Problem 65 - Algebra and Trigonometry

Question

write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the \(x\) -variable and the \(y\) -variable is at most \(4 .\) The \(y\) -variable added to the product of 3 and the \(x\) -variable does not exceed 6

Step-by-Step Solution

Answer

The system of inequalitiesbased on the given sentences is \(x + y \leq 4\) and \(3x + y \leq 6\). The graph will consist of the two lines representing these inequalities and a shaded common region containing points that satisfy both inequalities.

Step 1: Translate the Sentences into Inequalities

Translate the sentences into mathematical inequalities. The first sentence says that the sum of the \(x\)-variable and the \(y\)-variable is at most 4. This translates to \(x + y \leq 4\). The second sentence states that the \(y\)-variable added to the product of 3 and the \(x\)-variable does not exceed 6, which can be written as \(3x + y \leq 6\). Thus, we have a system of inequalities.

Step 2: Graph the Inequalities

To graph the inequalities, start by drawing a graph. For the first inequality \(x + y \leq 4\), choose some values for \(x\), find corresponding \(y\)-values using the inequality, and plot these points. Solutions for this inequality will lie on the line and below it. Repeat the same process for the second inequality \(3x + y \leq 6\). Its solutions lie on the line and below as well. The common region that satisfies both inequalities represents the solution to the system.

Key Concepts

System of Inequalities

When we talk about a system of inequalities, we refer to a set of two or more inequalities that are related to each other by having common variables. For example, in a system that involves x and y as variables, the solution must satisfy all the inequalities in the system simultaneously.

Considering our exercise, we have two inequalities that form our system:

• \(x + y \leq 4\)
• \(3x + y \leq 6\)

The solutions to this system are ordered pairs (that is, pairs of x and y values) that make both inequalities true at the same time. This system can represent a range of real-world situations, such as determining the feasible region of production quantities for two products under certain constraints, or figuring out how to allocate resources within a budget limit.

Linear Inequalities

Linear inequalities are similar to linear equations but with an inequality sign instead of an equality sign. They are used to represent ranges of possible solutions rather than a single solution. The inequality signs such as \(\leq\), \(<\), \(\geq\), and \(>\) tell us how the values of one variable relate to another.

For instance, the inequality \(x + y \leq 4\) from our exercise means that the sum of x and y should be less than or equal to 4. Unlike equations, which have a precise solution, inequalities depict a continuum of possibilities. The power of linear inequalities lies in their ability to capture a wide range of solutions that satisfy certain conditions.

Inequality Graphing

To visualize the solutions of inequalities, we can graph them on a coordinate plane. This is known as inequality graphing. The process can help us understand where the solutions lie in relation to each other, and it's especially useful for systems of inequalities.

In graphing the inequalities from our exercise, we would first plot the line corresponding to the equation part of the inequality (as if the \(\leq\) sign was an \(=\) sign). These lines, for \(x + y = 4\) and \(3x + y = 6\), act as the boundaries of the solution region. Next, we need to determine which side of these boundary lines the solutions lie. Since the inequalities include the boundary (due to the \(\leq\) sign), points on the lines are included in the solutions. Shading below these lines, as required by the \(\leq\) sign, reveals the area where both conditions are true – this is the solution set for the system. The overlapping shaded area points to the common solutions for both inequalities.

Algebraic Solution Methods

While graphing provides a visual representation of solutions, algebraic methods allow for a precise determination of solutions. Algebraic solution methods for systems of inequalities include testing points to determine the solution set's boundaries and substitution or elimination techniques for finding exact points of intersection.

The exercise provided doesn't need these methods since the graphing approach is sufficient. However, in more complex cases, substituting values from one inequality into another can help to solve for one of the variables, or adding or subtracting entire inequalities to eliminate a variable might be more efficient. The algebraic approach can also determine feasibility in optimization problems, such as linear programming, where the aim is to find the maximum or minimum values of a function subject to given constraints.