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 Depdendent Variable

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 Dependent Variable

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# Graphing Linear Inequalities

Why do we keep drawing graphs? When we solve 2x + 1 = 7, we donâ€™t bother to draw a graph showing 3 because the solution set is so simple. However, the solution set to a linear inequality is a very large set of ordered pairs. Graphing gives us a way to visualize the solution set.

Example 1

Inequalities with horizontal and vertical boundaries

Graph the inequalities.

a) y 5

b) x > 4

Solution

a) The line y = 5 is the horizontal line with y-intercept (0, 5). Draw a solid horizontal line and shade below it as in the figure below.

b) The points that satisfy x > 4 lie to the right of the vertical line x = 4. The solution set is shown in the following figure.

## The Test Point Method

The graph of any line Ax + By = C separates the xy-plane into two regions. Every point on one side of the line satisfies the inequality Ax + By < C, and every point on the other side satisfies the inequality Ax + By > C. We can use these facts to graph an inequality by the test point method:

1. Graph the corresponding equation.

2. Choose any point not on the line.

3. Test to see whether the point satisfies the inequality.

If the point satisfies the inequality, then the solution set is the region containing the test point. If not, then the solution set is the other region. With this method, it is not necessary to solve the inequality for y.

Example 2

Using the test point method Graph the inequality 3x - 4y > 7.

Solution

First graph the equation 3x - 4y = 7 using the x-intercept and the y-intercept. If x = 0, then . If y = 0, then . Use the x-intercept and the yintercept to graph the line as shown in figure (a) below

Select a point on one side of the line, say (0, 1), to test in the inequality. Because 3(0) - 4(1) > 7 is false, the region on the other side of the line satisfies the inequality. The graph of 3x - 4y > 7 is shown in figure (b) above.