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 Depdendent Variable

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 Dependent Variable

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# Graphing Linear Inequalities

You have probably already studied linear equations. We now turn our attention to linear inequalities.

Definition

A linear inequality is a linear equation with the equal sign replaced by an inequality symbol.

Linear Inequality

If A, B, and C are real numbers with A and B not both zero, then Ax + By C is called a linear inequality. In place of , we can also use , <, or >.

## Graphing Linear Inequalities

Consider the inequality -x + y > 1. If we solve the inequality for y, we get y > x + 1.

Which points in the xy-plane satisfy this inequality? We want the points where the y-coordinate is larger than the x-coordinate plus 1. If we locate a point on the line y = x + 1, say (2, 3), then the y-coordinate is equal to the x-coordinate plus 1. If we move upward from that point, to say (2, 4), the y-coordinate is larger than the x-coordinate plus 1. Because this argument can be made at every point on the line, all points above the line satisfy y > x + 1. Likewise, points below the line satisfy y < x + 1. The solution sets, or graphs, for the inequality y > x + 1 and the inequality y < x + 1 are the shaded regions shown in the figures below. In each case the line y = x + 1 is dashed to indicate that points on the line do not satisfy the inequality and so are not in the solution set. If the inequality symbol is or , then points on the boundary line also satisfy the inequality, and the line is drawn solid.

Every nonvertical line divides the xy-plane into two regions. One region is above the line, and the other is below the line. A vertical line also divides the plane into two regions, but one is on the left side of the line and the other is on the right side of the line. An inequality involving only x has a vertical boundary line, and its graph is one of those regions.

Graphing a Linear Inequality

1. Solve the inequality for y, then graph y = mx + b.

y > mx + b is satisfied above the line.

y = mx + b is satisfied on the line itself.

y < mx + b is satisfied below the line.

2. If the inequality involves x and not y, then graph the vertical line x = k.

x > k is satisfied to the right of the line.

x = k is satisfied on the line itself.

x < k is satisfied to the left of the line.

Example 1

Graphing linear inequalities

Graph each inequality.

a)

b) y ≥ - 2x + 1

c) 3x - 2y < 6

Solution

a) The set of points satisfying this inequality is the region below the line . To show this region, we first graph the boundary line . The slope of the line is and the y-intercept is (0, -1). Start at (0, -1) on the y-axis, then rise 1 and run 2 to get a second point of the line. We draw the line dashed because points on the line do not satisfy this inequality. The solution set to the inequality is the shaded region shown in the figure below.

b) Because the inequality symbol is , every point on or above the line satisfies this inequality. To show that the line y = - 2x + 1 is included, we make it a solid line. See the figuer below.

c) First solve for y:

 3x - 2y < 6 -2y < -3x + 6 y Divide by -2 and reverse the inequality.

To graph this inequality, use a dashed line for the boundary and shade the region above the line. See the figure below for the graph.

Caution In Example 1(c) we solved the inequality for y before graphing the line. We did that because < corresponds to the region below the line and > corresponds to the region above the line only when the inequality is solved for y.