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Chapter 3: Problem 28

Graph the inequality. $$ y \geq 0 $$

Short Answer

Expert verified
Draw a solid line along the x-axis and shade the region above it.

Step by step solution

01

Understand the inequality

The inequality given is \( y \geq 0 \). This means we are interested in the region where the value of \( y \) is greater than or equal to zero.
02

Plot the boundary line

Plot the line \( y = 0 \) on the graph. This line is the x-axis. Since the inequality includes \( y = 0 \), draw a solid line along the x-axis.
03

Determine the shaded region

Since the inequality is \( y \geq 0 \), we shade the region above the x-axis. This region represents all the points where the value of \( y \) is greater than or equal to zero.

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Key Concepts

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

Understanding Inequality
An inequality is a mathematical statement that shows the relationship between two values or expressions. It uses symbols like \(>\), \(<\), \(\geq\), and \(\leq\) to indicate if one value is greater than, less than, greater than or equal to, or less than or equal to the other value. In this exercise, we have the inequality \(y \geq 0\), which means every value of \(y\) must be zero or higher.

When graphing inequalities, our goal is to show all possible solutions visually. This involves plotting on a graph and shading the appropriate region. The challenge often lies in identifying the correct area to shade.
Graphing the Inequality
Graphing an inequality like \(y \geq 0\) involves a few straightforward steps. First, we start by understanding our inequality. Here, \(y \geq 0\) tells us we need to consider all the points where the value of \(y\) is zero or more.

Next, we plot the boundary line based on the related equation, which in this case is \(y = 0\). Because our inequality includes the equals sign (\( \geq \)), the boundary line is solid. If it were simply \(y > 0\), we'd use a dashed line instead.
Plotting the Boundary Line
The boundary line helps separate the graph into different regions. To find this, we convert the inequality into an equation. For \(y \geq 0\), it becomes \(y = 0\). This equation represents our boundary line.

On a standard graph, the line \(y = 0\) is the x-axis itself. We draw this line solid because the inequality \(\geq\) includes the boundary line itself.

If you're working with different inequalities, remember to use a dashed line for non-inclusive inequalities (\( > \) or \( < \)) and a solid line when the boundary is included (\( \geq \) or \( \leq \)).
Identifying the Shaded Region
The shaded region represents the set of solutions that satisfy the inequality. For \(y \geq 0\), we need to identify the area where \(y\) is zero or positive.

Once we've drawn our boundary line (here, the x-axis), the region above this line displays all the points where \(y\) is greater than or equal to zero. We shade this region to indicate it contains all possible solutions to our inequality.

For any inequality, always test a point to confirm your shaded region. For instance, choosing a point above the x-axis like (0, 1) should satisfy \(y \geq 0\). If it does, it's correctly shaded. If it doesn鈥檛, re-check your boundary line and region.

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Most popular questions from this chapter

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(2 x+3 y=9\)

For exercises 91-94, use a graphing calculator to graph each equation. Choose a window that shows the \(x\)-intercept and \(y\)-intercept. Sketch the graph; describe the window. \(y=-2 x+5\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((0,-7) ;(3,0)\)

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Use the slope formula to find the slope of the line that passes through the points. \((1,8) ;(3,15)\)
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