Certainly! Let's tackle this problem step by step.
### Step a: Identify the Restrictions
We need to determine the values of \( x \) that make the denominators zero because these values are not allowed in the solution.
The given equation is:
[tex]\[
\frac{3}{x} + 3 = \frac{5}{3x} + \frac{19}{6}
\][/tex]
The denominators in this equation are \( x \) and \( 3x \).
For \( \frac{3}{x} \), the denominator \( x \) cannot be zero.
For \( \frac{5}{3x} \), the denominator \( 3x \) also cannot be zero.
This restriction is because division by zero is undefined.
Therefore, the values that make the denominators zero are:
[tex]\[
x = 0
\][/tex]
### Step b: Solve the Equation
Next, we solve the rational equation while keeping in mind the restriction that \( x \neq 0 \).
The given equation is:
[tex]\[
\frac{3}{x} + 3 = \frac{5}{3x} + \frac{19}{6}
\][/tex]
First, let's find a common denominator for all terms, which is \( 6x \).
Rewriting each term with this common denominator:
[tex]\[
\frac{3 \cdot 6}{6x} + \frac{3 \cdot 6x}{6x} = \frac{5 \cdot 2}{6x} + \frac{19 \cdot x}{6x}
\][/tex]
Simplifying each term:
[tex]\[
\frac{18}{6x} + \frac{18x}{6x} = \frac{10}{6x} + \frac{19x}{6x}
\][/tex]
Now, combining the left-hand side and the right-hand side:
[tex]\[
\frac{18 + 18x}{6x} = \frac{10 + 19x}{6x}
\][/tex]
Since the denominators on both sides are the same, we can equate the numerators:
[tex]\[
18 + 18x = 10 + 19x
\][/tex]
Next, solve for \( x \):
Subtract \( 18x \) from both sides:
[tex]\[
18 = 10 + x
\][/tex]
Subtract 10 from both sides:
[tex]\[
8 = x
\][/tex]
So, the solution to the equation is:
[tex]\[
x = 8
\][/tex]
### Final Answer:
a. The value that makes the denominator zero is:
[tex]\[
x = 0
\][/tex]
b. The solution of the equation is:
[tex]\[
x = 8
\][/tex]
