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Chapter 8: Problem 6

Graph each inequality. $$y \leq \frac{1}{4} x$$

Short Answer

Expert verified
The graph of the inequality \(y \leq \frac{1}{4} x\) is a solid line passing through the points \((0, 0)\) and \((4, 1)\), with the area below this line shaded in.

Step by step solution

01

Understanding the Inequality

The inequality given is \(y \leq \frac{1}{4} x\). This means that \(y\) is less than or equal to \(1/4\) times \(x\). This can be interpreted as the line where \(y = \frac{1}{4} x\) and the area below the line, because \(y\) is less than or equal to \(1/4 x\).
02

Graphing the Line

To graph the line \(y = \frac{1}{4} x\), plot two points on the graph that satisfy this equation. For example, when \(x = 0\), \(y = 0\), and when \(x = 4\), \(y = 1\). Thus, the two points \((0, 0)\) and \((4, 1)\) are on the line. Draw the line passing through these points. In linear inequalities, if the symbol includes equality, the line should be solid. As the original inequality \(y \leq \frac{1}{4} x\) includes equality, the line you draw in this step should be solid.
03

Shading the Area

Since the original inequality is \(y \leq \frac{1}{4} x\), the area that satisfies this inequality is below the line. As a result, shade the area below the line drawn in Step 2. Therefore, any point in the shaded area, including the points on the line, satisfies the inequality.

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