What Is The Solution To $-4|-2x + 6| = -24$?A. $x = 0$ B. $x = 0$ Or $x = -6$ C. $x = 0$ Or $x = 6$ D. No Solution

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What is the Solution to the Equation $-4|-2x + 6| = -24$?

Understanding the Equation

The given equation is a linear equation involving absolute value. To solve it, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: When the Expression Inside the Absolute Value is Positive

When the expression inside the absolute value is positive, we have:

−4(2x+6)=−24-4(2x + 6) = -24

To solve for x, we need to isolate the variable x. We can start by dividing both sides of the equation by -4:

2x+6=62x + 6 = 6

Next, we can subtract 6 from both sides of the equation:

2x=02x = 0

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

x=0x = 0

Case 2: When the Expression Inside the Absolute Value is Negative

When the expression inside the absolute value is negative, we have:

−4(−2x+6)=−24-4(-2x + 6) = -24

To solve for x, we need to isolate the variable x. We can start by dividing both sides of the equation by -4:

−2x+6=6-2x + 6 = 6

Next, we can subtract 6 from both sides of the equation:

−2x=0-2x = 0

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

x=0x = 0

Combining the Results

In both cases, we found that x = 0. However, we need to consider the original equation and the definition of absolute value. When the expression inside the absolute value is negative, the absolute value is equal to the negation of the expression. Therefore, we need to consider the case when the expression inside the absolute value is negative.

Solving for x

When the expression inside the absolute value is negative, we have:

−2x+6<0-2x + 6 < 0

Solving for x, we get:

−2x<−6-2x < -6

Dividing both sides of the equation by -2, we get:

x>3x > 3

However, we also need to consider the case when the expression inside the absolute value is positive. When the expression inside the absolute value is positive, we have:

−2x+6>0-2x + 6 > 0

Solving for x, we get:

−2x>−6-2x > -6

Dividing both sides of the equation by -2, we get:

x<3x < 3

Conclusion

Combining the results from both cases, we can see that x = 0 is a solution to the equation. However, we also need to consider the case when the expression inside the absolute value is negative. In this case, we found that x > 3. Therefore, the solution to the equation is x = 0 or x = -6.

The Final Answer

The final answer is B. x=0x = 0 or x=−6x = -6.
Q&A: Understanding the Solution to the Equation $-4|-2x + 6| = -24$

Q: What is the main concept behind solving the equation $-4|-2x + 6| = -24$?

A: The main concept behind solving the equation is to consider two cases: when the expression inside the absolute value is positive and when it is negative. This is because the absolute value function has two different behaviors depending on the sign of the expression inside it.

Q: How do you handle the absolute value in the equation?

A: To handle the absolute value, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative. When the expression inside the absolute value is positive, we can simply remove the absolute value sign and solve the resulting equation. When the expression inside the absolute value is negative, we need to negate the expression and then remove the absolute value sign.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to isolate the absolute value expression. In this case, we can do this by dividing both sides of the equation by -4.

Q: How do you solve for x in the equation?

A: To solve for x, we need to isolate the variable x. We can do this by performing algebraic operations such as addition, subtraction, multiplication, and division.

Q: What is the solution to the equation?

A: The solution to the equation is x = 0 or x = -6. This is because we found that x = 0 is a solution to the equation, and we also found that x > 3 when the expression inside the absolute value is negative.

Q: Why is it important to consider both cases when solving the equation?

A: It is important to consider both cases because the absolute value function has two different behaviors depending on the sign of the expression inside it. If we only consider one case, we may miss some solutions to the equation.

Q: What are some common mistakes to avoid when solving equations with absolute value?

A: Some common mistakes to avoid when solving equations with absolute value include:

  • Not considering both cases when the expression inside the absolute value is positive and negative.
  • Not isolating the absolute value expression before solving for x.
  • Not performing algebraic operations correctly to isolate the variable x.

Q: How can I practice solving equations with absolute value?

A: You can practice solving equations with absolute value by working through examples and exercises in a textbook or online resource. You can also try solving equations with absolute value on your own by creating your own examples and solving them.

Q: What are some real-world applications of solving equations with absolute value?

A: Solving equations with absolute value has many real-world applications, including:

  • Modeling physical systems such as motion and vibration.
  • Solving problems in finance and economics.
  • Modeling population growth and decline.
  • Solving problems in engineering and physics.

Q: Can you provide more examples of solving equations with absolute value?

A: Yes, here are a few more examples:

  • ∣2x−3∣=5|2x - 3| = 5

  • ∣x+2∣=3|x + 2| = 3

  • ∣2x−1∣=4|2x - 1| = 4

These are just a few examples, and there are many more equations with absolute value that you can solve.