Let's start with the given equation:
[tex]\[
x + \frac{a}{b} = \frac{c}{d}
\][/tex]
Our goal is to isolate [tex]\( x \)[/tex]. First, let's rearrange this equation to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{c}{d} - \frac{a}{b}
\][/tex]
To subtract these fractions, we need a common denominator. The common denominator of [tex]\( \frac{c}{d} \)[/tex] and [tex]\( \frac{a}{b} \)[/tex] is [tex]\( bd \)[/tex]. Let's rewrite both fractions with this common denominator:
[tex]\[
x = \frac{c \cdot b}{d \cdot b} - \frac{a \cdot d}{b \cdot d}
\][/tex]
[tex]\[
x = \frac{cb}{bd} - \frac{ad}{bd}
\][/tex]
Now that both fractions have the same denominator, we can combine them into one fraction:
[tex]\[
x = \frac{cb - ad}{bd}
\][/tex]
This is a single fraction where both the numerator and the denominator are products of integers, hence [tex]\( x \)[/tex] is rational because it is the ratio of two integers.
Therefore, the correct statement that shows [tex]\( x \)[/tex] must be rational is:
[tex]\[
x = \frac{cb - ad}{bd}
\][/tex]
This corresponds to the third option in the given list:
[tex]\[
\boxed{\frac{cb - ad}{bd}}
\][/tex]
Thus, the correct option is the third one, and it proves that [tex]\( x \)[/tex] must be a rational number.
