Determine The Solutions Of The Equation:${ \left|\frac{1}{3} X+9\right|-3=21 }$A. { X=-11$}$ And { X=5$}$B. { X=-81$}$ And { X=45$}$C. { X=-99$}$ And { X=45$}$D.

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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, the absolute value of a number is always non-negative.

The Equation to Solve

The given equation is:

∣13x+9∣−3=21\left|\frac{1}{3} x+9\right|-3=21

Our goal is to solve for the variable xx.

Step 1: Isolate the Absolute Value Expression

To solve the equation, we need to isolate the absolute value expression on one side of the equation. We can do this by adding 3 to both sides of the equation:

∣13x+9∣=21+3\left|\frac{1}{3} x+9\right|=21+3

∣13x+9∣=24\left|\frac{1}{3} x+9\right|=24

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:

13x+9=24or13x+9=−24\frac{1}{3} x+9=24 \quad \text{or} \quad \frac{1}{3} x+9=-24

Step 3: Solve the First Equation

Let's solve the first equation:

13x+9=24\frac{1}{3} x+9=24

Subtract 9 from both sides:

13x=15\frac{1}{3} x=15

Multiply both sides by 3:

x=45x=45

Step 4: Solve the Second Equation

Now, let's solve the second equation:

13x+9=−24\frac{1}{3} x+9=-24

Subtract 9 from both sides:

13x=−33\frac{1}{3} x=-33

Multiply both sides by 3:

x=−99x=-99

Step 5: Check the Solutions

We have found two possible solutions: x=45x=45 and x=−99x=-99. To check if these solutions are correct, we need to plug them back into the original equation:

For x=45x=45:

∣13(45)+9∣−3=21\left|\frac{1}{3} (45)+9\right|-3=21

∣15+9∣−3=21\left|15+9\right|-3=21

∣24∣−3=21\left|24\right|-3=21

24−3=2124-3=21

21=2121=21

This solution checks out!

For x=−99x=-99:

∣13(−99)+9∣−3=21\left|\frac{1}{3} (-99)+9\right|-3=21

∣−33+9∣−3=21\left|-33+9\right|-3=21

∣−24∣−3=21\left|-24\right|-3=21

24−3=2124-3=21

21=2121=21

This solution also checks out!

Conclusion

We have successfully solved the absolute value equation:

∣13x+9∣−3=21\left|\frac{1}{3} x+9\right|-3=21

The solutions are x=45x=45 and x=−99x=-99.

Final Answer

The correct answer is:

C. x=−99x=-99 and x=45x=45

Note: The other options are incorrect.

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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 on one side of the equation. Then, you can write two separate equations: one where the expression inside the absolute value is positive, and one where it is negative. Solve each equation separately, and check the solutions to make sure they are correct.

Q: What are the steps to solve an absolute value equation?

A: The steps to solve an absolute value equation are:

  1. Isolate the absolute value expression on one side of the equation.
  2. Write two separate equations: one where the expression inside the absolute value is positive, and one where it is negative.
  3. Solve each equation separately.
  4. Check the solutions to make sure they are correct.

Q: How do I know which solution to choose?

A: When solving an absolute value equation, you need to check each solution to make sure it is correct. If a solution does not satisfy the original equation, it is not a valid solution.

Q: Can I use a calculator to solve an absolute value equation?

A: Yes, you can use a calculator to solve an absolute value equation. However, it's always a good idea to check your solutions by hand to make sure they are correct.

Q: What if I have a fraction inside the absolute value?

A: If you have a fraction inside the absolute value, you can multiply both sides of the equation by the denominator to eliminate the fraction. Then, you can proceed with solving the equation as usual.

Q: Can I have a negative number inside the absolute value?

A: Yes, you can have a negative number inside the absolute value. In this case, the absolute value will make the number positive, so you can proceed with solving the equation as usual.

Q: What if I have a variable inside the absolute value?

A: If you have a variable inside the absolute value, you can use the same steps as before to solve the equation. Just remember to check your solutions to make sure they are correct.

Q: Can I have a combination of numbers and variables inside the absolute value?

A: Yes, you can have a combination of numbers and variables inside the absolute value. In this case, you can use the same steps as before to solve the equation. Just remember to check your solutions to make sure they are correct.

Q: How do I know if an absolute value equation has one or two solutions?

A: An absolute value equation can have one or two solutions, depending on the equation. If the equation is of the form |x| = a, then it has two solutions: x = a and x = -a. If the equation is of the form |x| = a + b, then it has one solution: x = a + b.

Q: Can I have a quadratic equation inside the absolute value?

A: Yes, you can have a quadratic equation inside the absolute value. In this case, you can use the quadratic formula to solve the equation. Just remember to check your solutions to make sure they are correct.

Q: What if I have a system of absolute value equations?

A: If you have a system of absolute value equations, you can solve each equation separately and then check the solutions to make sure they are correct. You can also use substitution or elimination methods to solve the system.

Conclusion

We hope this article has helped you understand absolute value equations and how to solve them. Remember to always check your solutions to make sure they are correct, and don't be afraid to ask for help if you need it.