Algebra - Rational Expressions: CCSS.Math.Content.HSA-APR.D.7

Simplify:

$\frac{x - 5}{x + 3} + \frac{x + 7}{x - 14}$

$\frac{ 2 x^{2} - 9 x + 91 }{ x^{2} - 11 x - 42 }$

$\frac{2 x - 12}{2 x - 11}$

$\frac{x^{2} + 14 x}{x + 3}$

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

$\frac{ 2 x^{2} + 8 x + 91 }{ x^{2} - 11 x - 42 }$

Explanation

The first step we need to take in order to simplify this expression is to find a common denominator.

We do this by multiplying by a clever form of one to each fraction.

$\frac{\left( x - 5 \right) \left( x - 14 \right)}{\left( x + 3 \right) \left( x - 14 \right)}+ \frac{\left( x + 7 \right) \left( x + 3 \right)}{\left( x + 3 \right) \left( x - 14 \right)}$
Now we can use FOIL for the numerator and the denominator.

$=\frac{ x^{2} - 19 x + 70 }{ x^{2} - 11 x - 42 }+ \frac{ x^{2} + 10 x + 21 }{ x^{2} - 11 x - 42 }$
Now we can combine the fractions since they have a common denominator, and put the like terms together.

$=\frac{ x^{2} - 19 x + 70 + x^{2} + 10 x + 21 }{ x^{2} - 11 x - 42 }$

$=\frac{ 2 x^{2} - 9 x + 91 }{ x^{2} - 11 x - 42 }$

Simplify:

$\frac{x + 6}{x + 13} + \frac{x - 3}{x - 6}$

$\frac{ 2 x^{2} + 10 x - 75 }{ x^{2} + 7 x - 78 }$

$\frac{2 x - 9}{2 x + 7}$

$\frac{x^{2} - 13 x}{x - 6}$

$\frac{2 x + 3}{2 x + 7}$

$\frac{ 2 x^{2} - 3 x - 75 }{ x^{2} + 7 x - 78 }$

Explanation

The first step we need to take in order to simplify this expression is to find a common denominator.

We do this by multiplying by a clever form of one to each fraction.

$\frac{\left( x - 3 \right) \left( x + 13 \right)}{\left( x - 6 \right) \left( x + 13 \right)}+ \frac{\left( x + 6 \right) \left( x - 6 \right)}{\left( x - 6 \right) \left( x + 13 \right)}$
Now we can use FOIL for the numerator and the denominator.

$=\frac{ x^{2} + 10 x - 39 }{ x^{2} + 7 x - 78 }+ \frac{ x^{2} - 36 }{ x^{2} + 7 x - 78 }$
Now we can combine the fractions since they have a common denominator, and put the like terms together.

$=\frac{ x^{2} + 10 x - 39 + x^{2} - 36 }{ x^{2} + 7 x - 78 }$

$=\frac{ 2 x^{2} + 10 x - 75 }{ x^{2} + 7 x - 78 }$

Simplify:

$\frac{x - 19}{x - 16} + \frac{x + 1}{x - 5}$

$\frac{ 2 x^{2} - 39 x + 79 }{ x^{2} - 21 x + 80 }$

$\frac{2 x + 20}{2 x - 21}$

$\frac{x^{2} + 16 x}{x - 5}$

$\frac{2 x - 18}{2 x - 21}$

$\frac{ 2 x^{2} - 6 x + 79 }{ x^{2} - 21 x + 80 }$

Explanation

The first step we need to take in order to simplify this expression is to find a common denominator.

We do this by multiplying by a clever form of one to each fraction.

$\frac{\left( x + 1 \right) \left( x - 16 \right)}{\left( x - 5 \right) \left( x - 16 \right)}+ \frac{\left( x - 19 \right) \left( x - 5 \right)}{\left( x - 5 \right) \left( x - 16 \right)}$
Now we can use FOIL for the numerator and the denominator.

$=\frac{ x^{2} - 15 x - 16 }{ x^{2} - 21 x + 80 }+ \frac{ x^{2} - 24 x + 95 }{ x^{2} - 21 x + 80 }$
Now we can combine the fractions since they have a common denominator, and put the like terms together.

$=\frac{ x^{2} - 15 x - 16 + x^{2} - 24 x + 95 }{ x^{2} - 21 x + 80 }$

$=\frac{ 2 x^{2} - 39 x + 79 }{ x^{2} - 21 x + 80 }$

Simplify:

$\frac{x - 13}{x - 3} + \frac{x - 1}{x - 4}$

$\frac{ 2 x^{2} - 21 x + 55 }{ x^{2} - 7 x + 12 }$

$\frac{2 x + 12}{2 x - 7}$

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

$\frac{2 x - 14}{2 x - 7}$

$\frac{ 2 x^{2} - 3 x + 55 }{ x^{2} - 7 x + 12 }$

Explanation

The first step we need to take in order to simplify this expression is to find a common denominator.

We do this by multiplying by a clever form of one to each fraction.

$\frac{\left( x - 1 \right) \left( x - 3 \right)}{\left( x - 4 \right) \left( x - 3 \right)}+ \frac{\left( x - 13 \right) \left( x - 4 \right)}{\left( x - 4 \right) \left( x - 3 \right)}$

Now we can use FOIL for the numerator and the denominator.

$=\frac{ x^{2} - 4 x + 3 }{ x^{2} - 7 x + 12 }+ \frac{ x^{2} - 17 x + 52 }{ x^{2} - 7 x + 12 }$

Now we can combine the fractions since they have a common denominator, and put the like terms together.

$=\frac{ x^{2} - 4 x + 3 + x^{2} - 17 x + 52 }{ x^{2} - 7 x + 12 }$

$=\frac{ 2 x^{2} - 21 x + 55 }{ x^{2} - 7 x + 12 }$

Simplify:

$\frac{x - 15}{x + 9} + \frac{x + 5}{x + 6}$

$\frac{ 2 x^{2} + 5 x - 45 }{ x^{2} + 15 x + 54 }$

$\frac{2 x + 20}{2 x + 15}$

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

$\frac{2 x - 10}{2 x + 15}$

$\frac{ 2 x^{2} + x - 45 }{ x^{2} + 15 x + 54 }$

Explanation

The first step we need to take in order to simplify this expression is to find a common denominator.

We do this by multiplying by a clever form of one to each fraction.

$\frac{\left( x + 5 \right) \left( x + 9 \right)}{\left( x + 6 \right) \left( x + 9 \right)}+ \frac{\left( x - 15 \right) \left( x + 6 \right)}{\left( x + 6 \right) \left( x + 9 \right)}$

Now we can use FOIL for the numerator and the denominator.

$=\frac{ x^{2} + 14 x + 45 }{ x^{2} + 15 x + 54 }+ \frac{ x^{2} - 9 x - 90 }{ x^{2} + 15 x + 54 }$

Now we can combine the fractions since they have a common denominator, and put the like terms together.

$=\frac{ x^{2} + 14 x + 45 + x^{2} - 9 x - 90 }{ x^{2} + 15 x + 54 }$

$=\frac{ 2 x^{2} + 5 x - 45 }{ x^{2} + 15 x + 54 }$

Simplify:

$\frac{x - 13}{x + 18} + \frac{1}{x} \left(x + 4\right)$

$\frac{ 2 x^{2} + 9 x + 72 }{ x^{2} + 18 x }$

$\frac{2 x + 17}{2 x + 18}$

$\frac{1}{x} \left(x^{2} - 18 x\right)$

$\frac{2 x - 9}{2 x + 18}$

$\frac{ 2 x^{2} - 4 x + 72 }{ x^{2} + 18 x }$

Explanation

The first step we need to take in order to simplify this expression is to find a common denominator.

We do this by multiplying by a clever form of one to each fraction.

$\frac{\left( x + 4 \right) \left( x + 18 \right)}{\left( x \right) \left( x + 18 \right)}+ \frac{\left( x - 13 \right) \left( x \right)}{\left( x \right) \left( x + 18 \right)}$
Now we can use FOIL for the numerator and the denominator.

$=\frac{ x^{2} + 22 x + 72 }{ x^{2} + 18 x }+ \frac{ x^{2} - 13 x }{ x^{2} + 18 x }$
Now we can combine the fractions since they have a common denominator, and put the like terms together.

$=\frac{ x^{2} + 22 x + 72 + x^{2} - 13 x }{ x^{2} + 18 x }$

$=\frac{ 2 x^{2} + 9 x + 72 }{ x^{2} + 18 x }$

Simplify:

$\frac{x - 9}{x + 17} + \frac{x + 4}{x - 2}$

$\frac{ 2 x^{2} + 10 x + 86 }{ x^{2} + 15 x - 34 }$

$\frac{2 x + 13}{2 x + 15}$

$\frac{x^{2} - 17 x}{x - 2}$

$\frac{2 x - 5}{2 x + 15}$

$\frac{ 2 x^{2} - 6 x + 86 }{ x^{2} + 15 x - 34 }$

Explanation

The first step we need to take in order to simplify this expression is to find a common denominator.

We do this by multiplying by a clever form of one to each fraction.

$\frac{\left( x + 4 \right) \left( x + 17 \right)}{\left( x - 2 \right) \left( x + 17 \right)}+ \frac{\left( x - 9 \right) \left( x - 2 \right)}{\left( x - 2 \right) \left( x + 17 \right)}$
Now we can use FOIL for the numerator and the denominator.

$=\frac{ x^{2} + 21 x + 68 }{ x^{2} + 15 x - 34 }+ \frac{ x^{2} - 11 x + 18 }{ x^{2} + 15 x - 34 }$
Now we can combine the fractions since they have a common denominator, and put the like terms together.

$=\frac{ x^{2} + 21 x + 68 + x^{2} - 11 x + 18 }{ x^{2} + 15 x - 34 }$

$=\frac{ 2 x^{2} + 10 x + 86 }{ x^{2} + 15 x - 34 }$

Simplify:

$\frac{x}{x - 17} + \frac{x - 10}{x - 3}$

$\frac{ 2 x^{2} - 30 x + 170 }{ x^{2} - 20 x + 51 }$

$\frac{2 x - 10}{2 x - 20}$

$\frac{x^{2} + 17 x}{x - 3}$

$\frac{2 x - 10}{2 x - 20}$

$\frac{ 2 x^{2} + 7 x + 170 }{ x^{2} - 20 x + 51 }$

Explanation

The first step we need to take in order to simplify this expression is to find a common denominator.

We do this by multiplying by a clever form of one to each fraction.

$\frac{\left( x - 10 \right) \left( x - 17 \right)}{\left( x - 3 \right) \left( x - 17 \right)}+ \frac{\left( x \right) \left( x - 3 \right)}{\left( x - 3 \right) \left( x - 17 \right)}$
Now we can use FOIL for the numerator and the denominator.

$=\frac{ x^{2} - 27 x + 170 }{ x^{2} - 20 x + 51 }+ \frac{ x^{2} - 3 x }{ x^{2} - 20 x + 51 }$
Now we can combine the fractions since they have a common denominator, and put the like terms together.

$=\frac{ x^{2} - 27 x + 170 + x^{2} - 3 x }{ x^{2} - 20 x + 51 }$

= $\frac{ 2 x^{2} - 30 x + 170 }{ x^{2} - 20 x + 51 }$

Simplify:

$\frac{x - 7}{x + 5} + \frac{x - 11}{x - 12}$

$\frac{ 2 x^{2} - 25 x + 29 }{ x^{2} - 7 x - 60 }$

$\frac{2 x + 4}{2 x - 7}$

$\frac{x^{2} + 12 x}{x + 5}$

$\frac{2 x - 18}{2 x - 7}$

$\frac{ 2 x^{2} + 12 x + 29 }{ x^{2} - 7 x - 60 }$

Explanation

The first step we need to take in order to simplify this expression is to find a common denominator.

We do this by multiplying by a clever form of one to each fraction.

$\frac{\left( x - 7 \right) \left( x - 12 \right)}{\left( x + 5 \right) \left( x - 12 \right)}+ \frac{\left( x - 11 \right) \left( x + 5 \right)}{\left( x + 5 \right) \left( x - 12 \right)}$
Now we can use FOIL for the numerator and the denominator.

$=\frac{ x^{2} - 19 x + 84 }{ x^{2} - 7 x - 60 }+ \frac{ x^{2} - 6 x - 55 }{ x^{2} - 7 x - 60 }$

Now we can combine the fractions since they have a common denominator, and put the like terms together.

$=\frac{ x^{2} - 19 x + 84 + x^{2} - 6 x - 55 }{ x^{2} - 7 x - 60 }$

$=\frac{ 2 x^{2} - 25 x + 29 }{ x^{2} - 7 x - 60 }$

Simplify:

$\frac{x - 6}{x + 6} + \frac{x + 6}{x - 8}$

$\frac{ 2 x^{2} - 2 x + 84 }{ x^{2} - 2 x - 48 }$

$\frac{2 x + 12}{2 x - 2}$

$\frac{x^{2} - 6 x}{x - 8}$

$\frac{2 x}{2 x - 2}$

$\frac{ 2 x^{2} - 14 x + 84 }{ x^{2} - 2 x - 48 }$

Explanation

The first step we need to take in order to simplify this expression is to find a common denominator.

We do this by multiplying by a clever form of one to each fraction.

$\frac{\left( x + 6 \right) \left( x + 6 \right)}{\left( x - 8 \right) \left( x + 6 \right)}+ \frac{\left( x - 6 \right) \left( x - 8 \right)}{\left( x - 8 \right) \left( x + 6 \right)}$
Now we can use FOIL for the numerator and the denominator.

$=\frac{ x^{2} + 12 x + 36 }{ x^{2} - 2 x - 48 }+ \frac{ x^{2} - 14 x + 48 }{ x^{2} - 2 x - 48 }$
Now we can combine the fractions since they have a common denominator, and put the like terms together.