Simplifying Rational Expressions - Math

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Question

Simplify $\frac{x+2}{x-1}$+\frac{x-5}{x+3}$

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Answer

This is a more complicated form of $\frac{1}{2}$+\frac{1}{3}$= $\frac{3}{6}$+\frac{2}{6}$=\frac{5}{6}$

Find the least common denominator (LCD) and convert each fraction to the LCD, then add the numerators. Simplify as needed.

$\frac{x+2}{x-1}$+\frac{x-5}{x+3}$=\frac{x+2}{x-1}$\cdot $\frac{x+3}{x+3}$+\frac{x-5}{x+3}$\cdot $\frac{x-1}{x-1}$

which is equivalent to $\frac{(x+2)\cdot (x+3)+(x-5)\cdot (x-1)}{(x+3)\cdot (x-1)}$

Simplify to get

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Question

Divide and simplify the following rational expression:

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Answer

Multiply by the reciprocal of the second expression:

Factor the expressions:

$\frac{(x+5)(x+2)}{x+2}$\cdot $\frac{(x-3)(x-2)}{(x-3)(x+5)}$

Remove common terms:

$$\frac{\mathbf{(x+5)(x+2)}$}{\mathbf{x+2}}\cdot $\frac{\mathbf{(x-3)}$(x-2)}{\mathbf{(x-3)(x+5)}}$

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Question

Divide and simplify the following rational expression:

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Answer

Multiply by the inverse of the denominator:

Factor:

$\frac{(x-8)(x-3)}{(x-8)(x-10)}$\cdot $\frac{(x-5)(x-10)}{(x-5)(x-4)}$

Remove like terms:

$$\frac{(\mathbf{x-8}$)(x-3)}{(\mathbf{x-8})(\mathbf{x-10})}\cdot $\frac{(\mathbf{x-5}$)(\mathbf{x-10})}{(\mathbf{x-5})(x-4)}$

$\frac{x-3}{x-4}$

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Question

Add and simplify the following rational expression:

$3n+1+\frac{1}{3n-1}$$

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Answer

Begin by multiplying the left term by $\frac{3n-1}{3n-1}$ :

$3n+1+\frac{1}{3n-1}$$

$3n+1 ($\frac{3n-1}{3n-1}$)+\frac{1}{3n-1}$$

Simplify:

$\frac{(3n+1)(3n-1)+1}{3n-1}$

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Question

Simplify the following rational expression:

$$\frac{\frac{x+y}{x}$}{$\frac{1}{x}$+\frac{1}{y}$}$

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Answer

Begin by combining the terms in the denominator:

$$\frac{\frac{x+y}{x}$}{$\frac{1}{x}$+\frac{1}{y}$}$

$$\frac{\frac{x+y}{x}$}{$\frac{x+y}{xy}$}$

Multiply by the reciprocal of the denominator:

$\frac{x+y}{x}$ \cdot $\frac{xy}{x+y}$

Remove like terms:

$$\frac{\mathbf{x+y}$}{\mathbf{x}} \cdot $\frac{\boldsymbol{x}$y}{\mathbf{x+y}}$

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Question

Simplify the following rational expression:

$$\frac{n+5+\frac{3}{n+1}$}{n-1-$\frac{3}{n+1}$}$

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Answer

Create a common denominator of in both the numerator and denominator:

$$\frac{n+5+\frac{3}{n+1}$}{n-1-$\frac{3}{n+1}$}$

$$\frac{\frac{(n+5)(n+1)+3}{n+1}$}{$\frac{(n-1)(n+1)-3}{n+1}$}$

Multiply by the reciprocal of the denominator:

$\frac{(n+5)(n+1)+3}{n+1}$\cdot $\frac{n+1}{(n-1)(n+1)-3}$

Simplify:

$\frac{(n+4)(n+2)}{n+1}$\cdot $\frac{n+1}{(n-2)(n+2)}$

Remove common terms:

$$\frac{(n+4)\mathbf{(n+2)}$}{\mathbf{n+1}}\cdot $\frac{\mathbf{n+1}$}{(n-2)(\mathbf{n+2})}$

$\frac{n+4}{n-2}$

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Question

Multiply and simplify the following rational expression:

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Answer

Factor the expression:

Remove like terms:

$$\frac{a-b}{b-a}$=\frac{-1(b-a)}{b-a}$=-1$

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Question

Simplify $\frac{x+2}{x-1}$+\frac{x-5}{x+3}$

Tap to reveal answer

Answer

This is a more complicated form of $\frac{1}{2}$+\frac{1}{3}$= $\frac{3}{6}$+\frac{2}{6}$=\frac{5}{6}$

Find the least common denominator (LCD) and convert each fraction to the LCD, then add the numerators. Simplify as needed.

$\frac{x+2}{x-1}$+\frac{x-5}{x+3}$=\frac{x+2}{x-1}$\cdot $\frac{x+3}{x+3}$+\frac{x-5}{x+3}$\cdot $\frac{x-1}{x-1}$

which is equivalent to $\frac{(x+2)\cdot (x+3)+(x-5)\cdot (x-1)}{(x+3)\cdot (x-1)}$

Simplify to get

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Question

Divide and simplify the following rational expression:

Tap to reveal answer

Answer

Multiply by the reciprocal of the second expression:

Factor the expressions:

$\frac{(x+5)(x+2)}{x+2}$\cdot $\frac{(x-3)(x-2)}{(x-3)(x+5)}$

Remove common terms:

$$\frac{\mathbf{(x+5)(x+2)}$}{\mathbf{x+2}}\cdot $\frac{\mathbf{(x-3)}$(x-2)}{\mathbf{(x-3)(x+5)}}$

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Question

Divide and simplify the following rational expression:

Tap to reveal answer

Answer

Multiply by the inverse of the denominator:

Factor:

$\frac{(x-8)(x-3)}{(x-8)(x-10)}$\cdot $\frac{(x-5)(x-10)}{(x-5)(x-4)}$

Remove like terms:

$$\frac{(\mathbf{x-8}$)(x-3)}{(\mathbf{x-8})(\mathbf{x-10})}\cdot $\frac{(\mathbf{x-5}$)(\mathbf{x-10})}{(\mathbf{x-5})(x-4)}$

$\frac{x-3}{x-4}$

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Question

Add and simplify the following rational expression:

$3n+1+\frac{1}{3n-1}$$

Tap to reveal answer

Answer

Begin by multiplying the left term by $\frac{3n-1}{3n-1}$ :

$3n+1+\frac{1}{3n-1}$$

$3n+1 ($\frac{3n-1}{3n-1}$)+\frac{1}{3n-1}$$

Simplify:

$\frac{(3n+1)(3n-1)+1}{3n-1}$

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Question

Simplify the following rational expression:

$$\frac{\frac{x+y}{x}$}{$\frac{1}{x}$+\frac{1}{y}$}$

Tap to reveal answer

Answer

Begin by combining the terms in the denominator:

$$\frac{\frac{x+y}{x}$}{$\frac{1}{x}$+\frac{1}{y}$}$

$$\frac{\frac{x+y}{x}$}{$\frac{x+y}{xy}$}$

Multiply by the reciprocal of the denominator:

$\frac{x+y}{x}$ \cdot $\frac{xy}{x+y}$

Remove like terms:

$$\frac{\mathbf{x+y}$}{\mathbf{x}} \cdot $\frac{\boldsymbol{x}$y}{\mathbf{x+y}}$

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Question

Simplify the following rational expression:

$$\frac{n+5+\frac{3}{n+1}$}{n-1-$\frac{3}{n+1}$}$

Tap to reveal answer

Answer

Create a common denominator of in both the numerator and denominator:

$$\frac{n+5+\frac{3}{n+1}$}{n-1-$\frac{3}{n+1}$}$

$$\frac{\frac{(n+5)(n+1)+3}{n+1}$}{$\frac{(n-1)(n+1)-3}{n+1}$}$

Multiply by the reciprocal of the denominator:

$\frac{(n+5)(n+1)+3}{n+1}$\cdot $\frac{n+1}{(n-1)(n+1)-3}$

Simplify:

$\frac{(n+4)(n+2)}{n+1}$\cdot $\frac{n+1}{(n-2)(n+2)}$

Remove common terms:

$$\frac{(n+4)\mathbf{(n+2)}$}{\mathbf{n+1}}\cdot $\frac{\mathbf{n+1}$}{(n-2)(\mathbf{n+2})}$

$\frac{n+4}{n-2}$

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Question

Multiply and simplify the following rational expression:

Tap to reveal answer

Answer

Factor the expression:

Remove like terms:

$$\frac{a-b}{b-a}$=\frac{-1(b-a)}{b-a}$=-1$

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Question

Simplify $\frac{x+2}{x-1}$+\frac{x-5}{x+3}$

Tap to reveal answer

Answer

This is a more complicated form of $\frac{1}{2}$+\frac{1}{3}$= $\frac{3}{6}$+\frac{2}{6}$=\frac{5}{6}$

Find the least common denominator (LCD) and convert each fraction to the LCD, then add the numerators. Simplify as needed.

$\frac{x+2}{x-1}$+\frac{x-5}{x+3}$=\frac{x+2}{x-1}$\cdot $\frac{x+3}{x+3}$+\frac{x-5}{x+3}$\cdot $\frac{x-1}{x-1}$

which is equivalent to $\frac{(x+2)\cdot (x+3)+(x-5)\cdot (x-1)}{(x+3)\cdot (x-1)}$

Simplify to get

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Question

Divide and simplify the following rational expression:

Tap to reveal answer

Answer

Multiply by the reciprocal of the second expression:

Factor the expressions:

$\frac{(x+5)(x+2)}{x+2}$\cdot $\frac{(x-3)(x-2)}{(x-3)(x+5)}$

Remove common terms:

$$\frac{\mathbf{(x+5)(x+2)}$}{\mathbf{x+2}}\cdot $\frac{\mathbf{(x-3)}$(x-2)}{\mathbf{(x-3)(x+5)}}$

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Question

Add and simplify the following rational expression:

$3n+1+\frac{1}{3n-1}$$

Tap to reveal answer

Answer

Begin by multiplying the left term by $\frac{3n-1}{3n-1}$ :

$3n+1+\frac{1}{3n-1}$$

$3n+1 ($\frac{3n-1}{3n-1}$)+\frac{1}{3n-1}$$

Simplify:

$\frac{(3n+1)(3n-1)+1}{3n-1}$

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Question

Simplify the following rational expression:

$$\frac{\frac{x+y}{x}$}{$\frac{1}{x}$+\frac{1}{y}$}$

Tap to reveal answer

Answer

Begin by combining the terms in the denominator:

$$\frac{\frac{x+y}{x}$}{$\frac{1}{x}$+\frac{1}{y}$}$

$$\frac{\frac{x+y}{x}$}{$\frac{x+y}{xy}$}$

Multiply by the reciprocal of the denominator:

$\frac{x+y}{x}$ \cdot $\frac{xy}{x+y}$

Remove like terms:

$$\frac{\mathbf{x+y}$}{\mathbf{x}} \cdot $\frac{\boldsymbol{x}$y}{\mathbf{x+y}}$

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Question

Simplify the following rational expression:

$$\frac{n+5+\frac{3}{n+1}$}{n-1-$\frac{3}{n+1}$}$

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Answer

Create a common denominator of in both the numerator and denominator:

$$\frac{n+5+\frac{3}{n+1}$}{n-1-$\frac{3}{n+1}$}$

$$\frac{\frac{(n+5)(n+1)+3}{n+1}$}{$\frac{(n-1)(n+1)-3}{n+1}$}$

Multiply by the reciprocal of the denominator:

$\frac{(n+5)(n+1)+3}{n+1}$\cdot $\frac{n+1}{(n-1)(n+1)-3}$

Simplify:

$\frac{(n+4)(n+2)}{n+1}$\cdot $\frac{n+1}{(n-2)(n+2)}$

Remove common terms:

$$\frac{(n+4)\mathbf{(n+2)}$}{\mathbf{n+1}}\cdot $\frac{\mathbf{n+1}$}{(n-2)(\mathbf{n+2})}$

$\frac{n+4}{n-2}$

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Question

Multiply and simplify the following rational expression:

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Answer

Factor the expression:

Remove like terms:

$$\frac{a-b}{b-a}$=\frac{-1(b-a)}{b-a}$=-1$

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