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# Systems of Inequalities Help (page 2)

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By McGraw-Hill Professional
Updated on Oct 4, 2011

#### Examples

• yx 2x − 2
• The equality is y = x 2x − 2 = ( x − 2)( x + 1). The graph for this equation is a parabola.

Fig. 10.13

Because (0, 0) is not on the graph, we can use it to decide which side to shade; 0 ≤ 0 2 − 0 − 2 is false, so we shade below the graph, the side that does not contain (0, 0).

Fig. 10.14

• y > ( x + 2)( x − 2)( x − 4)
• When we check (0, 0) in the inequality, we get the false statement 0 > (0 + 2)(0 − 2)(0 − 4). We will shade above the graph, the region that does not contain (0, 0).

Fig. 10.15

## Graphing the Solution for Two or More Inequalities

The solution (if there is one) to a system of two or more inequalities is the region that is part of each solution for the individual inequalities. For example, if we have a system of two inequalities and shade the solution to one inequality in blue and the other in yellow, then the solution to the system would be the region in green.

#### Examples

• Sketch the solution for each inequality. The solution to xy < 3 is the region shaded vertically. The solution to x + 2 y > 1 is the region shaded horizontally.

Fig. 10.16

Fig. 10.17

The region that is in both solutions is above and between the lines.

Fig. 10.18

•

Fig. 10.19

The solution to y ≤ 4 − x 2 is the region shaded vertically. The solution to x − 7 y ≤ 4 is the region shaded horizontally. The region that is in both solutions is above the line and inside the parabola.

Fig. 10.20

Because a solid graph indicates that points on the graph are also solutions, to be absolutely accurate, the correct solution uses dashed graphs for the part of the graphs that are not on the border of the shaded region.

Fig. 10.21

We will not quibble with this technicality here.

The inequalities x ≥ 0 and y ≥ 0 mean that we only need the top right corner of the graph. These inequalities are common in word problems.

Fig. 10.22

The solution to the system is the region in the top right corner of the graph below the line 2x + y = 5 .

Fig. 10.23

Some systems of inequalities have no solution. In the following example, the regions do not overlap, so there are no ordered pairs (points) that make both inequalities true.

Fig. 10.24

## Graphing the Solution Region for Three or More Inequalities

It is easy to lose track of the solution for a system of three or more inequalities. There are a couple of things you can do to make it easier. First, make sure the graph is large enough, using graph paper if possible. Second, shade the solution for each inequality in a different way, with different colors or shaded with horizontal, vertical, and slanted lines. The solution (if there is one) would be shaded all different ways. You could also shade one region at a time, erasing the part of the previous region that is not part of the inequality.

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