What Is The Solution Set To The Equation $(3x - 9)(5x - 3) = 0$?A. { 3 5 , 3 } \left\{\frac{3}{5}, 3\right\} { 5 3 ​ , 3 } B. { 3 } \{3\} { 3 } C. { − 3 , − 3 5 } \left\{-3, -\frac{3}{5}\right\} { − 3 , − 5 3 ​ } D. { 3 5 , 5 3 } \left\{\frac{3}{5}, \frac{5}{3}\right\} { 5 3 ​ , 3 5 ​ }

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Understanding the Problem

To find the solution set to the equation (3x - 9)(5x - 3) = 0, we need to find the values of x that satisfy the equation. The equation is a quadratic equation in its factored form, and we can use the zero-product property to solve for x.

The Zero-Product Property

The zero-product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In this case, we have the equation (3x - 9)(5x - 3) = 0, and we can use the zero-product property to write:

3x - 9 = 0 or 5x - 3 = 0

Solving the First Equation

To solve the first equation, 3x - 9 = 0, we can add 9 to both sides of the equation to get:

3x = 9

Next, we can divide both sides of the equation by 3 to get:

x = 3

Solving the Second Equation

To solve the second equation, 5x - 3 = 0, we can add 3 to both sides of the equation to get:

5x = 3

Next, we can divide both sides of the equation by 5 to get:

x = 3/5

Finding the Solution Set

Now that we have found the values of x that satisfy the equation, we can write the solution set as:

{x | x = 3 or x = 3/5}

Simplifying the Solution Set

We can simplify the solution set by writing it in set notation as:

{x | x ∈ {3, 3/5}}

Conclusion

The solution set to the equation (3x - 9)(5x - 3) = 0 is {3, 3/5}. This means that the values of x that satisfy the equation are x = 3 and x = 3/5.

Final Answer

The final answer is {3,35}\boxed{\left\{3, \frac{3}{5}\right\}}.

Comparison with Options

Let's compare our solution with the options given:

A. {35,3}\left\{\frac{3}{5}, 3\right\}

B. {3}\{3\}

C. {−3,−35}\left\{-3, -\frac{3}{5}\right\}

D. {35,53}\left\{\frac{3}{5}, \frac{5}{3}\right\}

Our solution matches option A, which is {35,3}\left\{\frac{3}{5}, 3\right\}.

Final Conclusion

In conclusion, the solution set to the equation (3x - 9)(5x - 3) = 0 is {35,3}\left\{\frac{3}{5}, 3\right\}. This means that the values of x that satisfy the equation are x = 3 and x = 3/5.

Q: What is the zero-product property, and how is it used to solve the equation (3x - 9)(5x - 3) = 0?

A: The zero-product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In this case, we can use the zero-product property to write:

3x - 9 = 0 or 5x - 3 = 0

Q: How do I solve the first equation, 3x - 9 = 0?

A: To solve the first equation, 3x - 9 = 0, we can add 9 to both sides of the equation to get:

3x = 9

Next, we can divide both sides of the equation by 3 to get:

x = 3

Q: How do I solve the second equation, 5x - 3 = 0?

A: To solve the second equation, 5x - 3 = 0, we can add 3 to both sides of the equation to get:

5x = 3

Next, we can divide both sides of the equation by 5 to get:

x = 3/5

Q: What is the solution set to the equation (3x - 9)(5x - 3) = 0?

A: The solution set to the equation (3x - 9)(5x - 3) = 0 is {3, 3/5}. This means that the values of x that satisfy the equation are x = 3 and x = 3/5.

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

A: We can write the solution set in set notation as:

{x | x ∈ {3, 3/5}}

Q: What is the final answer to the equation (3x - 9)(5x - 3) = 0?

A: The final answer is {3,35}\boxed{\left\{3, \frac{3}{5}\right\}}.

Q: How do I compare the solution with the options given?

A: We can compare our solution with the options given:

A. {35,3}\left\{\frac{3}{5}, 3\right\}

B. {3}\{3\}

C. {−3,−35}\left\{-3, -\frac{3}{5}\right\}

D. {35,53}\left\{\frac{3}{5}, \frac{5}{3}\right\}

Our solution matches option A, which is {35,3}\left\{\frac{3}{5}, 3\right\}.

Q: What is the final conclusion about the solution set to the equation (3x - 9)(5x - 3) = 0?

A: In conclusion, the solution set to the equation (3x - 9)(5x - 3) = 0 is {35,3}\left\{\frac{3}{5}, 3\right\}. This means that the values of x that satisfy the equation are x = 3 and x = 3/5.

Q: What are some common mistakes to avoid when solving the equation (3x - 9)(5x - 3) = 0?

A: Some common mistakes to avoid when solving the equation (3x - 9)(5x - 3) = 0 include:

  • Not using the zero-product property to solve the equation
  • Not adding 9 to both sides of the first equation
  • Not adding 3 to both sides of the second equation
  • Not dividing both sides of the first equation by 3
  • Not dividing both sides of the second equation by 5

Q: How can I practice solving equations like (3x - 9)(5x - 3) = 0?

A: You can practice solving equations like (3x - 9)(5x - 3) = 0 by:

  • Working through practice problems in your textbook or online resources
  • Using online tools or calculators to check your work
  • Asking a teacher or tutor for help if you get stuck
  • Joining a study group or math club to work with others on math problems

Q: What are some real-world applications of solving equations like (3x - 9)(5x - 3) = 0?

A: Solving equations like (3x - 9)(5x - 3) = 0 has many real-world applications, including:

  • Modeling population growth or decline
  • Calculating the area or perimeter of a shape
  • Determining the cost or profit of a business
  • Solving problems in physics, engineering, or other fields that involve mathematical modeling.