Problem 52 - College Algebra

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&2 x+y \leq 6\\\&x+y \geq 2\\\&1 \leq x \leq 2\\\&y \leq 3\end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the region of the graph that is bounded by x=1 and x=2 on the left and right, by the line \(2x+y=6\) above, and by the line \(x+y=2\) below.

Step 1: Understand each inequality

Ensure to understand what each inequality represents. The constraints are: \(2x+y \leq 6\), \(x+y \geq 2\), \(1 \leq x \leq 2\), and \(y \leq 3\). These inequalities describe linear relationships between \(x\) and \(y\), and constraints for \(x\) and \(y\).

Step 2: Graph each inequality

Graph each of the inequalities on the same set of axes. The strategy is to first draw the line of the equivalent equation (e.g., for \(2x+y \leq 6\), draw \(2x + y = 6\) line). Afterward, choose which side of the line to shade. If the inequality is less or equal to (as in \(2x+y \leq 6\)), shade the area that includes the origin. But if it is greater or equal to (as in \(x+y \geq 2\)), shade the opposite side.

Step 3: Identify the solution set

The solution to the system of inequalities is the region where the shading for all inequalities overlaps. In this example, that would be the trapezoid bounded by x=1 and x=2 on the left and right, by \(2x+y=6\) above, and by \(x+y=2\) below.

Key Concepts

Linear Inequalities

When exploring linear inequalities, we essentially examine expressions in which two linear functions or values are compared using inequality symbols such as <, >, ≤, or ≥. For instance, in the inequality, 2x+y ≤ 6, '2x+y' represents the value of a linear function, and the ≤ symbol implies that this value is limited to being less than or equal to 6.

Linear inequalities can be used to model real-world constraints, such as budget limitations, space availability, or any situation where there is a range of possible solutions but with certain restrictions in place. Drawing a graph to represent these inequalities gives us a visual understanding of all possible solutions that satisfy the conditions imposed by them. Understanding the inequalities individually is the first step in solving a system involving multiple constraints. This includes analyzing the direction of the inequality, which can affect the shading direction during graphing.

Solution Set

The concept of a solution set is central to understanding systems of linear inequalities. A solution set represents the set of all possible values that satisfy all of the given inequalities in a system. When graphing the inequalities on the same set of axes, the solution set becomes the region where the shaded areas overlap.

For a single inequality, this could simply be the area under a line on a graph. However, in a system of inequalities, all conditions must be met simultaneously, which often leads to an intersection of shaded areas. In our exercise, the overlapping region is a trapezoidal shape, marking the feasible solutions that adhere to all four inequalities. It is important for students to learn how to identify this region, as it visually captures all the possible combinations of variables that are solutions to the system.

Inequality Shading

Inequality shading is a visual tool used in graphing to illustrate the range of possible solutions to an inequality on a Cartesian plane. It's important to know which side to shade since this is determined by the direction of the inequality symbol. If the symbol points to the variable (e.g., y ≥ x + 1), the area above the line will be shaded. Conversely, if the symbol points away from the variable (e.g., y ≤ x + 1), the area below the line will be shaded.

To ensure correct shading, a common technique is to test a point not on the boundary line—(0,0), the origin, is often a convenient choice. If the origin satisfies the inequality, then the area including the origin is shaded. Paying careful attention during shading helps to accurately visualize where the solution sets of different inequalities intersect.

Algebraic Constraints

Algebraic constraints refer to the mathematically imposed limits on the values that variables can assume in problems involving inequalities. In the given exercise, we encounter constraints like 1 ≤ x ≤ 2, which means that the value of x is restricted to being between 1 and 2, inclusive. These constraints may be compound, as in the given example, or simple, such as y ≤ 3.

The presence of algebraic constraints requires the solver to consider the permissible region of solutions while keeping these numerical bounds in mind. This leads to the development of a bounded region on a graph, and only the values within this bounded area are admissible. Recognizing these constraints is crucial, as they are foundational to reaching the correct solution set for the system.

Graphical Representation

Graphical representation is an essential step in understanding and solving systems of inequalities. It involves plotting each individual inequality on a coordinate plane and identifying the solution set by the area where the shaded regions intersect. The graph provides a clear and instant visual answer to the problem, allowing students to instantly recognize the range of permissible solutions.

In practice, students will draw boundary lines, perform shading for each inequality, and look for common shaded regions that satisfy the entire system. It is this graphical representation that brings together all the abstract algebraic concepts, translating them into a concrete visual form. It is a blend of art and mathematics that can significantly enhance a student's problem-solving skills.