Question

add or subtract. then simplify if possible.

1. $\frac{3}{2x^2} + \frac{1}{3x}$
2. $\frac{5x}{x-3} - \frac{2x+1}{x-3}$
3. $\frac{2}{x-1} - \frac{6}{2x^2 - x - 1}$
4. which of the following rational expressions has 4 as a restri

a.) $\frac{5}{3x-6}$
b.) $\frac{4}{x+1}$
c.) $\frac{20}{x-4}$

1. factor the d

2.find commo

1. multiply
2. add or subt
3. simplify, if

Explanation:

Step1: Q11: Find common denominator

The least common denominator of $2x^2$ and $3x$ is $6x^2$.

Step2: Q11: Rewrite fractions

$\frac{3}{2x^2} = \frac{3 \times 3}{2x^2 \times 3} = \frac{9}{6x^2}$, $\frac{1}{3x} = \frac{1 \times 2x}{3x \times 2x} = \frac{2x}{6x^2}$

Step3: Q11: Add fractions

$\frac{9}{6x^2} + \frac{2x}{6x^2} = \frac{9 + 2x}{6x^2}$

Step4: Q12: Subtract numerators (same denominator)

Since denominators are equal, subtract numerators directly:
$\frac{5x - (2x + 1)}{x - 3}$

Step5: Q12: Simplify numerator

$5x - 2x - 1 = 3x - 1$, so $\frac{3x - 1}{x - 3}$

Step6: Q13: Factor denominator

Factor $2x^2 - x - 1 = (2x + 1)(x - 1)$

Step7: Q13: Find common denominator

Common denominator is $(2x + 1)(x - 1)$. Rewrite $\frac{2}{x-1}$:
$\frac{2(2x + 1)}{(x - 1)(2x + 1)} = \frac{4x + 2}{(2x + 1)(x - 1)}$

Step8: Q13: Subtract fractions

$\frac{4x + 2 - 6}{(2x + 1)(x - 1)} = \frac{4x - 4}{(2x + 1)(x - 1)}$

Step9: Q13: Simplify expression

Factor numerator: $4x - 4 = 4(x - 1)$. Cancel $(x-1)$:
$\frac{4(x - 1)}{(2x + 1)(x - 1)} = \frac{4}{2x + 1}$

Step10: Q14: Identify restriction

A rational expression has $x=4$ as a restriction when denominator is 0 at $x=4$.

• For a: $3x-6=0 \implies x=2$
• For b: $x+1=0 \implies x=-1$
• For c: $x-4=0 \implies x=4$

Answer:

1. $\boldsymbol{\frac{9 + 2x}{6x^2}}$
2. $\boldsymbol{\frac{3x - 1}{x - 3}}$
3. $\boldsymbol{\frac{2}{2x + 1}}$
4. $\boldsymbol{c. \frac{20}{x-4}}$