What is the simplified form of the rational expression
4x−6

6−4x
?

Question

What is the simplified form of the rational expression 
4x−6

6−4x
 ?

BlackTom AI · Accepted Answer

To tackle the simplification, start by recognizing any common factors in the numerator and the denominator.
Option: rewrite 6 − 4x as −(4x − 6) to see the relationship between the numerator and denominator more clearly.
Step 1: Factor out the greatest common factor from each part: 4x − 6 = 2(2x − 3) and 6 − 4x = −2(2x − 3).
Step 2: Substitute these factors back into the expression: (4x − 6) / (6 − 4x) = [2(2x − 3)] / [−2(2x − 3)].
Step 3: Cancel the common factor (2x − 3) from numerator and denominator, keeping in mind the domain restriction that 2x − 3 ≠ 0 (i.e., x ≠ 3/2) to avoid division by zero.
Step 4: After cancellation, the expression reduces to 2 / (−2) = −1, as long as the canceled factor was not zero.
Step 5: Note the domain condition: the original expression is undefined when 4x − 6 = 0, which again gives x = 3/2. At all other x, the simplified value is −1.
Step 6: An alternate way to see this is to observe that 6 − 4x = −(4x − 6); thus the whole fraction is (4x − 6) / [−(4x − 6)] = −1 for all allowable x, reinforcing the same conclusion.
In summary, the expression simplifies to −1 for all x except x = 3/2, where the original expression is undefined.

ALGII141-027

What is the simplified form of the rational expression 4x−6 6−4x ?

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