Simplifying Rational Expressions - Math

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Question

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

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 $\frac{2x^{2}-x+11}{x^{2}+2x-3}$

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Question

Divide and simplify the following rational expression:

$\frac{x^2+7x+10}{x+2}\div \frac{x^2+2x-15}{x^2-5x+6}$

Answer

Multiply by the reciprocal of the second expression:

$\frac{x^2+7x+10}{x+2}\div \frac{x^2+2x-15}{x^2-5x+6}$

$\frac{x^2+7x+10}{x+2}\cdot \frac{x^2-5x+6}{x^2+2x-15}$

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:

$\frac{x^2-11x+24}{x^2-18x+80}\div \frac{x^2-9x+20}{x^2-15x+50}$

Answer

Multiply by the inverse of the denominator:

$\frac{x^2-11x+24}{x^2-18x+80}\div \frac{x^2-9x+20}{x^2-15x+50}$

$\frac{x^2-11x+24}{x^2-18x+80}\cdot \frac{x^2-15x+50}{x^2-9x+20}$

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:

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

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:

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

$\frac{9n^2-1+1}{3n-1}$

$\frac{9n^2}{3n-1}$

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Question

Simplify the following rational expression:

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

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:

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

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:

$\frac{n^2+6n+8}{n+1}\cdot \frac{n+1}{n^2-4}$

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

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:

$\frac{a^3-b^3}{b^2-a^2} \cdot \frac{a+b}{a^2+ab+b^2}$

Answer

Factor the expression:

$\frac{(a-b)(a^2+ab+b^2)(a+b)}{(b-a)(b+a)(a^2+ab+b^2)}$

Remove like terms:

$\frac{(a-b)(\mathbf{a^2+ab+b^2})(\mathbf{a+b})}{(b-a)(\mathbf{b+a})(\mathbf{a^2+ab+b^2})}$

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

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Question

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

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 $\frac{2x^{2}-x+11}{x^{2}+2x-3}$

Compare your answer with the correct one above

Question

Divide and simplify the following rational expression:

$\frac{x^2+7x+10}{x+2}\div \frac{x^2+2x-15}{x^2-5x+6}$

Answer

Multiply by the reciprocal of the second expression:

$\frac{x^2+7x+10}{x+2}\div \frac{x^2+2x-15}{x^2-5x+6}$

$\frac{x^2+7x+10}{x+2}\cdot \frac{x^2-5x+6}{x^2+2x-15}$

Factor the expressions:

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

Remove common terms:

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

Compare your answer with the correct one above

Question

Add and simplify the following rational expression:

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

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:

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

$\frac{9n^2-1+1}{3n-1}$

$\frac{9n^2}{3n-1}$

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Question

Simplify the following rational expression:

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

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:

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

Compare your answer with the correct one above

Question

Simplify the following rational expression:

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

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:

$\frac{n^2+6n+8}{n+1}\cdot \frac{n+1}{n^2-4}$

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

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:

$\frac{a^3-b^3}{b^2-a^2} \cdot \frac{a+b}{a^2+ab+b^2}$

Answer

Factor the expression:

$\frac{(a-b)(a^2+ab+b^2)(a+b)}{(b-a)(b+a)(a^2+ab+b^2)}$

Remove like terms:

$\frac{(a-b)(\mathbf{a^2+ab+b^2})(\mathbf{a+b})}{(b-a)(\mathbf{b+a})(\mathbf{a^2+ab+b^2})}$

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

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Question

Divide and simplify the following rational expression:

$\frac{x^2-11x+24}{x^2-18x+80}\div \frac{x^2-9x+20}{x^2-15x+50}$

Answer

Multiply by the inverse of the denominator:

$\frac{x^2-11x+24}{x^2-18x+80}\div \frac{x^2-9x+20}{x^2-15x+50}$

$\frac{x^2-11x+24}{x^2-18x+80}\cdot \frac{x^2-15x+50}{x^2-9x+20}$

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

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

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 $\frac{2x^{2}-x+11}{x^{2}+2x-3}$

Compare your answer with the correct one above

Question

Divide and simplify the following rational expression:

$\frac{x^2+7x+10}{x+2}\div \frac{x^2+2x-15}{x^2-5x+6}$

Answer

Multiply by the reciprocal of the second expression:

$\frac{x^2+7x+10}{x+2}\div \frac{x^2+2x-15}{x^2-5x+6}$

$\frac{x^2+7x+10}{x+2}\cdot \frac{x^2-5x+6}{x^2+2x-15}$

Factor the expressions:

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

Remove common terms:

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

Compare your answer with the correct one above

Question

Add and simplify the following rational expression:

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

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:

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

$\frac{9n^2-1+1}{3n-1}$

$\frac{9n^2}{3n-1}$

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Question

Simplify the following rational expression:

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

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:

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

Compare your answer with the correct one above

Question

Simplify the following rational expression:

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

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:

$\frac{n^2+6n+8}{n+1}\cdot \frac{n+1}{n^2-4}$

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

Remove common terms:

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

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

Compare your answer with the correct one above

Question

Multiply and simplify the following rational expression:

$\frac{a^3-b^3}{b^2-a^2} \cdot \frac{a+b}{a^2+ab+b^2}$

Answer

Factor the expression:

$\frac{(a-b)(a^2+ab+b^2)(a+b)}{(b-a)(b+a)(a^2+ab+b^2)}$

Remove like terms:

$\frac{(a-b)(\mathbf{a^2+ab+b^2})(\mathbf{a+b})}{(b-a)(\mathbf{b+a})(\mathbf{a^2+ab+b^2})}$

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

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