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Systems of Inequalities Help

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 “≤.”

Systems of Equations and Inequalities Examples

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.

Systems of Equations and Inequalities Examples

Fig. 10.7

 

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

Systems of Equations and Inequalities Examples

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.

Systems of Equations and Inequalities Examples

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.

Systems of Equations and Inequalities Examples

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.

Systems of Equations and Inequalities Examples

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.

Systems of Equations and Inequalities Examples

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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