Problem 65 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{4 x-6}{3-2 x}$$

Step-by-Step Solution

Answer

The simplified form of the rational expression is -2.

Step 1: Factoring out common terms

Look for any common factors in the numerator and the denominator. In this case, both \(4x - 6\) and \(3 - 2x\) have common factors. The numerator can be factored as \(2(2x - 3)\) and the denominator can be factored as \(1(3 - 2x)\)

Step 2: Reverse the signs

The denominator has the term \(2x\) with a negative sign. To make this similar to the numerator, reverse the signs in the denominator, yielding \(-1(-3 + 2x)\). This does not change the value of the expression because multiplication by \(-1\) just changes the sign, which is equivalent to switching the order of subtraction.

Step 3: Simplify the expression

Now the numerator and the denominator have similar terms: \(2(2x - 3)\) and \(-1(2x - 3)\). These terms can be cancelled out, so the simplified expression becomes \(\frac{-2}{1}\).