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Chapter 7: Problem 15

Graph each linear inequality. \(y<-\frac{1}{4} x\)

Short Answer

Expert verified
To graph the linear inequality \(y < -\frac{1}{4}x\), draw a dashed line representing \(y = -\frac{1}{4}x\), then shade all the area below the line because 'y' is less than \(-\frac{1}{4}x\).

Step by step solution

01

Identify values

Identify the slope and the y-intercept of the inequality. The inequality given is \(y < -\frac{1}{4}x\). The slope \(m = -\frac{1}{4}\) and there is no y-intercept in this inequality, so it is 0 (:y = mx + c; c = 0)
02

Draw the graph

Draw the line \(y = -\frac{1}{4}x\) on the graph. Since the y-intercept is 0, the line goes through the origin. The slope is negative, which means the line descends from left to right. Because of the '<' sign in the inequality, the line is dashed. A dashed line means the points on the line are not included in the solution of the inequality: it's a strict inequality.
03

Shade the appropriate region

To determine which region to shade, pick a test point not on the line and substitute its coordinates in the inequality. If the inequality is true, you shade the part which includes that point; if it is false, you shade the other part. The easiest point to pick is (0,0), but since the line goes through the origin, a different point must be picked. A good point to pick would be (0,1). When these values are substituted into the inequality we get \(1 < 0\) which is false, so shade the area underneath the line to represent the solution range for the inequality.

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