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Chapter 12: Problem 30

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

Short Answer

Expert verified
Shade the region where \( x \geq -2 \) and \( y \leq 4 \) overlap.

Step by step solution

01

Understand the Inequalities

There are two inequalities to consider: 1. The first inequality is given by: \( x \geq -2 \) 2. The second inequality is given by: \( y \leq 4 \)
02

Graph the First Inequality

To graph \( x \geq -2 \), draw a vertical line at \( x = -2 \). Then shade the region to the right of this line because the inequality indicates that \( x \) is greater than or equal to -2.
03

Graph the Second Inequality

To graph \( y \leq 4 \), draw a horizontal line at \( y = 4 \). Then shade the region below this line because the inequality indicates that \( y \) is less than or equal to 4.
04

Identify the Solution Region

The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap. This represents all the points that satisfy both \( x \geq -2 \) and \( y \leq 4 \).

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Key Concepts

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

graphing inequalities
Learning to graph inequalities might initially feel daunting, but it becomes simple with practice. To graph an inequality like \(x \geq -2\), you first draw a vertical line at \(x = -2\). This line separates the coordinates where \(x \geq -2\). Next, because the inequality is 'greater than or equal to,' you shade the entire area to the right of this line. By shading to the right, you indicate all possible values for \(x\) that are greater than or equal to \(-2\). Similarly, for \(y \leq 4\), draw a horizontal line at \(y = 4\). Shade the area below this line to indicate all points where \(y\) is less than or equal to \(4\).
shading regions
Shading regions is critical in graphing systems of inequalities. It helps visualize the set of possible solutions. For the inequality \(x \geq -2\), after drawing the vertical line at \(x = -2\), shade the region to its right. The shading represents all potential values of \(x\) that satisfy the inequality. Similarly, for \(y \leq 4\), after drawing the horizontal line at \(y = 4\), shade the region below it. For both inequalities, the shaded regions are semi-infinite in nature. This means they stretch infinitely in one direction. The areas you shade give a clear indication of where each individual inequality holds true on the graph.
solution region
The solution region of a system of inequalities is the area where all individual shaded regions overlap. This region represents all the points that satisfy every inequality in the system. To find the solution region in our example, look where the shading for \(x \geq -2\) and \(y \leq 4\) overlaps. This common area shows all coordinates \(x, y\) that meet both conditions. Graphically, it鈥檚 the region to the right of \(x = -2\) and below \(y = 4\). The boundaries of the solution region are included unless the inequalities are strict (e.g., < or >), which would require dashed boundary lines.

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