Question

in this problem, (a), (b), (c), and (d) are nonzero integers. if (\frac{a}{b}) is added to (x), the sum is (\frac{c}{d}). which statement can be used to prove that (x) must be a rational number?

(x = \frac{c - a}{d - b})

(x = \frac{cb + ad}{bd})

(x = \frac{cb - ad}{bd})

(x = \frac{c + a}{d - b})

Explanation:

Step1: Start with the given equation

We know that \(\frac{a}{b}+x = \frac{c}{d}\), where \(a,b,c,d\) are non - zero integers. Our goal is to solve for \(x\) by isolating it on one side of the equation.

Step2: Subtract \(\frac{a}{b}\) from both sides

To solve for \(x\), we perform the operation \(x=\frac{c}{d}-\frac{a}{b}\).

Step3: Find a common denominator

The common denominator of \(d\) and \(b\) is \(bd\). So we rewrite the fractions with the common denominator: \(\frac{c}{d}=\frac{cb}{bd}\) and \(\frac{a}{b}=\frac{ad}{bd}\).

Step4: Subtract the two fractions

Now, \(x = \frac{cb}{bd}-\frac{ad}{bd}=\frac{cb - ad}{bd}\). Since \(a,b,c,d\) are non - zero integers, \(cb - ad\) and \(bd\) are also integers (the product and difference of integers are integers), and \(bd
eq0\) (because \(b\) and \(d\) are non - zero). A number that can be written as the ratio of two integers (where the denominator is non - zero) is a rational number. So the expression \(x=\frac{cb - ad}{bd}\) can be used to prove that \(x\) is rational.

Answer:

\(x = \frac{cb - ad}{bd}\) (the third option among the given options)