Question

Solving Rational Equations Quick Check
1 of 51 of 5 Items
Question
Solve for x: 4xx−3=2+12x−3.(1 point)
Responses

no solutions
no solutions

72
Start Fraction 7 over 2 End Fraction

92
Start Fraction 9 over 2 End Fraction

2

Answer 1

To solve the given rational equation:

\[
\frac{4}{x - 3} = 2 + \frac{12}{x - 3}
\]

First, let's eliminate the fractions by multiplying both sides by \(x - 3\) (assuming \(x \neq 3\)):

\[
4 = 2(x - 3) + 12
\]

Now distribute the 2 on the right side:

\[
4 = 2x - 6 + 12
\]

Combine like terms:

\[
4 = 2x + 6
\]

Now, isolate \(2x\) by subtracting 6 from both sides:

\[
4 - 6 = 2x
\]
\[
-2 = 2x
\]

Divide by 2:

\[
x = -1
\]

Finally, let's check if \(x = -1\) causes any issues (like making the denominator zero):

\[
x - 3 = -1 - 3 = -4 \quad \text{(not zero)}
\]

Thus, the solution is valid.

Therefore, the answer is:

\[
\boxed{-1}
\]

It seems there might've been a misunderstanding with available responses, as -1 was not listed. If you need a particular response from the options like \( \frac{7}{2}, \frac{9}{2}, 2 \) or state "no solutions," the correct chosen response would be none of the above.

So to summarize, the correct solution is \( -1 \), which represents none in the selections you provided.