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Chapter 7: Problem 16

Graph each linear inequality. \(y<-\frac{1}{3} x\)

Short Answer

Expert verified
The graph of the inequality \(y < -\frac{1}{3} x\) consists of a dashed line representing \(y = -\frac{1}{3} x\) and a shaded area below the line.

Step by step solution

01

Understanding the Inequality

The given inequality is \(y < -\frac{1}{3} x\). This inequality can be broken down into two parts: the line given by \(y = -\frac{1}{3} x\) and the region defined by \(y\), which is less than this line. Meaning, the solution includes the points that make the y-values less than the y-values on the line.
02

Graphing the Line

Start by graphing the line \(y = -\frac{1}{3} x\). Because it is y <, this indicates we are not including the values on the line, so it's graphed as a dashed line. The line passes through the origin (0, 0) and has a slope of -1/3, which means that for every positive step of 3 units horizontally (x direction), the line drops 1 unit vertically (y direction).
03

Shading the Correct Area

The inequality is \(y < -\frac{1}{3} x\), so we shade the area below the line. Any point in this shaded area, if substituted into the inequality will make it a true statement, which is how we know that it represents the inequality.

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