Given The System Of Inequalities:$\[ \begin{array}{l} y \ \textless \ |x-1| \\ y \leq -2x \end{array} \\]Which Of The Following Describes The Boundary Line And Shading For The First Inequality In The System?A. Dashed Line, Shade Below B.

Feb 27, 2025 by ADMIN 241 views

**Understanding the System of Inequalities**

In mathematics, a system of inequalities is a set of two or more inequalities that are related to each other. These inequalities can be linear or non-linear, and they can be combined using various operations such as addition, subtraction, multiplication, and division. In this article, we will focus on a system of inequalities involving absolute value and linear inequalities.

The First Inequality: Absolute Value

The first inequality in the system is $y < |x-1|$ . This inequality involves an absolute value expression, which means that the distance between $x$ and $1$ is always non-negative. The absolute value function $|x-1|$ can be rewritten as a piecewise function:

|x-1| = \begin{cases} x-1 & \text{if } x \geq 1 \\ -(x-1) & \text{if } x < 1 \end{cases}

This means that the graph of the absolute value function consists of two line segments: one with a positive slope and one with a negative slope.

Graphing the Absolute Value Function

To graph the absolute value function, we can start by plotting the two line segments. For $x \geq 1$ , the line segment has a positive slope and passes through the point $(1,0)$ . For $x < 1$ , the line segment has a negative slope and passes through the point $(1,0)$ .

Since the absolute value function is always non-negative, the graph of the function will be above the x-axis. The graph will also be symmetric about the vertical line $x=1$ .

Boundary Line and Shading

The boundary line for the first inequality is the graph of the absolute value function. Since the inequality is strict ( $y < |x-1|$ ), the boundary line should be dashed. The shading for the inequality should be below the boundary line, since the inequality is less than.

Conclusion

In conclusion, the boundary line for the first inequality in the system is a dashed line that represents the graph of the absolute value function. The shading for the inequality is below the boundary line, since the inequality is less than.

Answer

The correct answer is A. Dashed line, shade below.

Additional Examples

Here are a few additional examples of systems of inequalities involving absolute value:

$y < |x-2|$
$y \leq |x-3|$
$y < |x-4|$

In each of these examples, the boundary line is a dashed line that represents the graph of the absolute value function. The shading for the inequality is below the boundary line, since the inequality is less than.

Tips and Tricks

Here are a few tips and tricks for working with systems of inequalities involving absolute value:

Always start by graphing the absolute value function.
Use a dashed line to represent the boundary line.
Shade below the boundary line, since the inequality is less than.
Use a solid line to represent the boundary line for equalities.
Shade on both sides of the boundary line for equalities.

By following these tips and tricks, you can easily graph and solve systems of inequalities involving absolute value.

Common Mistakes

Here are a few common mistakes to avoid when working with systems of inequalities involving absolute value:

Failing to graph the absolute value function.
Using a solid line to represent the boundary line.
Shading above the boundary line, since the inequality is less than.
Failing to use a dashed line to represent the boundary line for strict inequalities.

By avoiding these common mistakes, you can ensure that your graphs and solutions are accurate and complete.

Conclusion

In the previous article, we discussed the basics of systems of inequalities involving absolute value. In this article, we will answer some common questions about these systems.

Q: What is the boundary line for the first inequality in the system?

A: The boundary line for the first inequality in the system is a dashed line that represents the graph of the absolute value function.

Q: How do I graph the absolute value function?

A: To graph the absolute value function, start by plotting the two line segments. For $x \geq 1$ , the line segment has a positive slope and passes through the point $(1,0)$ . For $x < 1$ , the line segment has a negative slope and passes through the point $(1,0)$ .

Q: What is the shading for the first inequality in the system?

A: The shading for the first inequality in the system is below the boundary line, since the inequality is less than.

Q: How do I determine the shading for the inequality?

A: To determine the shading for the inequality, look at the inequality sign. If the inequality sign is less than ( $<$ ), the shading should be below the boundary line. If the inequality sign is less than or equal to ( $\leq$ ), the shading should be on both sides of the boundary line.

Q: What is the difference between a solid line and a dashed line?

A: A solid line represents an equality, while a dashed line represents a strict inequality.

Q: How do I graph a system of inequalities involving absolute value?

A: To graph a system of inequalities involving absolute value, start by graphing the absolute value function. Then, graph the second inequality in the system. The solution to the system is the region where the two inequalities overlap.

Q: What are some common mistakes to avoid when working with systems of inequalities involving absolute value?

A: Some common mistakes to avoid when working with systems of inequalities involving absolute value include:

Failing to graph the absolute value function.
Using a solid line to represent the boundary line.
Shading above the boundary line, since the inequality is less than.
Failing to use a dashed line to represent the boundary line for strict inequalities.

Q: How do I determine the solution to a system of inequalities involving absolute value?

A: To determine the solution to a system of inequalities involving absolute value, look at the two inequalities in the system. The solution is the region where the two inequalities overlap.

Q: What are some tips and tricks for working with systems of inequalities involving absolute value?

A: Some tips and tricks for working with systems of inequalities involving absolute value include:

Always start by graphing the absolute value function.
Use a dashed line to represent the boundary line.
Shade below the boundary line, since the inequality is less than.
Use a solid line to represent the boundary line for equalities.
Shade on both sides of the boundary line for equalities.

By following these tips and tricks, you can easily graph and solve systems of inequalities involving absolute value.

Conclusion

In conclusion, systems of inequalities involving absolute value can be challenging to graph and solve. However, by following the tips and tricks outlined in this article, you can easily graph and solve these systems. Remember to always graph the absolute value function, use a dashed line to represent the boundary line, and shade below the boundary line for strict inequalities.