Simplify each rational expression. State all non-permissive values (restrictions)

Question

(𝑥+5)(2𝑥−3) / 𝑥(𝑥+5) ÷ 𝑥^2(2𝑥−3) / (𝑥+5)(𝑥−2)

Answer 1

To simplify the expression, we will first write it as a multiplication of fractions rather than division by inverting the second fraction.

(𝑥+5)(2𝑥−3) / 𝑥(𝑥+5) ÷ 𝑥^2(2𝑥−3) / (𝑥+5)(𝑥−2)

= [(𝑥+5)(2𝑥−3) / 𝑥(𝑥+5)] * [(𝑥+5)(𝑥−2) / 𝑥^2(2𝑥−3)]

= (𝑥+5)(2𝑥−3)(𝑥+5)(𝑥−2) / 𝑥(𝑥+5) * 𝑥^2(2𝑥−3)

= (𝑥+5)(2𝑥−3)(𝑥+5)(𝑥−2) / 𝑥 * 𝑥(𝑥+5) * 𝑥^2(2𝑥−3)

= (2𝑥−3)(𝑥+5)(𝑥−2) / 𝑥^3(2𝑥−3)

= [(2𝑥−3)(𝑥+5)(𝑥−2)] / [𝑥^3(2𝑥−3)]

The non-permissive values (restrictions) are the values of 𝑥 that make the denominator(s) equal to zero. In this case, the denominator of the original expression is 𝑥(𝑥+5)(𝑥−2), so the non-permissive values are 𝑥 = 0, 𝑥 = -5, and 𝑥 = 2.

Therefore, the simplified expression is [(2𝑥−3)(𝑥+5)(𝑥−2)] / [𝑥^3(2𝑥−3)] with non-permissive values 𝑥 = 0, 𝑥 = -5, and 𝑥 = 2.