Question

learning goal i can describe the similarities between the set of rational numbers and rational expressions. i can add and subtract rational expressions. independent practice lesson 9.1 checkpoint how i did (circle one) starting... getting there... got it once you have completed the above problems and checked your solutions, complete the lesson checkpoint below. complete the lesson reflection above by circling your current understanding of the learning goal. given the rational expression shown below, select all of the statements that are true about the completely simplified form of this rational expression. 1. $\frac{-3}{3x + 2}+\frac{x - 4}{3x^{2}+2x}$ a. the denominator is $3x^{2}+5x + 2$ b. the denominator is $3x^{2}+2x$ c. the numerator is $x - 7$ d. the numerator is $-2x - 4$ put the numbers into the boxes to represent the sum of the rational expression given below. 2. $\frac{-10x + 1}{3x - 1}+\frac{-16x - 6}{3x - 1}$ $\frac{square x+square}{square x+square}$ solution box -2 -6 3 5 -26 -1 6 -5 combine the rational expression into a single fraction and reduce it in lowest terms. 3. $\frac{-11}{x^{2}-3x - 28}-\frac{x}{x^{2}-2x - 24}$

Explanation:

Step1: Find common denominator for $\frac{-3}{3x + 2}+\frac{x - 4}{3x^{2}+2x}$

Factor $3x^{2}+2x=x(3x + 2)$. The common - denominator is $x(3x + 2)$.

Step2: Rewrite fractions with common denominator

$\frac{-3}{3x + 2}\times\frac{x}{x}=\frac{-3x}{x(3x + 2)}$ and $\frac{x - 4}{3x^{2}+2x}=\frac{x - 4}{x(3x + 2)}$. Then $\frac{-3x+(x - 4)}{x(3x + 2)}=\frac{-3x+x - 4}{x(3x + 2)}=\frac{-2x - 4}{x(3x + 2)}=\frac{-2x - 4}{3x^{2}+2x}$. The numerator is $-2x - 4$ and the denominator is $3x^{2}+2x$. So for question 1, the correct answer is D.

Step3: Add $\frac{-10x + 1}{3x-1}+\frac{-16x - 6}{3x - 1}$

Since the denominators are the same, we add the numerators: $(-10x + 1)+(-16x - 6)=-10x+1 - 16x - 6=-26x - 5$. So the sum is $\frac{-26x - 5}{3x - 1}$, and the numbers in the boxes are -26, -5, 3, -1.

Step4: Combine $\frac{-11}{x^{2}-3x - 28}-\frac{x}{x^{2}-2x - 24}$

Factor $x^{2}-3x - 28=(x - 7)(x+4)$ and $x^{2}-2x - 24=(x - 6)(x + 4)$. The common - denominator is $(x - 7)(x + 4)(x - 6)$.
Rewrite the fractions: $\frac{-11(x - 6)}{(x - 7)(x + 4)(x - 6)}-\frac{x(x - 7)}{(x - 7)(x + 4)(x - 6)}=\frac{-11x+66-(x^{2}-7x)}{(x - 7)(x + 4)(x - 6)}=\frac{-11x + 66 - x^{2}+7x}{(x - 7)(x + 4)(x - 6)}=\frac{-x^{2}-4x + 66}{(x - 7)(x + 4)(x - 6)}$.

Answer:

1. D. The numerator is $-2x - 4$
2. $\frac{-26x - 5}{3x - 1}$ (numbers in boxes: -26, -5, 3, -1)
3. $\frac{-x^{2}-4x + 66}{(x - 7)(x + 4)(x - 6)}$