Isolate the absolute value term

Oct 13, 2025 by windailys.com 130 views

Introduction

When dealing with absolute value equations, it's essential to understand how to represent the solutions on a number line. In this case, we're given the equation $-2|x|=-6$ , and we need to determine which number line represents the solutions to this equation. To solve this problem, we'll first analyze the equation and then use the properties of absolute value to find the solutions.

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 it's always non-negative. For example, the absolute value of $-3$ is $3$ , and the absolute value of $4$ is also $4$ . When dealing with absolute value equations, we need to consider both the positive and negative cases.

Solving the Equation

To solve the equation $-2|x|=-6$ , we can start by isolating the absolute value term. We can do this by dividing both sides of the equation by $-2$ . This gives us:

|x| = 3

Now, we need to consider both the positive and negative cases. When $x$ is positive, the absolute value of $x$ is equal to $x$ itself. So, we have:

x = 3

When $x$ is negative, the absolute value of $x$ is equal to $-x$ . So, we have:

-x = 3

Solving for $x$ in both cases, we get:

x = 3 \text{ or } x = -3

Representing the Solutions on a Number Line

Now that we have the solutions to the equation, we can represent them on a number line. A number line is a graphical representation of the real numbers, with each point on the line corresponding to a real number. To represent the solutions on a number line, we can use a horizontal line with equally spaced points.

Case 1: $x = 3$

To represent the solution $x = 3$ on a number line, we can place a point at $x = 3$ . This point will be to the right of the origin, since $3$ is a positive number.

Case 2: $x = -3$

To represent the solution $x = -3$ on a number line, we can place a point at $x = -3$ . This point will be to the left of the origin, since $-3$ is a negative number.

Conclusion

In conclusion, the number line that represents the solutions to $-2|x|=-6$ is a horizontal line with two points: one at $x = 3$ and the other at $x = -3$ . These points are equally spaced from the origin, with the point at $x = 3$ to the right of the origin and the point at $x = -3$ to the left of the origin.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

Isolate the absolute value term: Divide both sides of the equation by $-2$ to get $|x| = 3$ .
Consider both the positive and negative cases: When $x$ is positive, the absolute value of $x$ is equal to $x$ itself. When $x$ is negative, the absolute value of $x$ is equal to $-x$ .
Solve for $x$ in both cases: We get $x = 3$ or $x = -3$ .
Represent the solutions on a number line: Place a point at $x = 3$ and another point at $x = -3$ on a horizontal line.

Frequently Asked Questions

What is the absolute value of a number?: The absolute value of a number is its distance from zero on the number line.
How do I represent the solutions to an absolute value equation on a number line?: To represent the solutions on a number line, you need to consider both the positive and negative cases and place points at the corresponding values on the number line.
What is the difference between the positive and negative cases in an absolute value equation?: In an absolute value equation, the positive case refers to the situation where the absolute value of $x$ is equal to $x$ itself, while the negative case refers to the situation where the absolute value of $x$ is equal to $-x$ .

Final Answer

The final answer is: $\boxed{3}$ and $\boxed{-3}$

Introduction

Absolute value equations are a fundamental concept in mathematics, and understanding how to solve and represent them on a number line is crucial for success in algebra and beyond. In this article, we'll delve into the world of absolute value equations, exploring the solutions and representations of various equations, including the classic $-2|x|=-6$ .

Q&A: Absolute Value Equations

Q: What is an absolute value equation?

A: An absolute value equation is a mathematical equation that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider both the positive and negative cases. When the absolute value of a number is positive, the number itself is positive. When the absolute value of a number is negative, the number itself is negative.

Q: What is the difference between the positive and negative cases in an absolute value equation?

A: In an absolute value equation, the positive case refers to the situation where the absolute value of $x$ is equal to $x$ itself, while the negative case refers to the situation where the absolute value of $x$ is equal to $-x$ .

Q: How do I represent the solutions to an absolute value equation on a number line?

A: To represent the solutions on a number line, you need to place points at the corresponding values on the number line. For example, if the solution to an absolute value equation is $x = 3$ , you would place a point at $x = 3$ on the number line.

Q: What is the significance of the absolute value in an absolute value equation?

A: The absolute value in an absolute value equation represents the distance of the variable or expression from zero on the number line. This distance is always non-negative.

Q: Can I have multiple solutions to an absolute value equation?

A: Yes, it is possible to have multiple solutions to an absolute value equation. For example, the equation $|x| = 3$ has two solutions: $x = 3$ and $x = -3$ .

Q: How do I determine the number of solutions to an absolute value equation?

A: To determine the number of solutions to an absolute value equation, you need to consider the equation itself and the properties of absolute value. If the absolute value of a number is equal to a positive number, there are two solutions. If the absolute value of a number is equal to a negative number, there are no solutions.

Q: Can I have no solutions to an absolute value equation?

A: Yes, it is possible to have no solutions to an absolute value equation. For example, the equation $|x| = -3$ has no solutions, since the absolute value of a number is always non-negative.

Q: How do I graph an absolute value equation on a number line?

A: To graph an absolute value equation on a number line, you need to place points at the corresponding values on the number line. For example, if the equation is $|x| = 3$ , you would place a point at $x = 3$ and another point at $x = -3$ on the number line.

Q: What is the relationship between absolute value equations and inequalities?

A: Absolute value equations and inequalities are closely related. An absolute value inequality is an inequality that involves the absolute value of a variable or expression. To solve an absolute value inequality, you need to consider both the positive and negative cases.

Q: Can I have multiple inequalities in an absolute value inequality?

A: Yes, it is possible to have multiple inequalities in an absolute value inequality. For example, the inequality $|x| > 3$ has two cases: $x > 3$ and $x < -3$ .

Q: How do I determine the number of inequalities in an absolute value inequality?

A: To determine the number of inequalities in an absolute value inequality, you need to consider the inequality itself and the properties of absolute value. If the absolute value of a number is greater than a positive number, there are two inequalities. If the absolute value of a number is less than a positive number, there are two inequalities.

Q: Can I have no inequalities in an absolute value inequality?

A: Yes, it is possible to have no inequalities in an absolute value inequality. For example, the inequality $|x| < -3$ has no inequalities, since the absolute value of a number is always non-negative.

Conclusion

In conclusion, absolute value equations are a fundamental concept in mathematics, and understanding how to solve and represent them on a number line is crucial for success in algebra and beyond. By considering both the positive and negative cases and using the properties of absolute value, you can solve and represent absolute value equations with ease. Whether you're dealing with a simple equation like $-2|x|=-6$ or a more complex inequality like $|x| > 3$ , the principles of absolute value equations will guide you to the solution.