The following rational equation has denominators that contain variables. For this equation:

[tex]\[ \frac{3}{x} + 3 = \frac{5}{3x} + \frac{19}{6} \][/tex]

a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable.

What is/are the value or values of the variable that make(s) the denominators zero?

[tex]\[ x = \][/tex]

[tex]\[ \square \][/tex] (Simplify your answer. Use a comma to separate answers as needed.)

b. Keeping the restrictions in mind, solve the equation.



Answer :

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]