2x + 1.

    -2 < 1/2(5) + 1

    -2 < 2.5 + 1

    -2 < 3.5

    This is also true, so (5, -2) satisfies the second inequality.

Since (5, -2) satisfies both inequalities, it is a solution to the system.

Testing the Ordered Pair (3, 1)

  • For inequality 1: Substitute x = 3 and y = 1 into y ≥ -1/2x.

    1 ≥ -1/2(3)

    1 ≥ -1.5

    This is true, so (3, 1) satisfies the first inequality.

  • For inequality 2: Substitute x = 3 and y = 1 into y < 1/2x + 1.

    1 < 1/2(3) + 1

    1 < 1.5 + 1

    1 < 2.5

    This is also true, so (3, 1) satisfies the second inequality.

Since (3, 1) satisfies both inequalities, it is a solution to the system.

Testing the Ordered Pair (-4, 2)

  • For inequality 1: Substitute x = -4 and y = 2 into y ≥ -1/2x.

    2 ≥ -1/2(-4)

    2 ≥ 2

    This is true, so (-4, 2) satisfies the first inequality.

  • For inequality 2: Substitute x = -4 and y = 2 into y < 1/2x + 1.

    2 < 1/2(-4) + 1

    2 < -2 + 1

    2 < -1

    This is false, so (-4, 2) does not satisfy the second inequality.

Since (-4, 2) does not satisfy both inequalities, it is not a solution to the system.

Testing the Ordered Pair (3, -1)

  • For inequality 1: Substitute x = 3 and y = -1 into y ≥ -1/2x.

    -1 ≥ -1/2(3)

    -1 ≥ -1.5

    This is true, so (3, -1) satisfies the first inequality.

  • For inequality 2: Substitute x = 3 and y = -1 into y < 1/2x + 1.

    -1 < 1/2(3) + 1

    -1 < 1.5 + 1

    -1 < 2.5

    This is also true, so (3, -1) satisfies the second inequality.

Since (3, -1) satisfies both inequalities, it is a solution to the system.

Testing the Ordered Pair (4, -3)

  • For inequality 1: Substitute x = 4 and y = -3 into y ≥ -1/2x.

    -3 ≥ -1/2(4)

    -3 ≥ -2

    This is false, so (4, -3) does not satisfy the first inequality.

Since (4, -3) does not satisfy the first inequality, it is not a solution to the system.

Conclusion: Identifying the Solution Set

By systematically testing each ordered pair, we can determine which ones belong to the solution set of the system of linear inequalities. In this case, the ordered pairs (5, -2) and (3, 1) satisfy both inequalities, making them solutions to the system. The ordered pairs (-4, 2) and (4, -3) were found not to be solutions, as they did not satisfy both inequalities. The ordered pair (3, -1) satisfies both inequalities and is also a solution to the system.

Understanding how to solve systems of linear inequalities is crucial in various fields, including economics, engineering, and computer science. This method of substituting ordered pairs is a fundamental technique that provides a solid foundation for more advanced mathematical concepts. Mastering this skill allows for effective problem-solving and a deeper understanding of linear relationships and their graphical representations.

In summary, to determine if an ordered pair is a solution to a system of linear inequalities, substitute the coordinates into each inequality. If the ordered pair satisfies all inequalities, it is part of the solution set. If it fails even one inequality, it is not a solution. This methodical approach ensures accurate identification of solutions and a comprehensive understanding of the system's behavior.