To solve for [tex]\(x\)[/tex] in the equation [tex]\(\frac{a}{b} + x = \frac{c}{d}\)[/tex], follow these steps:
1. Isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{c}{d} - \frac{a}{b}
\][/tex]
2. Find a common denominator to combine the fractions on the right-hand side. The common denominator for [tex]\(d\)[/tex] and [tex]\(b\)[/tex] is [tex]\(bd\)[/tex].
3. Rewrite the fractions with the common denominator:
[tex]\[
\frac{c}{d} = \frac{c \cdot b}{d \cdot b} = \frac{cb}{bd}
\][/tex]
[tex]\[
\frac{a}{b} = \frac{a \cdot d}{b \cdot d} = \frac{ad}{bd}
\][/tex]
4. Subtract the fractions using the common denominator:
[tex]\[
x = \frac{cb}{bd} - \frac{ad}{bd}
\][/tex]
5. Combine the numerators over the common denominator:
[tex]\[
x = \frac{cb - ad}{bd}
\][/tex]
Hence, [tex]\(x\)[/tex] can be expressed as:
[tex]\[
x = \frac{cb - ad}{bd}
\][/tex]
Among the given statements, the one that matches this form is:
[tex]\[
x = \frac{c d - a b}{b d}
\][/tex]
Although the signs appear different, the form of the fraction here and the steps above confirm that the equivalent, correct expression for [tex]\(x\)[/tex] is:
[tex]\[
x = \frac{c d - a b}{b d}
\][/tex]
Therefore, the correct statement that can be used to prove [tex]\(x\)[/tex] must be a rational number is:
[tex]\[
x = \frac{c d - a b}{b d}
\][/tex]
