To tackle this problem, let's break it down into steps:
### Part a: Finding the Restrictions on the Variable
The given equation is:
[tex]\[
\frac{2}{x} = \frac{13}{8x} + 3
\][/tex]
We need to determine the values of [tex]\( x \)[/tex] that make any of the denominators equal to zero.
1. Denominator from the first term: [tex]\(\frac{2}{x}\)[/tex]
[tex]\[ x \neq 0 \][/tex]
2. Denominator from the second term: [tex]\(\frac{13}{8x}\)[/tex]
[tex]\[ 8x \neq 0 \][/tex]
[tex]\[ x \neq 0 \][/tex] (since 8 is a non-zero constant)
Thus, the value of [tex]\( x \)[/tex] that makes the denominator zero is:
[tex]\[ x = 0 \][/tex]
### Part b: Solving the Equation
Knowing that [tex]\( x = 0 \)[/tex] is a restriction and cannot be a solution, let's solve the equation:
[tex]\[
\frac{2}{x} = \frac{13}{8x} + 3
\][/tex]
To eliminate the denominators, we can multiply every term in the equation by the least common multiple (LCM) of [tex]\( x \)[/tex] and [tex]\( 8x \)[/tex], which is [tex]\( 8x \)[/tex]:
[tex]\[
8x \cdot \frac{2}{x} = 8x \cdot \frac{13}{8x} + 8x \cdot 3
\][/tex]
Simplify each term:
[tex]\[
8 \cdot 2 = 8 \cdot \frac{13}{8} + 8x \cdot 3
\][/tex]
[tex]\[
16 = 13 + 24x
\][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[
16 = 13 + 24x
\][/tex]
Subtract 13 from both sides:
[tex]\[
16 - 13 = 24x
\][/tex]
[tex]\[
3 = 24x
\][/tex]
Divide by 24:
[tex]\[
x = \frac{3}{24}
\][/tex]
[tex]\[
x = \frac{1}{8}
\][/tex]
### Conclusion
a. What is/are the value or values of the variable that make a denominator zero?
[tex]\[
x = 0
\][/tex]
b. Solution to the equation keeping the restrictions in mind:
[tex]\[
x = \frac{1}{8}
\][/tex]
So, the correct solution to the problem is [tex]\( x = \frac{1}{8} \)[/tex], which is within the allowable range since it does not make the denominators zero.
