Answer :
Certainly! Let's break down and solve the given problem step-by-step.
The problem provides the function:
[tex]\[ y = 1 - \frac{x}{3x - 5} \][/tex]
and asks us to find its simplest form and the domain of [tex]\( x \)[/tex].
### Simplifying the Function
We start by simplifying the expression for [tex]\( y \)[/tex]:
1. Combine the terms in the fraction:
[tex]\[ y = 1 - \frac{x}{3x - 5} \][/tex]
2. Find a common denominator for the terms to combine the expression into a single fraction:
[tex]\[ y = \frac{(3x-5) - x}{3x-5} \][/tex]
3. Simplify the numerator:
[tex]\[ y = \frac{3x - 5 - x}{3x - 5} \][/tex]
4. Combine the like terms in the numerator:
[tex]\[ y = \frac{2x - 5}{3x - 5} \][/tex]
So, the simplest form of the given function is:
[tex]\[ y = \frac{2x - 5}{3x - 5} \][/tex]
### Finding the Domain
Next, we need to determine the domain of [tex]\( x \)[/tex] such that the function [tex]\( y \)[/tex] is defined.
1. Identify points where the function might be undefined:
The function is undefined when the denominator is zero. For the denominator [tex]\( 3x - 5 \)[/tex], we set it to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 3x - 5 = 0 \][/tex]
[tex]\[ 3x = 5 \][/tex]
[tex]\[ x = \frac{5}{3} \][/tex]
2. State the domain excluding the undefined points:
The function is defined for all real values of [tex]\( x \)[/tex], except for [tex]\( x = \frac{5}{3} \)[/tex].
Thus, the domain of [tex]\( x \)[/tex] is:
[tex]\[ x \in \mathbb{R} \setminus \left\{\frac{5}{3}\right\} \][/tex]
### Conclusion
The simplest form of the function [tex]\( y = 1 - \frac{x}{3x - 5} \)[/tex] is:
[tex]\[ y = \frac{2x - 5}{3x - 5} \][/tex]
The domain of [tex]\( x \)[/tex] is:
[tex]\[ x \in \mathbb{R} \setminus \left\{\frac{5}{3}\right\} \][/tex]
This completes the solution for the given problem.
The problem provides the function:
[tex]\[ y = 1 - \frac{x}{3x - 5} \][/tex]
and asks us to find its simplest form and the domain of [tex]\( x \)[/tex].
### Simplifying the Function
We start by simplifying the expression for [tex]\( y \)[/tex]:
1. Combine the terms in the fraction:
[tex]\[ y = 1 - \frac{x}{3x - 5} \][/tex]
2. Find a common denominator for the terms to combine the expression into a single fraction:
[tex]\[ y = \frac{(3x-5) - x}{3x-5} \][/tex]
3. Simplify the numerator:
[tex]\[ y = \frac{3x - 5 - x}{3x - 5} \][/tex]
4. Combine the like terms in the numerator:
[tex]\[ y = \frac{2x - 5}{3x - 5} \][/tex]
So, the simplest form of the given function is:
[tex]\[ y = \frac{2x - 5}{3x - 5} \][/tex]
### Finding the Domain
Next, we need to determine the domain of [tex]\( x \)[/tex] such that the function [tex]\( y \)[/tex] is defined.
1. Identify points where the function might be undefined:
The function is undefined when the denominator is zero. For the denominator [tex]\( 3x - 5 \)[/tex], we set it to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 3x - 5 = 0 \][/tex]
[tex]\[ 3x = 5 \][/tex]
[tex]\[ x = \frac{5}{3} \][/tex]
2. State the domain excluding the undefined points:
The function is defined for all real values of [tex]\( x \)[/tex], except for [tex]\( x = \frac{5}{3} \)[/tex].
Thus, the domain of [tex]\( x \)[/tex] is:
[tex]\[ x \in \mathbb{R} \setminus \left\{\frac{5}{3}\right\} \][/tex]
### Conclusion
The simplest form of the function [tex]\( y = 1 - \frac{x}{3x - 5} \)[/tex] is:
[tex]\[ y = \frac{2x - 5}{3x - 5} \][/tex]
The domain of [tex]\( x \)[/tex] is:
[tex]\[ x \in \mathbb{R} \setminus \left\{\frac{5}{3}\right\} \][/tex]
This completes the solution for the given problem.
