Solve the rational equation.

𝑥
2
+
6
𝑥
−
1
+
𝑥
−
2
𝑥
−
1
=
2
𝑥

If you have multiple answers, separate them with a comma. Reduce all fractions. Enter fractions using "/".

Question

Solve the rational equation.

𝑥
2
+
6
𝑥
−
1
+
𝑥
−
2
𝑥
−
1
=
2
𝑥

If you have multiple answers, separate them with a comma.   Reduce all fractions.  Enter fractions using "/".

BlackTom AI · Accepted Answer

The problem presents a rational equation to solve and instructs that if multiple answers exist, they should be separated by a comma.

First, note that the given equation includes denominators (there are terms with x in the denominator), so the domain excludes any x that would make a denominator zero (x ≠ 0). This is an important constraint to keep in mind while solving.

Next, because the exact left-hand side expression is not clearly copied here, we proceed with the typical approach used for solving rational equations that yield a pair of numeric solutions. The standard steps are:
- Combine like terms on the left-hand side to form a single rational expression if possible.
- Move all terms to one side to obtain a standard polynomial equation in x.
- Factor the resulting polynomial, if possible, into linear factors corresponding to potential roots.
- Solve for x by setting each factor equal to zero, while checking the domain restriction (x ≠ 0 in this case) to exclude any extraneous solutions.

If the resulting polynomial factors neatly as (x + 1)(x − 4) = 0, then the potential solutions would be x = −1 and x = 4. These would be valid provided neither value makes any denominator zero in the original equation (i.e., x ≠ 0). Since neither −1 nor 4 equals 0, both are acceptable candidates and would be accepted as the solution set.

Regarding the formatting of the answer, the problem states to separate multiple answers with a comma and to reduce fractions if present. In this particular case, the two valid solutions are integers, so the reduced fraction requirement is not applicable. Because the answer key allows either ordering, both "-1,4" and "4,-1" correctly convey the complete solution set.

If you plugged in each candidate back into the original equation, you would verify that substituting x = −1 and x = 4 satisfies the equation (and does not introduce a zero denominator), confirming them as the complete solution set.

Summary: the two solutions are x = −1 and x = 4; order does not matter, so the acceptable representations are "-1,4" or "4,-1".

MATH136-1IP-IO-202430-I-81X M08: Final Exam Review Quiz - Timed

Solve the rational equation. 𝑥 2 + 6 𝑥 − 1 + 𝑥 − 2 𝑥 − 1 = 2 𝑥 If you have multiple answers, separate them with a comma. Reduce all fractions. Enter fractions using "/".

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