Which Number Line Represents The Solutions To { -2|x|=-6$}$?

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Introduction

When dealing with absolute value equations, it's essential to understand how to represent their solutions on a number line. In this article, we will explore the process of solving absolute value equations and graphing their solutions on a number line. We will focus on the equation −2∣x∣=−6{-2|x|=-6} and determine which number line represents its solutions.

Understanding Absolute Value Equations

Absolute value equations involve the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering direction. For example, the absolute value of -3 is 3, and the absolute value of 4 is 4.

When solving absolute value equations, we need to consider two cases: one where the expression inside the absolute value is positive, and another where it is negative. This is because the absolute value of a negative number is positive, and the absolute value of a positive number is also positive.

Solving the Equation −2∣x∣=−6{-2|x|=-6}

To solve the equation −2∣x∣=−6{-2|x|=-6}, we need to isolate the absolute value expression. We can do this by dividing both sides of the equation by -2.

-2|x| = -6
|x| = -6 / -2
|x| = 3

Now, we have the equation ∣x∣=3{|x| = 3}. This means that the distance of x from zero on the number line is 3.

Graphing the Solutions on a Number Line

To graph the solutions on a number line, we need to consider two cases: one where x is positive, and another where x is negative.

Case 1: x is Positive

When x is positive, the absolute value of x is equal to x. So, we can write the equation as x = 3.

x = 3

This means that the solution to the equation is x = 3.

Case 2: x is Negative

When x is negative, the absolute value of x is equal to -x. So, we can write the equation as -x = 3.

-x = 3
x = -3

This means that the solution to the equation is x = -3.

Determining Which Number Line Represents the Solutions

Now that we have the solutions to the equation, we need to determine which number line represents them. We can do this by graphing the solutions on a number line.

  -6  -3  0  3  6

The number line above represents the solutions to the equation −2∣x∣=−6{-2|x|=-6}. The solutions are x = -3 and x = 3.

Conclusion

In this article, we explored the process of solving absolute value equations and graphing their solutions on a number line. We focused on the equation −2∣x∣=−6{-2|x|=-6} and determined which number line represents its solutions. We found that the solutions to the equation are x = -3 and x = 3, and we graphed them on a number 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, without considering direction.
  • How do you solve absolute value equations? To solve absolute value equations, you need to consider two cases: one where the expression inside the absolute value is positive, and another where it is negative.
  • How do you graph the solutions to an absolute value equation on a number line? To graph the solutions to an absolute value equation on a number line, you need to consider two cases: one where the variable is positive, and another where it is negative.

Further Reading

  • Absolute Value Equations: A Comprehensive Guide
  • Graphing Absolute Value Functions on a Number Line
  • Solving Absolute Value Inequalities

References

  • [1] "Absolute Value Equations" by Math Open Reference
  • [2] "Graphing Absolute Value Functions" by Khan Academy
  • [3] "Solving Absolute Value Inequalities" by Purplemath

Introduction

Absolute value equations can be a challenging topic for many students. However, with a clear understanding of the concepts and a step-by-step approach, solving absolute value equations can become a breeze. In this article, we will provide a comprehensive Q&A guide to help you understand and solve absolute value equations.

Q&A

Q1: What is the absolute value of a number?

A1: The absolute value of a number is its distance from zero on the number line, without considering direction.

Q2: How do you solve absolute value equations?

A2: To solve absolute value equations, you need to consider two cases: one where the expression inside the absolute value is positive, and another where it is negative.

Q3: What is the difference between an absolute value equation and an absolute value inequality?

A3: An absolute value equation is an equation that involves the absolute value of a variable or expression, whereas an absolute value inequality is an inequality that involves the absolute value of a variable or expression.

Q4: How do you graph the solutions to an absolute value equation on a number line?

A4: To graph the solutions to an absolute value equation on a number line, you need to consider two cases: one where the variable is positive, and another where it is negative.

Q5: What is the formula for solving absolute value equations?

A5: The formula for solving absolute value equations is:

|x| = a

x = a or x = -a

Q6: How do you solve absolute value equations with variables on both sides?

A6: To solve absolute value equations with variables on both sides, you need to isolate the absolute value expression and then solve for the variable.

Q7: What is the difference between a linear equation and an absolute value equation?

A7: A linear equation is an equation that involves a linear expression, whereas an absolute value equation is an equation that involves the absolute value of a variable or expression.

Q8: How do you solve absolute value equations with fractions?

A8: To solve absolute value equations with fractions, you need to isolate the absolute value expression and then solve for the variable.

Q9: What is the formula for solving absolute value inequalities?

A9: The formula for solving absolute value inequalities is:

|x| < a

x < a or x > -a

Q10: How do you graph the solutions to an absolute value inequality on a number line?

A10: To graph the solutions to an absolute value inequality on a number line, you need to consider two cases: one where the variable is positive, and another where it is negative.

Examples

Example 1: Solving an Absolute Value Equation

Solve the equation: |x| = 5

A: x = 5 or x = -5

Example 2: Solving an Absolute Value Inequality

Solve the inequality: |x| < 3

A: x < 3 or x > -3

Example 3: Solving an Absolute Value Equation with Variables on Both Sides

Solve the equation: |x + 2| = 4

A: x + 2 = 4 or x + 2 = -4

x = 2 or x = -6

Conclusion

In this article, we provided a comprehensive Q&A guide to help you understand and solve absolute value equations. We covered topics such as the definition of absolute value, solving absolute value equations, and graphing the solutions on a number line. We also provided examples to help illustrate the concepts.

Frequently Asked Questions

  • What is the absolute value of a number?
  • How do you solve absolute value equations?
  • How do you graph the solutions to an absolute value equation on a number line?
  • What is the formula for solving absolute value equations?
  • How do you solve absolute value equations with variables on both sides?

Further Reading

  • Absolute Value Equations: A Comprehensive Guide
  • Graphing Absolute Value Functions on a Number Line
  • Solving Absolute Value Inequalities

References

  • [1] "Absolute Value Equations" by Math Open Reference
  • [2] "Graphing Absolute Value Functions" by Khan Academy
  • [3] "Solving Absolute Value Inequalities" by Purplemath