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Chapter 9: Problem 84

When graphing \(3 x-4 y<12,\) it's not necessary for me to graph the linear equation \(3 x-4 y=12\) because the inequality contains a \(<\) symbol, in which equality is not included.

Short Answer

Expert verified
In this case, you need to first graph the boundary line \(3x - 4y = 12\) with a dashed line because it's not included in the solution. Then a test point, for example the origin (0,0), is used to decide which side of the graph to shade. The shaded section marks the solution of the inequality.

Step by step solution

01

Identifying The Line

First, you need to graph the boundary line. Even though the exercise suggests that it's not necessary, this line aids in visualizing the inequality. The line refers to the equation \(3x - 4y = 12\). To graph the line, you need to find the intercepts. When \(x = 0\), solve for \(y\) to find the y-intercept. When \(y = 0\), solve for \(x\) to find the x-intercept. Then, draw the line through these two points.
02

Checking The Inequality Symbol

The inequality symbol '<' tells you about the region you should shade. If the symbol was `<=` or `>=`, you'd use a solid line, showing that points on the line are included in the solution. Since the symbol here is '<', you'll use a dashed line to show that points on the line aren't included in the solution.
03

Selecting Test Points

A test point is used to determine which side of the line to shade. Since we've determined that the line itself isn't part of the solution, test points from either side will let you know which area represents valid solutions. The simplest point to choose is often the origin (0,0), provided the line doesn't pass through it.
04

Shading The Graph

To determine which side to shade, substitute the coordinates of your test point into the inequality: if it makes the inequality true, shade that side of the line. If the test point doesn't satisfy the inequality, shade the other side of the line. That shaded region is the solution.

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