What Is The Solution Of $|x-1|=-8$?A. $x=-9$ Or $x=7$ B. No Solution C. $x=-7$ Or $x=9$ D. $x=-8$ Or $x=8$

What is the Solution of $|x-1|=-8$?

Understanding Absolute Value Equations

Absolute value equations are a type of 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, without considering direction. In other words, it is the magnitude of the number. When dealing with absolute value equations, we need to consider both the positive and negative possibilities of the expression inside the absolute value bars.

The Equation $|x-1|=-8$

The given equation is $|x-1|=-8$. To solve this equation, we need to consider the two possible cases:

• Case 1: $x-1$ is positive, which means $x-1 \geq 0$.
• Case 2: $x-1$ is negative, which means $x-1 < 0$.

Case 1: $x-1$ is Positive

If $x-1$ is positive, then the absolute value of $x-1$ is equal to $x-1$ itself. So, we have:

$$|x-1| = x-1$$

Substituting this into the original equation, we get:

$$x-1 = -8$$

Solving for $x$, we get:

$$x = -7$$

However, this solution does not satisfy the condition that $x-1$ is positive, since $-7-1=-8$ is negative. Therefore, this solution is not valid.

Case 2: $x-1$ is Negative

If $x-1$ is negative, then the absolute value of $x-1$ is equal to $-(x-1)$. So, we have:

$$|x-1| = -(x-1)$$

Substituting this into the original equation, we get:

$$-(x-1) = -8$$

Solving for $x$, we get:

$$x-1 = 8$$

$$x = 9$$

However, this solution does not satisfy the condition that $x-1$ is negative, since $9-1=8$ is positive. Therefore, this solution is not valid.

Conclusion

Since neither of the two cases yields a valid solution, we conclude that the equation $|x-1|=-8$ has no solution.

Answer

The correct answer is B. no solution.

Additional Tips and Tricks

When dealing with absolute value equations, it's essential to consider both the positive and negative possibilities of the expression inside the absolute value bars. Additionally, be careful when solving for the variable, as the solution may not satisfy the condition that the expression inside the absolute value bars is positive or negative.

Common Mistakes to Avoid

• Failing to consider both the positive and negative possibilities of the expression inside the absolute value bars.
• Not checking if the solution satisfies the condition that the expression inside the absolute value bars is positive or negative.
• Not being careful when solving for the variable.

Real-World Applications

Absolute value equations have numerous real-world applications, such as:

• Modeling distance and time problems in physics and engineering.
• Solving problems involving financial transactions, such as calculating the absolute value of a stock's price change.
• Modeling population growth and decline in biology and ecology.

Conclusion

In conclusion, the equation $|x-1|=-8$ has no solution. When dealing with absolute value equations, it's essential to consider both the positive and negative possibilities of the expression inside the absolute value bars and be careful when solving for the variable.
Frequently Asked Questions (FAQs) About Absolute Value Equations

Q: What is an absolute value equation?

A: An absolute value equation is a type of 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, without considering direction.

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 possibilities of the expression inside the absolute value bars. You can do this by setting up two separate equations, one for each possibility, and then solving for the variable.

Q: What is the difference between a positive and negative absolute value equation?

A: A positive absolute value equation is one where the expression inside the absolute value bars is positive, while a negative absolute value equation is one where the expression inside the absolute value bars is negative.

Q: How do I determine whether an absolute value equation is positive or negative?

A: To determine whether an absolute value equation is positive or negative, you need to consider the sign of the expression inside the absolute value bars. If the expression is positive, the absolute value equation is positive. If the expression is negative, the absolute value equation is negative.

Q: What is the solution to the equation $|x-1|=8$?

A: To solve the equation $|x-1|=8$, you need to consider both the positive and negative possibilities of the expression inside the absolute value bars. This gives you two separate equations:

$$x-1=8$$

and

$$-(x-1)=8$$

Solving for $x$ in both equations, you get:

$$x=9$$

and

$$x=-7$$

Therefore, the solution to the equation $|x-1|=8$ is $x=9$ or $x=-7$.

Q: What is the solution to the equation $|x+2|=0$?

A: To solve the equation $|x+2|=0$, you need to consider both the positive and negative possibilities of the expression inside the absolute value bars. This gives you two separate equations:

$$x+2=0$$

and

$$-(x+2)=0$$

Solving for $x$ in both equations, you get:

$$x=-2$$

and

$$x=-2$$

Therefore, the solution to the equation $|x+2|=0$ is $x=-2$.

Q: What is the solution to the equation $|x-3|=x-3$?

A: To solve the equation $|x-3|=x-3$, you need to consider both the positive and negative possibilities of the expression inside the absolute value bars. This gives you two separate equations:

$$x-3=x-3$$

and

$$-(x-3)=x-3$$

Solving for $x$ in both equations, you get:

$$x=3$$

and

$$x=3$$

Therefore, the solution to the equation $|x-3|=x-3$ is $x=3$.

Q: What is the solution to the equation $|x+1|=-(x+1)$?

A: To solve the equation $|x+1|=-(x+1)$, you need to consider both the positive and negative possibilities of the expression inside the absolute value bars. This gives you two separate equations:

$$x+1=-(x+1)$$

and

$$-(x+1)=-(x+1)$$

Solving for $x$ in both equations, you get:

$$x=-1$$

and

$$x=-1$$

Therefore, the solution to the equation $|x+1|=-(x+1)$ is $x=-1$.

Conclusion

In conclusion, absolute value equations are a type of mathematical equation that involves the absolute value of a variable or expression. To solve an absolute value equation, you need to consider both the positive and negative possibilities of the expression inside the absolute value bars. By following the steps outlined in this article, you can solve a wide range of absolute value equations and gain a deeper understanding of this important mathematical concept.