What Is The Solution Set For ∣ 4 − X 2 ∣ = 8 \left|4-\frac{x}{2}\right|=8 ​ 4 − 2 X ​ ​ = 8 ?A. X = − 24 X=-24 X = − 24 And X = − 8 X=-8 X = − 8 B. X = − 24 X=-24 X = − 24 And X = 24 X=24 X = 24 C. X = − 8 X=-8 X = − 8 And X = 8 X=8 X = 8 D. X = − 8 X=-8 X = − 8 And X = 24 X=24 X = 24

Introduction

When dealing with 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. In this case, we have the equation $\left|4-\frac{x}{2}\right|=8$, and we want to find the solution set for $x$. This means we need to find all the values of $x$ that satisfy the equation.

Understanding Absolute Value Equations

Absolute value equations are equations that contain the absolute value of an expression. The absolute value of a number is its distance from zero on the number line, without considering whether it is positive or negative. For example, the absolute value of $-3$ is $3$, and the absolute value of $3$ is also $3$.

When dealing with 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 number is always non-negative, so if the expression inside the absolute value is negative, we need to make it positive by multiplying it by $-1$.

Case 1: $4-\frac{x}{2} \geq 0$

In this case, we can simply remove the absolute value signs and solve the equation $4-\frac{x}{2}=8$. To do this, we can first subtract $4$ from both sides of the equation, which gives us $-\frac{x}{2}=4$. Then, we can multiply both sides of the equation by $-2$ to get $x=-8$.

Case 2: $4-\frac{x}{2} < 0$

In this case, we need to make the expression inside the absolute value negative by multiplying it by $-1$. This gives us the equation $-(4-\frac{x}{2})=8$. To simplify this equation, we can distribute the negative sign to get $-4+\frac{x}{2}=8$. Then, we can add $4$ to both sides of the equation, which gives us $\frac{x}{2}=12$. Finally, we can multiply both sides of the equation by $2$ to get $x=24$.

Combining the Results

We have found two possible values for $x$: $x=-8$ and $x=24$. However, we need to check if these values satisfy the original equation. If we plug $x=-8$ into the original equation, we get $\left|4-\frac{-8}{2}\right|=\left|4+4\right|=8$, which is true. Similarly, if we plug $x=24$ into the original equation, we get $\left|4-\frac{24}{2}\right|=\left|4-12\right|=8$, which is also true.

Conclusion

In conclusion, the solution set for the equation $\left|4-\frac{x}{2}\right|=8$ is $x=-8$ and $x=24$. This means that both $x=-8$ and $x=24$ satisfy the equation, and we can write the solution set as $\boxed{x=-8 \text{ and } x=24}$.

Final Answer

The final answer is: $\boxed{x=-8 \text{ and } x=24}$

Introduction

In our previous article, we discussed the solution set for the equation $\left|4-\frac{x}{2}\right|=8$. We found that the solution set is $x=-8$ and $x=24$. In this article, we will answer some frequently asked questions about the solution set and provide additional insights.

Q&A

Q: What is the definition of an absolute value equation?

A: An absolute value equation is an equation that contains the absolute value of an expression. The absolute value of a number is its distance from zero on the number line, without considering whether it is positive or negative.

Q: How do I 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 positive, and another where it is negative. You can then solve each case separately and combine the results.

Q: What is the difference between the two cases in an absolute value equation?

A: In the first case, the expression inside the absolute value is positive, so you can simply remove the absolute value signs and solve the equation. In the second case, the expression inside the absolute value is negative, so you need to make it positive by multiplying it by $-1$.

Q: How do I check if a value satisfies an absolute value equation?

A: To check if a value satisfies an absolute value equation, you can plug the value into the equation and see if it is true. If the value satisfies the equation, then it is part of the solution set.

Q: What is the solution set for the equation $\left|4-\frac{x}{2}\right|=8$?

A: The solution set for the equation $\left|4-\frac{x}{2}\right|=8$ is $x=-8$ and $x=24$.

Q: How do I write the solution set in a box notation?

A: To write the solution set in a box notation, you can use the following format: $\boxed{x=-8 \text{ and } x=24}$.

Q: What is the final answer for the equation $\left|4-\frac{x}{2}\right|=8$?

A: The final answer for the equation $\left|4-\frac{x}{2}\right|=8$ is $\boxed{x=-8 \text{ and } x=24}$.

Conclusion

In conclusion, we have discussed the solution set for the equation $\left|4-\frac{x}{2}\right|=8$ and answered some frequently asked questions about the solution set. We hope that this article has provided additional insights and helped you to better understand the solution set.

Final Answer

The final answer is: $\boxed{x=-8 \text{ and } x=24}$