Understanding Rational Expressions - Algebra II

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Question

Which of the following fractions is NOT equivalent to $- \frac{x-5}{2x + 3}$ ?

Answer

We know that $-\frac{a}{b}$ is equivalent to $\frac{-a}{b}$ or $\frac{a}{-b}$ .

By this property, there is no way to get $\frac{x+5}{2x+3}$ from $-\frac{x-5}{2x+3}$ .

Therefore the correct answer is $\frac{x+5}{2x+3}$ .

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Question

Simplify:

$\frac{x^2-1}{x^2+4x+2}\div\frac{x^2+2x+1}{x^2+3x+2}$

Answer

This problem is a lot simpler if we factor all the expressions involved before proceeding:

$\frac{x^2-1}{x^2+4x+2}\div\frac{x^2+2x+1}{x^2+3x+2} \vspace{5mm}\ = \frac{(x+1)(x-1)}{(x+2)(x+2)}\div\frac{(x+1)(x+1)}{(x+2)(x+1)}$

Next let's remember how we divide one fraction by another—by multiplying by the reciprocal:

$\frac{(x+1)(x-1)}{(x+2)(x+2)}\div\frac{(x+1)(x+1)}{(x+2)(x+1)} \vspace{5mm}\ = \frac{(x+1)(x-1)}{(x+2)(x+2)}\times\frac{(x+2)(x+1)}{(x+1)(x+1)}$

In this form, we can see that a lot of terms are going to start canceling with each other. All that we're left with is just $\frac{x-1}{x+2}$ .

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Question

Determine the domain of

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

Answer

Because the denominator cannot be zero, the domain is all other numbers except for 1, or

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Question

Which of the following is the best definition of a rational expression?

Answer

The rational expression is a ratio of two polynomials.

$\frac{A(x)}{B(x)}$

The denominator cannot be zero.

An example of a rational expression is:

$\frac{x^2+x-5}{x+8}$

The answer is:

$\textup{The ratio of two polynomials that cannot have a zero denominator.}$

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Question

$f(x)=\frac{2x-5}{x+3}-\frac{-3x-1}{x-1}$

Which of the following equations is equivalent to ?

Answer

By looking at the answer choices, we can assume that the problem wants us to simplify . To do that, we need to combine the two terms within into one fraction.

First, let's remember how to add or subtract fractions:

Make sure the fractions have the same denominator.

Add or subtract the numerators, leaving the denominator alone.

The process looks like this:

$\frac{a}{b}+\frac{c}{d}=\frac{a}{b}\left( \frac{d}{d} \right) + \frac{c}{d}\left( \frac{b}{b} \right) = \frac{ad}{bd}+\frac{cb}{bd}=\frac{ad+cb}{bd}$

This is exactly what we're going to have to do to .

$f(x)=\frac{2x-5}{x+3}-\frac{-3x-1}{x-1}$

First, we find a common denominator between the two terms. No matter what ends up being equal to, a common denominator can always be found by multiplying the two terms together. In other words, we can use as our common denominator.

$f(x)=\frac{2x-5}{x+3}\left ( \frac{x-1}{x-1} \right )-\frac{-3x-1}{x-1} \left ( \frac{x+3}{x+3} \right )$

$f(x)=\frac{(2x-5)(x-1)}{(x+3)(x-1)}-\frac{(-3x-1)(x+3)}{(x-1)(x+3)}$

$f(x)=\frac{(2x-5)(x-1)-(-3x-1)(x+3)}{(x+3)(x-1)}$

Now, all that's left is getting rid of these parentheses.

$f(x)=\frac{(2x^{2}-2x-5x+5)-(-3x^{2}-9x-x-3)}{x^{2}-x+3x-3}$

$f(x)=\frac{2x^{2}-7x+5+3x^{2}+10x+3}{x^{2}+2x-3}$

$f(x)=\frac{5x^{2}+3x+8}{x^{2}+2x-3}$

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Question

$\frac{(x^{2}-1x-6)}{x+3}+\frac{(x^{2}-6x+9)}{x+2}$

What is the least common denominator of the above expression?

Answer

The least common denominator is the least common multiple of the denominators of a set of fractions.

Simply multiply the two denominators together to find the LCD:

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Question

Simplify the expression:

$\frac{5}{x+ 6} + \frac{2}{x^{2}+ 4x-12}$

Answer

Factor the second denominator, then simplify:

$\frac{5}{x+ 6} + \frac{2}{x^{2}+ 4x-12}$

$=\frac{5}{x+ 6} + \frac{2}{(x-2)(x+6)}$

$=\frac{5 (x-2)}{ (x-2)(x+6)} + \frac{2}{(x-2)(x+6)}$

$=\frac{5x-10}{ (x-2)(x+6)} + \frac{2}{(x-2)(x+6)}$

$=\frac{5x-10 + 2}{ (x-2)(x+6)}$

$=\frac{5x-8}{ (x-2)(x+6)}$

$=\frac{5x-8}{x^{2}+ 4x-12}$

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Question

Find the least common denominator of the following fractions:

$\frac{6}{7} , \frac{4}{3} , \frac{5}{9}$

Answer

The denominators are 7, 3, and 9. We have to find the common multiple of 7, 3, and 9.

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63

Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90

The least common multiple of the 3 denominators is 63.

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Question

What is the least common denominator of the following fractions?

$\frac{7}{19} , \frac{3}{5}$

Answer

Solution 1

The least common denominator is the least common multiple of the denominators.

We list the multiples of each denominator and we find the lowest common multiple.

Multiples of 19: 19, 38, 57, 76, 95, 114, 133, 152, 171, 190

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100

The lowest common multiple in both lists is 95.

Solution 2

19 and 5 are prime numbers. They have no positive divisors other than 1 and themselves.

The least common denominator of two prime numbers is their product.

$LCD= 19\times5= 95$

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Question

Find the least common denominator of $\frac{x-2}{x-1}$ and $\frac{3}{x+9}$ .

Answer

To find the least common denominator for these two fractions, multiply the denominators together.

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Question

Find the least common denominator for $\frac{x^2-2x-1}{x+8}$ and $\frac{x^3-2x-2}{x-9}$

Answer

To find the least common denominator for these two fractions, multiply the denominators together.

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Question

Find the least common denominator between $\frac{x^2-2x-3}{x^2-5x+6}$ and $\frac{x^2+x-12}{x^2+2x-8}$ .

Answer

Start by factoring the numerator and denominator for each fraction.

$\frac{x^2-2x-3}{x^2-5x+6}=\frac{(x-3)(x+1)}{(x-3)(x-2)}=\frac{x+1}{x-2}$

$\frac{x^2+x-12}{x^2+2x-8}=\frac{(x+4)(x-3)}{(x+4)(x-2)}=\frac{x-3}{x-2}$

So when the two simplified fractions are compared, they actually have the same denominator, which will be the least common denominator.

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Question

Find the least common denominator of $\frac{x^2-8x+12}{x^2-2x+24}$ and $\frac{x^2+7x+12}{x^2+10x+24}$

Answer

Start by simplifying both fractions.

$\frac{x^2-8x+12}{x^2-2x+24}=\frac{(x-2)(x-6)}{(x-6)(x-4)}=\frac{x-2}{x-4}$

$\frac{x^2+7x+12}{x^2+10x+24}=\frac{(x+3)(x+4)}{(x+6)(x+4)}=\frac{x+3}{x+6}$

Now, to find the least common denominator, multiply the denominators together.

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Question

Find the least common denominator of $\frac{x-2}{2x-3}$ and $\frac{6-x}{x+8}$

Answer

To find the least common denominator, multply the two denominators together.

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Question

Find the least common denominator between $\frac{x-9}{2x+1}$ and $\frac{x^2}{3x+9}$

Answer

To find the least common denominator, multply the two denominators together.

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Question

Find the least common denominator between $\frac{x^2-2}{x+4}$ and $\frac{3x^2-2}{x-9}$

Answer

To find the least common denominator, multply the two denominators together.

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Question

Find the least common denominator for $\frac{x-2}{x^2-4}$ and $\frac{x-3}{x^2-9}$ .

Answer

Start by simplifying both fractions.

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

$\frac{x-3}{x^2-9}=\frac{x-3}{(x-3)(x+3)}=\frac{1}{x+3}$

Now, to find the least common denominator for the two simplified fractions, multiply the denominators together.

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Question

What is the least common denominator for the following fractions?

$\small \frac{1}{4},\frac{2}{3},\frac{1}{6},\frac{3}{8},\frac{5}{12}$

Answer

Remember that the least common denominator is the smallest number such that all fractions' denominators divide it evenly.

Always start with the largest number. In this case, it is 12. 3, 4, 6, and 12 all divide evenly into 12, but 8 does not. Think of the next multiple of the largest number. In this case, 24. Here we see that 3, 4, 6, 8, and 12 all divide 24 evenly. Thus, our least common denominator is 24.

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Question

Simplify:

$\frac{4}{x}+\frac{3}{x+5}+\frac{1}{x^2+5x}$

Answer

To simplify the expression, we must find the common denominator, which in this case is

Note that when we distribute the x, we get the denominator of the third term, .

Now, we multiply each term by the LCD divided by itself:

$\frac{x(x+5)}{x(x+5)}\cdot \frac{4}{x}+\frac{x(x+5)}{x(x+5)}\cdot \frac{3}{x+5}+\frac{x(x+5)}{x(x+5)}\cdot \frac{1}{x^2+5x}$

Which, after some canceling, becomes

$\frac{4(x+5)}{x(x+5)}+\frac{3(x)}{x(x+5)}+\frac{1}{x^2+5x}$

Now that all of the terms have the same denominator, we can add their numerators together:

$\frac{4(x+5)+3x+1}{x^2+5x}=\frac{4x+20+3x+1}{x^2+5x}=\frac{7x+21}{x^2+5x}$

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Question

What is the least common denominator of the following fractions?

$\frac{2}{3x}+\frac{3}{2+x}-\frac{1}{6}$

Answer

In order to determine the least common denominator, multiply all the uncommon denominators together.

Use the distributive property to simplify these terms.

The least common denominator is:

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