Question

Create a rational expression that simplifies to 2x/(x+1)

and that has the following restrictions on x:

x ≠ −1, 0, 2, 3. Write your expression here.

-Contains multiplication of two rational
expressions
-Contains division of two rational expressions

Answer 1

One possible expression that meets the given requirements is:

(2x)/(x+1) = (2x/[(x-2)(x-3)]) / ([(x+1)/(x-2)(x-3)])

This expression simplifies to 2x/(x+1) when x is not equal to -1, 0, 2, or 3, as required.

Explanation: We can rewrite 2x/(x+1) as (2x/(x-2)(x-3)) * ((x-2)(x-3)/(x+1)). The first term in this expression is a division of two rational expressions, while the second term is a multiplication of two rational expressions. Then, we can simplify the first term by cancelling the (x-2)(x-3) terms in the numerator and denominator, which gives 2x/[(x-2)(x-3)]. We can also simplify the second term by expanding the denominator, which gives (x-2)(x-3)/(x-2)(x-3)(x+1) = 1/[(x-2)(x-3)] * 1/(x+1). Then, we can combine the two simplified terms to get the expression given above.

Explanation:

Answer 2

One possible rational expression that simplifies to 2x/(x+1) and satisfies the given restrictions on x is:

(2x)/(x+1) = (4x/4)/((x-2)(x-3)/(4(x-2)(x-3)))

= 4x/(x^2 - 5x + 6)

This expression contains multiplication of two rational expressions (4x/4 and (x-2)(x-3)/(4(x-2)(x-3))) and division of two rational expressions (4x/4 divided by (x-2)(x-3)/(4(x-2)(x-3))).

The restrictions on x are satisfied because the denominator x^2 - 5x + 6 factors as (x-2)(x-3) and thus the expression is undefined at x = -1, 0, 2, and 3 but simplifies to 2x/(x+1) for all other values of x.