Solve For $x$.$|2x + 14| = 0$

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Introduction

In mathematics, equations involving absolute values can be challenging to solve. The absolute value of a number is its distance from zero on the number line, and it can be positive or negative. When we have an equation with an absolute value, we need to consider both the positive and negative cases. In this article, we will focus on solving the equation $|2x + 14| = 0$, which involves an absolute value.

Understanding Absolute Values

Before we dive into solving the equation, let's review what absolute values are. The absolute value of a number $a$, denoted by $|a|$, is defined as:

∣a∣={a,if a≥0−a,if a<0|a| = \begin{cases} a, & \text{if } a \geq 0 \\ -a, & \text{if } a < 0 \end{cases}

In other words, if $a$ is non-negative, its absolute value is simply $a$, and if $a$ is negative, its absolute value is the opposite of $a$, which is $-a$.

Solving the Equation

Now that we understand absolute values, let's solve the equation $|2x + 14| = 0$. To do this, we need to consider two cases:

Case 1: $2x + 14 \geq 0$

In this case, the absolute value of $2x + 14$ is simply $2x + 14$, since it is non-negative. So, we can write the equation as:

2x+14=02x + 14 = 0

To solve for $x$, we can subtract 14 from both sides of the equation:

2x=−142x = -14

Next, we can divide both sides of the equation by 2:

x=−7x = -7

Case 2: $2x + 14 < 0$

In this case, the absolute value of $2x + 14$ is the opposite of $2x + 14$, which is $-2x - 14$. So, we can write the equation as:

−2x−14=0-2x - 14 = 0

To solve for $x$, we can add 14 to both sides of the equation:

−2x=14-2x = 14

Next, we can divide both sides of the equation by -2:

x=−7x = -7

Conclusion

In this article, we solved the equation $|2x + 14| = 0$, which involves an absolute value. We considered two cases: when $2x + 14 \geq 0$ and when $2x + 14 < 0$. In both cases, we found that the solution to the equation is $x = -7$. This shows that when we have an equation with an absolute value, we need to consider both the positive and negative cases to find the solution.

Frequently Asked Questions

  • What is the absolute value of a number?
  • How do we solve an equation with an absolute value?
  • What are the two cases we need to consider when solving an equation with an absolute value?

Final Thoughts

Solving equations involving absolute values can be challenging, but by considering both the positive and negative cases, we can find the solution. In this article, we solved the equation $|2x + 14| = 0$ and found that the solution is $x = -7$. We hope this article has provided you with a better understanding of how to solve equations involving absolute values.

Additional Resources

  • Khan Academy: Absolute Value Equations
  • Mathway: Absolute Value Equations
  • Wolfram Alpha: Absolute Value Equations

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "College Algebra" by James Stewart
  • "Mathematics for the Nonmathematician" by Morris Kline

Introduction

In our previous article, we solved the equation $|2x + 14| = 0$, which involves an absolute value. Absolute value equations can be challenging to solve, but with the right approach, we can find the solution. In this article, we will answer some frequently asked questions about absolute value equations.

Q&A

Q: What is the absolute value of a number?

A: The absolute value of a number is its distance from zero on the number line. It can be positive or negative.

Q: How do we solve an equation with an absolute value?

A: To solve an equation with an absolute value, we need to consider two cases: when the expression inside the absolute value is non-negative and when it is negative.

Q: What are the two cases we need to consider when solving an equation with an absolute value?

A: The two cases are:

  • Case 1: The expression inside the absolute value is non-negative.
  • Case 2: The expression inside the absolute value is negative.

Q: How do we handle the absolute value in each case?

A: In Case 1, we can simply remove the absolute value sign and solve the equation. In Case 2, we need to multiply the expression inside the absolute value by -1 to make it non-negative.

Q: Can we always find a solution to an absolute value equation?

A: No, we cannot always find a solution to an absolute value equation. If the expression inside the absolute value is always non-negative, then the equation has no solution.

Q: How do we know if the expression inside the absolute value is always non-negative?

A: We can check if the expression inside the absolute value is always non-negative by plugging in different values of the variable.

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

A: An absolute value equation is an equation that involves an absolute value, while an absolute value inequality is an inequality that involves an absolute value.

Q: How do we solve an absolute value inequality?

A: To solve an absolute value inequality, we need to consider two cases: when the expression inside the absolute value is non-negative and when it is negative.

Q: Can we use the same approach to solve absolute value inequalities as we do for absolute value equations?

A: No, we cannot use the same approach to solve absolute value inequalities as we do for absolute value equations. We need to use a different approach to solve absolute value inequalities.

Conclusion

In this article, we answered some frequently asked questions about absolute value equations. We hope this article has provided you with a better understanding of how to solve absolute value equations and inequalities.

Frequently Asked Questions

  • What is the absolute value of a number?
  • How do we solve an equation with an absolute value?
  • What are the two cases we need to consider when solving an equation with an absolute value?
  • Can we always find a solution to an absolute value equation?
  • How do we know if the expression inside the absolute value is always non-negative?
  • What is the difference between an absolute value equation and an inequality?
  • How do we solve an absolute value inequality?

Final Thoughts

Solving absolute value equations and inequalities can be challenging, but with the right approach, we can find the solution. We hope this article has provided you with a better understanding of how to solve absolute value equations and inequalities.

Additional Resources

  • Khan Academy: Absolute Value Equations and Inequalities
  • Mathway: Absolute Value Equations and Inequalities
  • Wolfram Alpha: Absolute Value Equations and Inequalities

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "College Algebra" by James Stewart
  • "Mathematics for the Nonmathematician" by Morris Kline