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Chapter 12: Problem 29

Graph each system of inequalities. \(x \leq 5\) \(y \leq 4\)

Short Answer

Expert verified
The solution region is the rectangle in the xy-plane where \( x \leq 5 \) and \( y \leq 4 \).

Step by step solution

01

- Graph the first inequality

Start by graphing the line corresponding to the equation \( x = 5 \). Draw a vertical line through the point where \( x = 5 \). Since the inequality is \( x \leq 5 \), shade the area to the left of this line. This represents all the points where \( x \) is less than or equal to 5.
02

- Graph the second inequality

Next, graph the line corresponding to the equation \( y = 4 \). Draw a horizontal line through the point where \( y = 4 \). Since the inequality is \( y \leq 4 \), shade the area below this line. This represents all the points where \( y \) is less than or equal to 4.
03

- Identify the solution region

The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap. Look at the graph and identify the region where the shaded area from \( x \leq 5 \) overlaps with the shaded area from \( y \leq 4 \). This overlapping region represents the set of all points that satisfy both inequalities.
04

- Finalize the graph

Outline the overlapping region clearly. This region is a rectangular area that includes all points where \( x \leq 5 \) and \( y \leq 4 \). Any point within this shaded rectangular area is a solution to the system of inequalities.

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Key Concepts

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

linear inequalities
Linear inequalities are expressions that show the relationship between two variables using inequality symbols like <, >, 鈮�, and 鈮�. They are similar to linear equations, but instead of equality, there's a range of values for the variables. For example, in the inequality \( x \leq 5 \), we are looking at all values of \( x \) that are less than or equal to 5. Linear inequalities can be graphed on a coordinate plane just like linear equations, but with shading to indicate the solution region.
graphing inequalities
Graphing inequalities involves several steps. First, you graph the equation as if it were an equality. For example, to graph \( x \leq 5 \), you graph the line \( x = 5 \). This is a vertical line passing through all points where the x-coordinate is 5. Since the inequality is \( \leq \), you shade the area to the left of this line. This shading shows the region where the inequality holds true. For \( y \leq 4 \), you would draw a horizontal line at \( y = 4 \) and shade below it. Graphing inequalities helps visualize all the solutions to the inequality, making it easier to understand.
solution region
The solution region is where the shaded areas of multiple inequalities overlap. This region represents all the points that satisfy all given inequalities. For instance, if we have \( x \leq 5 \) and \( y \leq 4 \), the solution region is where the shaded areas of these two inequalities intersect. This region is a rectangle where every point within it satisfies both inequalities. To finalize, outline this overlapping region clearly, helping identify the set of all possible solutions to the system of inequalities.

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Most popular questions from this chapter

Match each inequality in parts (a)-(d) with its graph in choices A-D. (a) \(y \geq x^{2}+4\) (b) \(y \leq x^{2}+4\) (c) \(yx^{2}+4\)

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