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.

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Identify the variables: let the x-variable be \(x\) and the y-variable be \(y\).

Translate the first sentence "The sum of the x-variable and the y-variable is at most 4" into the inequality: \(x + y \leq 4\).

Translate the second sentence "The y-variable added to the product of 3 and the x-variable does not exceed 6" into the inequality: \(y + 3x \leq 6\).

Write the system of inequalities as: \[ \begin{cases} x + y \leq 4 \\ y + 3x \leq 6 \end{cases} \]

To graph the system, first graph the boundary lines \(x + y = 4\) and \(y + 3x = 6\) by finding intercepts or using slope-intercept form, then shade the regions that satisfy each inequality (below or on the lines). The solution to the system is the overlapping shaded region.

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주요 개념

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Translating Word Problems into Inequalities

This involves converting verbal statements into mathematical inequalities. Key phrases like 'at most' or 'does not exceed' indicate 'less than or equal to' (≤). Identifying variables and their relationships is essential to form accurate inequalities representing the problem.

Systems of Inequalities

A system of inequalities consists of two or more inequalities with the same variables. Solutions must satisfy all inequalities simultaneously. Understanding how to write and interpret these systems is crucial for analyzing constraints in problems.

Graphing Inequalities in Two Variables

Graphing inequalities involves shading regions on the coordinate plane that satisfy the inequality. The boundary line is drawn using equality, solid if ≤ or ≥, dashed if < or >. The solution to a system is the overlapping shaded region of all inequalities.