To solve this problem, we need to isolate [tex]\( x \)[/tex] in the given equation, [tex]\(\frac{a}{b} + x = \frac{c}{d}\)[/tex], and then express [tex]\( x \)[/tex] to determine if it is a rational number.
1. Start with the given equation:
[tex]\[ \frac{a}{b} + x = \frac{c}{d} \][/tex]
2. Isolate [tex]\( x \)[/tex] by subtracting [tex]\(\frac{a}{b}\)[/tex] from both sides:
[tex]\[ x = \frac{c}{d} - \frac{a}{b} \][/tex]
3. To subtract these fractions, we need a common denominator. The common denominator of [tex]\( b \)[/tex] and [tex]\( d \)[/tex] is [tex]\( b \times d \)[/tex].
4. Convert both fractions to have this common denominator:
[tex]\[
\frac{c}{d} = \frac{c \cdot b}{d \cdot b} = \frac{c \cdot b}{bd}
\][/tex]
[tex]\[
\frac{a}{b} = \frac{a \cdot d}{b \cdot d} = \frac{a \cdot d}{bd}
\][/tex]
5. Now, substitute these equivalent fractions back into the equation:
[tex]\[
x = \frac{c \cdot b}{bd} - \frac{a \cdot d}{bd}
\][/tex]
6. Since the denominators are the same, we can combine the numerators:
[tex]\[
x = \frac{c \cdot b - a \cdot d}{bd}
\][/tex]
7. Therefore, we have:
[tex]\[
x = \frac{c b - a d}{b d}
\][/tex]
This form of [tex]\( x \)[/tex] tells us that [tex]\( x \)[/tex] is expressed as a fraction where both the numerator [tex]\( c b - a d \)[/tex] and the denominator [tex]\( b d \)[/tex] are products and differences of the given integers [tex]\( a, b, c, \)[/tex] and [tex]\( d \)[/tex]. Since the numerator and the denominator are both integers, this confirms that [tex]\( x \)[/tex] is a rational number, as a ratio of two integers.
Thus, the statement that corresponds to the equation to prove that [tex]\( x \)[/tex] is rational is:
[tex]\[ \boxed{x = \frac{c b - a d}{bd}} \][/tex]
This corresponds to the third option provided in the original question.
