Problem 66 - College Algebra

Question

In Exercises 65–68, 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 \(3 .\) The \(y\)-variable added to the product of 4 and the \(x\)-variable does not exceed 6.

Step-by-Step Solution

Answer

The system of inequalities is \(x+y \leq 3\) and \(4x + y \leq 6\). The solution of the system is the region which lies in the first quadrant bounded by parts of the lines \(x+y = 3\), \(4x + y = 6\) and the axes.

Step 1: Formulate the inequalities

The first sentence says that the sum of the \(x\)-variable and the \(y\)-variable is at most \(3\). This means \(x+y \leq 3\). The second sentence says that the \(y\)-variable added to the product of \(4\) and the \(x\)-variable does not exceed \(6\), which translates to \(4x + y \leq 6\).

Step 2: Graph the inequalities

The graph is divided into four quadrants. The inequality \(x+y \leq 3\) represents all the points below the line \(x+y=3\). The inequality \(4x + y \leq 6\) represents all the points below the line \(4x + y = 6\). The system of these two inequalities is the intersection of the regions represented by them, which lies in the first quadrant and is bounded by parts of the two lines and the axes.

Key Concepts

Understanding Graphing Inequalities

Graphing inequalities involves plotting regions on a coordinate plane that represent solutions to an inequality.
When we work with inequalities, rather than equalities, we're not just looking at one line or point, but an area.
For instance, the inequality \(x + y \leq 3\) implies that every point \((x, y)\) on the plane, as long as when you add the \(x\) and \(y\) values together, they equal 3 or less, is included.

• We start by plotting the boundary line, like with an equation.
• The line \(x + y = 3\) is plotted as a dashed line, because the points on the line could still be solutions.
• Once the line is drawn, we shade the side where points satisfy the inequality, which means checking whether to shade above or below the line.

Knowing which region to shade can be tested by picking any test point (usually the origin \((0,0)\) is the simplest choice) and checking if this point satisfies the inequality.

Working with Two-Variable Inequalities

Two-variable inequalities like \(x + y \leq 3\) involve two inputs, usually represented by \(x\) and \(y\), making the solution more than just a simple number.

With these inequalities:

• We're looking for a combination of \(x\) and \(y\) that makes the inequality true.
• Solutions aren't single numbers but sets of numbers (x, y pairs).
• These pairs create regions on the graph.

Visualizing these solutions means understanding that instead of finding just one solution, you find a range or area of solutions.
Identifying the particular region of solutions involves determining where inequalities overlap.

Exploring Linear Inequalities

Linear inequalities such as \(x + y \leq 3\) are quite common in algebra.
They are a version of linear equations which you might know as y = mx + b, where you deal with simple straight lines.
However, the inequality introduces a range of possible solutions.

• The line is known as the boundary of the inequality.
• Every point on one side of the line is a solution, depending on the direction of the inequality.
• For example, in \(4x + y \leq 6\), any point within the shaded area on the graph is important.

In this context, boundaries are inclusive, which means if the inequality is \(\leq\), the points on the line are in the solution set, otherwise you use a dotted line to show exclusion of the boundary itself.

Deciphering Algebraic Expressions in Context

Algebraic expressions form the backbone of these types of problems and are a combination of variables, numbers, and operations.
To effectively handle such problems, understanding how to translate sentences into algebraic form is vital.
For example:

• The sentence "the sum of the \(x\)-variable and the \(y\)-variable is at most 3" results in \(x + y \leq 3\).
• "The \(y\)-variable added to the product of 4 and the \(x\)-variable" translates to \(4x + y\).

The key is correctly interpreting words like "at most" which means \(\leq\) or "exceeds" which would mean \(>\).
Understanding this language allows a student to visualize and solve these problems efficiently.