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# Systems of Inequalities Help

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By — McGraw-Hill Professional
Updated on Oct 4, 2011

## Introduction to Systems of Inequalities

The solution (if any) for a system of inequalities is usually a region in the plane. The solution to a polynomial inequality (the only kind in this book) is the region above or below the curve. We will begin with linear inequalities.

### Graphing the Solution Region for Linear Inequalities

When sketching the graph for an inequality, we will use a solid graph for “≤” and “≥” inequalities, and a dashed graph for “<” and “>” inequalities. We can decide which side of the graph to shade by choosing any point not on the graph itself. We will put this point into the inequality. If it makes the inequality true, we will shade the side that has that point. If it makes the inequality false, we will shade the other side. Every point in the shaded region is a solution to the inequality.

#### Examples

• 2 x + 3 y ≤ 6
• We will sketch the line 2 x + 3 y = 6, using a solid line because the inequality is “≤.”

Fig. 10.6

We will always use the origin, (0, 0) in our inequalities unless the graph goes through the origin. Does x = 0 and y = 0 make 2 x + 3 y ≤ 6 true? 2(0) + 3(0) ≤ 6 is a true statement, so we will shade the side that has the origin.

Fig. 10.7

• x − 2 y > 4
• We will sketch the line x − 2 y = 4 using a dashed line because the inequality is “>.”

Fig. 10.8

Now we need to decide which side of the line to shade. When we put (0, 0) in x − 2 y > 4, we get the false statement 0 − 2(0) > 4. We need to shade the side of the line that does not have the origin.

Fig. 10.9

• y < 3 x
• We use a dashed line to sketch the line y = 3 x . Because the line goes through (0, 0), we cannot use it to determine which side of the line to shade. This is because any point on the line makes the equality true. We want to know where the inequality is true. The point (1, 0) is not on the line, so we can use it. 0 < 3(1) is true so we will shade the side of the line that has the point (1,0), which is the right side.

Fig. 10.10

• x ≥ −3
• The line x = −3 is a vertical line through x = −3. Because we want x ≥ −3 we will shade to the right of the line.

Fig. 10.11

• y < 2
• The line y = 2 is a horizontal line at y = 2. Because we want y < 2, we will shade below the line.

Fig. 10.12

## Graphing the Solution Region for Nonlinear Inequalities

Graphing the solution region for nonlinear inequalities is done the same way—graph the inequality, using a solid graph for “≤” and “≥” inequalities and a dashed graph for “<” and “>” inequalities, then checking a point to see which side of the graph to shade.

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