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Chapter 13: Problem 30

Graph each system of inequalities. \(x \geq-2\) \(y \leq 4\)

Short Answer

Expert verified
Shade the region to the right of \(x = -2\) and below \(y = 4\).

Step by step solution

01

Graph the Boundary Line for Inequality 1

First, consider the inequality \(x \geq -2\). Graph the line \(x = -2\). This will be a vertical line passing through \(x = -2\). Since the inequality is \(\geq\), shade the entire region to the right of this line.
02

Graph the Boundary Line for Inequality 2

Next, consider the inequality \(y \leq 4\). Graph the line \(y = 4\). This will be a horizontal line passing through \(y = 4\). Since the inequality is \(\leq\), shade the entire region below this line.
03

Identify the Solution Region

The solution to the system of inequalities is the region where the shaded areas overlap. This will be the area to the right of the line \(x = -2\) and below the line \(y = 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

graphing inequalities
Graphing inequalities involves drawing a line that represents the inequality on a coordinate plane.
For example, if we have the inequality \(x \geq -2\), we first draw its corresponding line, \(x = -2\).
This line divides the plane into two regions: one where \(x\) is less than \(-2\) and one where \(x\) is greater than \(-2\).
The side we shade depends on whether the inequality is greater than (or equal to) or less than (or equal to). To graph this correctly:
  • For inequalities like \(x \geq -2\), we shade to the right of the boundary line.
  • For inequalities like \(y \leq 4\), we shade below the boundary line.
boundary lines
Boundary lines are critical in graphing inequalities because they show where the inequality changes from true to false.
When graphing the inequality \(x \geq -2\), we start with the boundary line \(x = -2\).
This is a vertical line that we draw on the coordinate plane. For the inequality \(y \leq 4\), we graph \(y = 4\) as a horizontal boundary line.
  • Always use a solid line for inequalities that include equality (\
solution region
Once we have drawn all boundary lines and shaded the appropriate regions, the solution region is where the shaded areas overlap.
For our example, the shaded region to the right of \(x = -2\) and below \(y = 4\) is the solution region.
This overlapping area contains all the pairs (\()x, y) \), which satisfy both inequalities.
  • The solution region can be a finite or infinite area on the graph.
  • It is important because it represents all the possible solutions to the system of inequalities.
overlapping regions
Overlapping regions occur when we graph more than one inequality, and their solution areas intersect.
In our case, the regions that overlap are:
  • The region to the right of \(x = -2\) for \(x \geq -2\).
  • The region below \(y = 4\) for \(y \leq 4\).
This overlap is the solution area where both conditions are met. Overlapping regions are crucial in solving systems of inequalities as they show the common solutions.

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