Problem 41 - Thinking Mathematically

Question

In Exercises 41-42, 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 \(x+y \leq 4\) and \(y+3x \leq 6\). The graphical representation of the system is the region where the shaded areas under the lines \(x+y=4\) and \(y+3x=6\) overlap.

Step 1: Translate Sentences into Inequalities

The first sentence translates to the inequality \(x+y \leq 4\). The second sentence translates to the inequality \(y+3x \leq 6\).

Step 2: Graph the Inequalities

For each inequality, draw a line on a graph representing the 'boundary line'. For \(x+y=4\), the line passes through points (0,4) and (4,0). For \(y+3x=6\), the line passes through points (0,6) and (2,0). Since the original inequalities are 'less than or equal to', the areas represented are under the lines (including the lines). Shade the region under each of these lines to represent the inequalities.

Step 3: Identify the Overlapping Region

The system of inequalities is represented by the region where the shaded areas of both inequalities overlap. This region satisfies both inequalities simultaneously.

Key Concepts

Graphing Inequalities

When graphing inequalities, you're not just drawing a line—you're revealing a whole area that holds all the solutions to the inequality. For example, the inequality \(x + y \leq 4\) translates to a line when equality is used, \(x + y = 4\), dividing the plane into a half-plane where the condition is true.
This technique helps visualize regions that satisfy the inequality condition. Here's how to do it:

• First, draw the 'boundary line' by temporarily replacing the inequality sign with an equality sign.
• The line for \(x + y = 4\) passes through the intercept points (0, 4) and (4, 0).
• Since the inequality is 'less than or equal to', shade the region below the line, as this area satisfies \(x + y \leq 4\).

It’s essential to shade correctly because the shaded area indicates solutions—every point in this area satisfies the inequality. Repeat this for each inequality in your system.

Linear Inequalities

Linear inequalities are similar to linear equations but with inequality signs like \(\leq, \geq, <, >\) instead of an equal sign. They represent a range of possible solutions, instead of one precise solution like a linear equation.
To convert a statement into a linear inequality:

• Express conditions in terms of mathematical inequalities. For example, "The sum of the \(x\)-variable and the \(y\)-variable is at most 4" translates to \(x+y \leq 4\). This shows that result can't be more than 4.
• "The \(y\)-variable added to the product of 3 and the \(x\)-variable does not exceed 6" translates to \(y+3x \leq 6\), indicating another range of conditions.

Recognizing the context of "at most" or "does not exceed" is crucial to correctly formulating linear inequalities. It's the first step in solving systems of inequalities.

Inequality Solutions

Finding inequality solutions involves identifying the regions on the graph that meet all given conditions. For the system with inequalities as discussed, \(x+y \leq 4\) and \(y+3x \leq 6\), you'll solve by determining where these conditions overlap on your graph.
Here’s what to do:

• Graph each inequality separately, shading the appropriate area that represents solutions to the inequality.
• Identify where the shaded regions overlap—this common area is your solution.
• Ensure you include the boundary lines when inequalities are inclusive (\(\leq\) or \(\geq\)).

The overlapping region is crucial as it visualizes where both stipulations are true. Points in this area aren't just possible solutions—they’re the only valid ones in terms of the problem statement. This method assists in comprehending complex systems by simplifying them into manageable visual parts.