Add and Subtract Unlike Denominators

4.6 Add/Subtract Unlike denominators

Notes

Multiple Choice

$$\frac{u-v}{8v}+\frac{6u-3v}{8v}$$  

• $$\frac{7u-4v}{16v}$$  

• $$\frac{7u-4v}{8v}$$  

• $$\frac{7u+2v}{8v}$$  

• $$\frac{7u+2v}{16v}$$  

Like Denominators are required to add/subtract Rational Expressions

• These denominators are called LCD or least common Denominator.

• How we find them is First identify what factors are in the denominator

Multiple Select

What factors are in the denominator of the following? Select ALL that apply.

$$\frac{4}{5m+4}+\frac{2m}{m-3}$$  

• 5m+4

• 4

• 2m

• m-3

 $\frac{4}{5m+4}+\frac{2m}{m-3}$  

• Factors in the denomiator are 5m+4 and m-3

• The LCD is (5m+4)(m-3)

• We do this by multiplying by 1 aka  $\frac{m-3}{m-3\ }\ and\ \frac{5m+4}{5m+4}$  

• Lets take this step by step

Multiple Choice

Lets multiply, Only distribute the NUMERATOR (top)

$$\frac{m-3}{m-3\ }\cdot\frac{4}{5m+4}$$  

• $$\frac{4m-12}{\left(m-3\right)\left(5m+4\right)}$$  

• $$\frac{4m-12}{m-3}$$  

• $$\frac{4m-12}{5m+4}$$  

• $$\frac{m-3}{5m+4}$$  

• $$\frac{4}{m-3}$$  

Multiple Choice

Now lets do the other fraction. Remember only distribute the Numerator.

$$\frac{2m}{m-3}\cdot\frac{5m+4}{5m+4}$$  

• $$10m^2+8m$$  

• $$\frac{2m}{5m+4}$$  

• $$\frac{10m^2+8m}{\left(m-3\right)\left(5m+4\right)}$$  

• $$\frac{5m+4}{m-3}$$  

So you found that the two expressions are now

 $\frac{4m-12}{\left(m-3\right)\left(5m+4\right)}+\frac{10m^2+8m}{\left(m-3\right)\left(5m+4\right)}$  

Now we can add these together by combining their numerators

Multiple Choice

Simplify.

$$\frac{4m-12}{\left(m-3\right)\left(5m+4\right)}+\frac{10m^2+8m}{\left(m-3\right)\left(5m+4\right)}$$  

• $$\frac{3m+3}{m-1}$$  

• $$\frac{10m^2+12m-12}{\left(m-3\right)\left(5m+4\right)}$$  

• $$\frac{4+2m}{6m+1}$$  

• $$\frac{21m-56}{5\left(m-6\right)}$$  

Now lets try a subtraction

 $\frac{5x}{3}-\frac{3x}{3x\left(x+6\right)}$  
Pay attention if you can sipmplify here do it.

Multiple Select

What factors are in the denominator?

$$\frac{5x}{3}-\frac{3x}{3x\left(x+6\right)}$$  

• 3

• x

• x+6

• 5x

Why isn't x a factor in the denominator?

 $\frac{5x}{3}-\frac{3x}{3x\left(x+6\right)}$  

• Becuase  3x is in the numerator and denominator of the second fraction so it cancels out.

• So our LCD is 3(x+6)

• Now lets multiply

Multiple Choice

$$\frac{x+6}{x+6}\cdot\frac{5x}{3}$$  

• $$\frac{x+6}{3}$$  

• $$\frac{5x}{x+6}$$  

• $$\frac{5x+30}{3\left(x+6\right)}$$  

• $$\frac{5x^2+30x}{3\left(x+6\right)}$$  

Multiple Choice

 Remember that 3x canceled out, which equals 1.

$$\frac{1}{x+6}\cdot\frac{3}{3}$$  

• $$\frac{3}{3\left(x+6\right)}$$  

• $$\frac{9x}{3\left(x+6\right)}$$  

• $$\frac{3}{x+6}$$  

• $$\frac{1}{x+6}$$  

What does the problem look like now?

 $\frac{5x^2+30x}{3\left(x+6\right)}-\frac{3}{3\left(x+6\right)}$  

Multiple Choice

$$\frac{5x^2+30x}{3\left(x+6\right)}-\frac{3}{3\left(x+6\right)}$$  

• $$\frac{5x^2+31x+4}{3\left(x+6\right)}$$  

• $$\frac{5x^2+31x+6}{2}$$  

• $$\frac{x}{1-x^2-6x}$$  

• $$\frac{5x^2+30x-3}{3\left(x+6\right)}$$  

Lets try some!

Multiple Choice

• A
• B
• C
• D

Multiple Choice

Multiple Choice

• A.

• B.

• C.

• D.

Multiple Choice

Simplify the expression.