Which Number Line Represents The Solutions To − 2 ∣ X ∣ = − 6 -2|x| = -6 − 2∣ X ∣ = − 6 ?

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Introduction

When dealing with absolute value equations, it's essential to understand the concept of absolute value and how it affects the solutions to an equation. In this case, we're given the equation $-2|x| = -6$ and asked to determine which number line represents the solutions to this equation. To solve this problem, we need to understand the properties of absolute value and how to work with absolute value equations.

Understanding Absolute Value

Absolute value is a mathematical concept that represents the distance of a number from zero on the number line. It's denoted by the symbol $|x|$ and is defined as:

$$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$

In other words, if $x$ is a non-negative number, then $|x|$ is equal to $x$. If $x$ is a negative number, then $|x|$ is equal to the opposite of $x$, or $-x$.

Solving Absolute Value Equations

To solve an absolute value equation, we need to consider two cases: one where the expression inside the absolute value is non-negative, and one where the expression inside the absolute value is negative.

For the equation $-2|x| = -6$, we can start by dividing both sides by $-2$ to get:

$$|x| = 3$$

Now, we need to consider two cases:

Case 1: $x \geq 0$

If $x$ is non-negative, then $|x|$ is equal to $x$. So, we can write:

$$x = 3$$

This is one possible solution to the equation.

Case 2: $x < 0$

If $x$ is negative, then $|x|$ is equal to $-x$. So, we can write:

$$-x = 3$$

Solving for $x$, we get:

$$x = -3$$

This is another possible solution to the equation.

Number Line Representation

Now that we have found the solutions to the equation, we need to determine which number line represents these solutions. To do this, we can plot the solutions on a number line.

The number line for the equation $-2|x| = -6$ is shown below:

  -6  |  -3  |  0  |  3  |  6

In this number line, the solutions to the equation are represented by the points $-3$ and $3$.

Conclusion

In conclusion, the number line that represents the solutions to the equation $-2|x| = -6$ is the one shown above, with the solutions $-3$ and $3$ marked on the number line. This number line represents the two possible solutions to the equation, which are $x = 3$ and $x = -3$.

Frequently Asked Questions

Q: What is the definition of absolute value?

A: Absolute value is a mathematical concept that represents the distance of a number from zero on the number line.

Q: How do you solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases: one where the expression inside the absolute value is non-negative, and one where the expression inside the absolute value is negative.

Q: What are the solutions to the equation $-2|x| = -6$?

A: The solutions to the equation $-2|x| = -6$ are $x = 3$ and $x = -3$.

Q: Which number line represents the solutions to the equation $-2|x| = -6$?

A: The number line that represents the solutions to the equation $-2|x| = -6$ is the one shown above, with the solutions $-3$ and $3$ marked on the number line.

Introduction

Absolute value equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in algebra and beyond. In this article, we'll delve into the world of absolute value equations, exploring the definition, properties, and solutions to these equations. We'll also provide a comprehensive guide to help you master absolute value equations.

Frequently Asked Questions

Q: What is the definition of absolute value?

A: Absolute value is a mathematical concept that represents the distance of a number from zero on the number line. It's denoted by the symbol $|x|$ and is defined as:

$$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$

Q: How do you solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases: one where the expression inside the absolute value is non-negative, and one where the expression inside the absolute value is negative.

Q: What are the properties of absolute value?

A: Absolute value has the following properties:

• Non-negativity: The absolute value of any number is always non-negative.
• Identity of Indiscernibles: If $|x| = |y|$, then $x = y$ or $x = -y$.
• Triangle Inequality: $|x + y| \leq |x| + |y|$.

Q: How do you graph absolute value functions?

A: To graph an absolute value function, you need to consider two cases: one where the expression inside the absolute value is non-negative, and one where the expression inside the absolute value is negative.

Q: What are the solutions to the equation $|x| = 3$?

A: The solutions to the equation $|x| = 3$ are $x = 3$ and $x = -3$.

Q: What are the solutions to the equation $|x - 2| = 1$?

A: The solutions to the equation $|x - 2| = 1$ are $x = 3$ and $x = 1$.

Q: How do you solve absolute value inequalities?

A: To solve an absolute value inequality, you need to consider two cases: one where the expression inside the absolute value is non-negative, and one where the expression inside the absolute value is negative.

Q: What are the solutions to the inequality $|x| < 3$?

A: The solutions to the inequality $|x| < 3$ are $x < -3$ and $x > 3$.

Q: What are the solutions to the inequality $|x - 2| > 1$?

A: The solutions to the inequality $|x - 2| > 1$ are $x < 1$ and $x > 3$.

Advanced Topics

Q: What is the concept of absolute value in real-world applications?

A: Absolute value has numerous real-world applications, including:

• Distance and speed: Absolute value is used to calculate distance and speed in physics and engineering.
• Finance: Absolute value is used to calculate profit and loss in finance.
• Computer science: Absolute value is used in algorithms and data structures.

Q: How do you use absolute value in programming?

A: Absolute value is used in programming to:

• Calculate distance and speed: Absolute value is used to calculate distance and speed in physics and engineering.
• Check for errors: Absolute value is used to check for errors in programming.
• Optimize algorithms: Absolute value is used to optimize algorithms and data structures.

Conclusion

In conclusion, absolute value equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in algebra and beyond. We've provided a comprehensive guide to help you master absolute value equations, including frequently asked questions, properties, and real-world applications. Whether you're a student or a professional, this guide will help you navigate the world of absolute value equations with confidence.

Additional Resources

• Absolute Value Equations Worksheet: A comprehensive worksheet with exercises and solutions to help you practice absolute value equations.
• Absolute Value Functions Graphing Guide: A step-by-step guide to graphing absolute value functions.
• Absolute Value Inequalities Solutions: A comprehensive guide to solving absolute value inequalities.

Final Thoughts

Absolute value equations are a powerful tool in mathematics, and understanding how to solve them is crucial for success in algebra and beyond. With this guide, you'll be able to master absolute value equations and apply them to real-world problems. Whether you're a student or a professional, this guide will help you navigate the world of absolute value equations with confidence.