Tag: adding/subtractingrationalexpressions

Week 15 – Pre Calc 11

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This week in Pre-Calculus 11 we continued the rational expressions unit. This week we learned how to add and subtract rational expressions and how to solve equations.

How To Add and Subtract Rational Expressions: When you add rational expressions you always have to find a common denominator. To find a common denominator it is best to figure out the lowest common multiple by factoring both of the denominators.

Ex. \frac{3}{5}+\frac{7}{8}  

(\frac{3}{5}\times\frac{8}{8})+(\frac{7}{8}\times\frac{5}{5})

\frac{24}{40}+\frac{35}{40} 

\frac{59}{40} or 1\frac{19}{40}

In the example above, I had to add two fractions together. The first step was to find a common denominator, the lowest common multiple between 5 and 8 is 40. Then I added the numerators together. After adding the fractions together you must check if the fractions simplify any further.

Ex. \frac{13}{9}-\frac{4}{3} 

(\frac{13}{9}\times\frac{2}{2})-(\frac{4}{3}\times\frac{6}{6})

\frac{26}{18}-\frac{24}{18}

\frac{2}{18}

\frac{1}{9} 

In the example above I subtracted two fractions together. The steps were the exact same as when I added. The first step was to find a common denominator, the lowest common multiple between 9 and 3 is 18. After changing the denominators I subtracted the numerators. Then I simplified the fraction.

How To Add and Subtract Rational Expressions with Variables: Similarly, when dealing with variables while adding and subtracting you still need to find a common denominator.

Ex. \frac{12}{5x}+\frac{7x}{2}

(\frac{12}{5x}\times\frac{2}{2})+(\frac{7x}{2}\times\frac{5x}{5x})

\frac{24}{10x}+\frac{35x^2}{10x}

\frac{35x^2+24}{10x} 

x\neq0

In the example above, I found a common denominator of 10x and then I added the fractions together.

Ex. \frac{8}{6x+9}+\frac{3}{4x-4}

(\frac{8}{6x+9}\times\frac{4(x-1)}{4(x-1))}+(\frac{3}{4x-4}\times\frac{3(2x+3)}{3(2x+3)})

\frac{32x-32}{3(2x+3)(4(x-1))}+\frac{18x+27}{3(2x+3)(4(x-1))}

\frac{32x-32+18x+27}{3(2x+3)(4(x-1))}

\frac{50x-5}{12(2x+3)(x-1)}

\frac{5(10x-1)}{12(2x+3)(x-1)}

2x+3\neq0

2x\neq-3

x\neq\frac{-3}{2}

x-1\neq0

x\neq1

In the example above, the first thing I did was factor both of the fractions. After, I found a common denominator by multiplying both of the denominators together. Then, I made a big fraction and added the like terms together and simplified. One key step is to remember to find the non-permissible values.

How To Solve Fractional Equations: There are a number of different ways to solve fractional equations that involve variables. If the equation has one fraction on both sides of the equals sign then you can cross multiply the fractions. Because we are solving, you have to isolate the variable and find out it’s value.

Ex. \frac{x+2}{x-3}=\frac{x-1}{x-2}

(x+2)(x-2)=(x-1)(x-3)

x^2-2x+2x-4=x^2-3x-x+3

x^2-4=x^2-4x+3

x^2-x^2+4x-4-3=0

4x-7=0

4x=7

x=\frac{7}{4} 

x-3\neq0
x\neq3

x-2\neq0

x\neq2

In the example above, the first step was to cross multiply. After cross multiplying I collected all the like terms. Then, I isolated x and found x’s value.