Problem 542 Graph the linear inequality \(x&... [FREE SOLUTION]

Chapter 4: Problem 542

Graph the linear inequality \(x>-3 .\)

Short Answer

Expert verified

Draw a dashed line at \(x = -3\) and shade the area to the right.

Step by step solution

Understand the Inequality

The inequality given is in the form of a linear inequality. Here, the variable is compared to a number using a greater than sign: \(x > -3\). This means that all values of x that are greater than -3 satisfy the inequality.

Plot the Boundary Line

To graph the inequality, first plot the boundary line represented by the equation \(x = -3\). Because the inequality is strictly greater than (and not \(\geq\)), draw a dashed vertical line at \(x = -3\) to indicate that points on the line are not included in the solution set.

Shade the Solution Area

Since the inequality is \(x > -3\), shade the region to the right of the dashed line. This is because all the points in this region will have x-values that are greater than -3 and thus satisfy the inequality.

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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 where a linear function is related to a constant using an inequality sign such as <, 鈮�, >, or 鈮�. When we say a linear inequality, we're comparing a linear expression to a constant and determining which set of values makes the inequality true. For example, in the inequality \(x > -3\), we are interested in all values of \(x\) which are greater than -3. These types of inequalities help us understand ranges of solutions rather than specific points, offering a broader perspective in mathematics. Linear inequalities can have one or more variables and can be visualized using a number line or graph.

Graphing Inequalities

Graphing inequalities involves showcasing the solution set of the inequality on a coordinate plane. Here's a step-by-step method to graph linear inequalities:

First, convert the inequality into an equation to find the boundary line.
Plot the boundary line on the graph. If the inequality uses > or <, draw a dashed line to indicate that the points on the line are not included in the solution. Use a solid line for 鈮� or 鈮�.
Determine which side of the boundary line contains the solutions. You can do this by picking a test point not on the line and checking if it satisfies the inequality.
Shade the region where the inequality is true. For instance, in our case, with \(x > -3\), after plotting the dashed vertical line at \(x = -3\), you would shade the region to the right.

This visual representation helps in understanding and solving inequalities more effectively.

Boundary Line

The boundary line is a crucial part of graphing linear inequalities. It represents the points where the inequality transforms into an equation. For the inequality \(x > -3\), our boundary line is given by \(x = -3\). Here are some points to note about boundary lines:

Drawing the Boundary: For strict inequalities (>, <), the boundary line is dashed to indicate that points on the line do not satisfy the inequality. For inclusive inequalities (鈮�, 鈮�), draw a solid line.
Position: The position of the boundary line and the side you shade matters a lot. In \(x > -3\), the line \(x = -3\) is vertical, and you shade to the right because this region contains the values of \(x\) that are greater than -3.
Shading: Always ensure that you shade the correct side of the boundary line, which denotes the solution region. This is where the inequality holds true.

Understanding the role of the boundary line is essential for accurately graphing any linear inequality.

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91影视

Graph the linear inequality \(x>-3 .\)

Short Answer

Step by step solution

Understand the Inequality

Plot the Boundary Line

Shade the Solution Area

Key Concepts

Linear Inequalities

Graphing Inequalities

Boundary Line

One App. One Place for Learning.

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