Select The Correct Answer.Solve:${ |x-3|-10=-5 }$A. { X=2 $}$ Or { X=8 $}$ B. { X=-2 $}$ Or { X=8 $}$ C. { X=8 $}$ Or { X=18 $}$ D. { X=-8 $}$ Or [$ X=2

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Understanding Absolute Value Equations

Absolute value equations are a type of algebraic 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. Absolute value equations are used to solve problems that involve distances, temperatures, and other quantities that can be positive or negative.

The Given Equation

The given equation is:

∣x−3∣−10=−5{ |x-3|-10=-5 }

This equation involves the absolute value of the expression x−3x-3. Our goal is to solve for xx.

Step 1: Isolate the Absolute Value Expression

To solve the equation, we need to isolate the absolute value expression. We can do this by adding 10 to both sides of the equation:

∣x−3∣=−5+10{ |x-3| = -5 + 10 }

∣x−3∣=5{ |x-3| = 5 }

Step 2: Write Two Separate Equations

Since the absolute value of an expression can be positive or negative, we need to write two separate equations:

x−3=5{ x-3 = 5 }

x−3=−5{ x-3 = -5 }

Step 3: Solve Each Equation

Now, we need to solve each equation separately.

Equation 1: x-3 = 5

To solve for xx, we need to add 3 to both sides of the equation:

x=5+3{ x = 5 + 3 }

x=8{ x = 8 }

Equation 2: x-3 = -5

To solve for xx, we need to add 3 to both sides of the equation:

x=−5+3{ x = -5 + 3 }

x=−2{ x = -2 }

Conclusion

Based on our solutions, we have two possible values for xx: x=8x=8 and x=−2x=-2. Therefore, the correct answer is:

A. x=2x=2 or x=8x=8

However, this is not among the options. Let's re-evaluate our solutions.

We have two possible values for xx: x=8x=8 and x=−2x=-2. Therefore, the correct answer is:

B. x=−2x=-2 or x=8x=8

This is among the options. Therefore, the correct answer is:

B. x=−2x=-2 or x=8x=8

Why is this the correct answer?

This is the correct answer because we have two possible values for xx: x=8x=8 and x=−2x=-2. These values satisfy the original equation:

∣x−3∣−10=−5{ |x-3|-10=-5 }

When x=8x=8, we have:

∣8−3∣−10=−5{ |8-3|-10=-5 }

∣5∣−10=−5{ |5|-10=-5 }

5−10=−5{ 5-10=-5 }

−5=−5{ -5=-5 }

This is true.

When x=−2x=-2, we have:

∣−2−3∣−10=−5{ |-2-3|-10=-5 }

∣−5∣−10=−5{ |-5|-10=-5 }

5−10=−5{ 5-10=-5 }

−5=−5{ -5=-5 }

This is also true.

Therefore, both x=8x=8 and x=−2x=-2 are solutions to the original equation.

Final Answer

The final answer is:

Q: What is an absolute value equation?

A: An absolute value equation is a type of algebraic 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 isolate the absolute value expression and then write two separate equations. One equation will have a positive value, and the other will have a negative value. You then solve each equation separately.

Q: What is the first step in solving an absolute value equation?

A: The first step in solving an absolute value equation is to isolate the absolute value expression. This means getting the absolute value expression by itself on one side of the equation.

Q: How do I isolate the absolute value expression?

A: To isolate the absolute value expression, you need to add or subtract the same value from both sides of the equation. This will get the absolute value expression by itself.

Q: What happens if the absolute value expression is equal to a negative number?

A: If the absolute value expression is equal to a negative number, then the equation has no solution. This is because the absolute value of a number is always non-negative.

Q: Can I have multiple solutions to an absolute value equation?

A: Yes, you can have multiple solutions to an absolute value equation. This happens when the absolute value expression is equal to a positive number, and there are two possible values for the variable.

Q: How do I know which solution to choose?

A: To choose the correct solution, you need to check which solution satisfies the original equation. You can do this by plugging the solution back into the original equation and checking if it is true.

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

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

  • Not isolating the absolute value expression
  • Not writing two separate equations
  • Not solving each equation separately
  • Not checking which solution satisfies the original equation

Q: Can I use absolute value equations to solve real-world problems?

A: Yes, you can use absolute value equations to solve real-world problems. Absolute value equations are used to model problems that involve distances, temperatures, and other quantities that can be positive or negative.

Q: What are some examples of real-world problems that can be solved using absolute value equations?

A: Some examples of real-world problems that can be solved using absolute value equations include:

  • Finding the distance between two points on a number line
  • Calculating the temperature difference between two locations
  • Determining the amount of money that needs to be paid or received in a financial transaction

Conclusion

Solving absolute value equations can be a challenging task, but with practice and patience, you can become proficient in solving these types of equations. Remember to isolate the absolute value expression, write two separate equations, and solve each equation separately. By following these steps and avoiding common mistakes, you can solve absolute value equations and apply them to real-world problems.

Final Tips

  • Make sure to check which solution satisfies the original equation
  • Use absolute value equations to model real-world problems
  • Practice solving absolute value equations to become proficient

Common Absolute Value Equations

Here are some common absolute value equations that you may encounter:

  • ∣x−2∣=3|x-2|=3
  • ∣x+1∣=4|x+1|=4
  • ∣x−5∣=2|x-5|=2
  • ∣x+3∣=1|x+3|=1

Solving Absolute Value Equations: A Step-by-Step Guide

Here is a step-by-step guide to solving absolute value equations:

  1. Isolate the absolute value expression
  2. Write two separate equations
  3. Solve each equation separately
  4. Check which solution satisfies the original equation

By following these steps and practicing solving absolute value equations, you can become proficient in solving these types of equations and apply them to real-world problems.