Problem 65 - College Algebra Essentials

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 inequalities is \(x + y ≤ 4\) and \(y + 3x ≤ 6\). The graph is a shaded area satisfying both inequalities bounded by the lines \(x + y = 4\) and \(y + 3x = 6\).

Step 1: Formulate Inequalities

To form the system of inequalities, we translate the given sentences. The 'at most' suggests a less than or equal to sign (≤). So, the first sentence becomes \(x + y ≤ 4\). For the second sentence, 'does not exceed' also suggests a less than or equal to sign (≤). So, it becomes \(y + 3x ≤ 6\).

Step 2: Graph the Inequalities

To graph the inequalities, we first draw a line for each inequality. For \(x + y ≤ 4\), when \(x=0\), \(y=4\) and when \(y=0\), \(x=4\). For \(y + 3x ≤ 6\), when \(x=0\), \(y=6\) and when \(y=0\), \(x=2\). Then we shade the area that satisfies both inequalities. Notice that the shaded areas should be under the line because it is less than or equal to (≤).

Key Concepts

graphing inequalities

Graphing inequalities involves representing a range of possible values on a coordinate plane. Begin by taking the inequality equation and treating it as if it were an equation. Suppose you're working with the inequality \(x + y \leq 4\).
You'll first convert this into an equation by using an equal sign: \(x + y = 4\). Plot this as a straight line on the graph. You find points by choosing simple values like \(x = 0\), then solve for \(y\), giving you \(y = 4\); repeat this by switching to \(y = 0\) to solve for \(x = 4\).

Next, since it's a "less than or equal to" inequality, the area below or on the line represents all solutions. Use shading to visually capture these solutions under the line. Remember, for inequalities with just "less than" or "greater than" signs, use dashed lines to indicate points ON the line aren't solutions.
When graphing, ensure each line corresponds with its mathematical operation. This method applies to multiple inequalities, allowing for regions of overlap, which complete the solution set.

two-variable systems

When dealing with two-variable systems, you work with equations or inequalities that involve two different variables, often seen as \(x\) and \(y\). These systems can represent multiple constraints in a problem which, when graphed, depict a set of permissible solutions.
In the system given, you have:

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

These inequalities describe boundaries on a coordinate plane. When graphed, their intersection represents solutions that meet all inequalities simultaneously. Each line distinguishes a boundary where one region is valid based on the inequality's conditions.

Understanding this concept is crucial because it helps visualize how two factors, the variables, constrain a scenario. Solutions to such systems are often represented graphically as the area where shaded regions overlap, representing solutions common to both inequalities.

algebraic translation

Algebraic translation involves converting verbal statements into mathematical inequalities or equations. This process is essential for accurately representing practical problems within an algebraic framework. Let's translate sentences into inequalities:

• "The sum of the \(x\)-variable and the \(y\)-variable is at most 4" translates to the inequality \(x + y \leq 4\).
• "The \(y\)-variable added to the product of 3 and the \(x\)-variable does not exceed 6" translates to \(y + 3x \leq 6\).

Key phrases to watch for include "at most," suggesting less than or equal to (≤), and "does not exceed," which also translates as ≤.

Being able to transition between language and symbols is foundational in algebra, allowing you to create precise mathematical models of real-world scenarios. This skill is particularly useful in forming systems of equations or inequalities, making complex word problems manageable and solvable through algebraic techniques.