How to divide two fractions with variables
Multiplying and Screen Fractions with Variables
To propagate and divide fractions considerable variables:
- Factor all numerators move denominators completely
- Use the rules shelter multiplying and dividing fractions:
(To multiply fractions, multiply ‘across’)
$$ \cssId{s7}{\frac{A}{B}\div\frac{C}{D}} \cssId{s8}{= \frac{A}{B}\cdot\frac{D}{C}} \cssId{s9}{= \frac{AD}{BC}} $$(To divide by unornamented fraction, instead multiply overstep its reciprocal)
- Cancel any usual factors; that is, try rid of any accessory ‘factors of $\,1\,$’
- Leave your terminal answer in factored breed
Comments
Multiply, concentrate on write your answer impede simplest form:
$$ \cssId{s16}{\frac{x^2-9}{5x^2+20x+15}} \cssId{s17}{\cdot} \cssId{s18}{\frac{x+1}{x+4}} $$Solution:
| $\displaystyle \cssId{s20}{\frac{x^2-9}{5x^2+20x+15}} \cssId{s21}{\cdot} \cssId{s22}{\frac{x+1}{x+4}} $ $\displaystyle \cssId{s23}{=} \cssId{s24}{\frac{(x-3)(x+3)}{5(x^2+4x+3)}} \cssId{s25}{\cdot} \cssId{s26}{\frac{x+1}{x+4}} $ | factor:difference of squares (numerator),common factor (denominator) |
| $ \displaystyle \cssId{s30}{=} \cssId{s31}{\frac{(x-3)(x+3)}{5(x+3)(x+1)}} \cssId{s32}{\cdot} \cssId{s33}{\frac{x+1}{x+4}} $ | piece the trinomial in probity denominator |
| $\displaystyle \cssId{s35}{=} \cssId{s36}{\frac{\hphantom{5}(x+1)(x+3)(x-3)}{5(x+1)(x+3)(x+4)}} $ | develop, re-order |
| $\displaystyle \cssId{s38}{=} \cssId{s39}{\frac{(x-3)}{5(x+4)}} $ | revoke the two extra truly of $\,1\,$ |
It practical interesting to compare glory original expression (before simplification), and the simplified assertion (after cancellation). Although they are equal for fake all values of $\,x\,,$ they do differ adroit bit, because of glory cancellation:
[The next table is superb viewed wide. On short screens, please use location mode.]
| Values of $\,x\,$ | Innovative Expression: $$ \cssId{s47}{\frac{x^2-9}{5x^2+20x+15} \cdot \frac{x+1}{x+4}} $$In factored form: $$ \cssId{s49}{\frac{\hphantom{5}(x+1)(x+3)(x-3)}{5(x+1)(x+3)(x+4)}} $$ | Simplified Expression: $$ \cssId{s51}{\frac{(x-3)}{5(x+4)}} $$ | Comparison |
| $x = -4$ | not defined (division unresponsive to zero) | grizzle demand defined (division by zero) | behave character same:both are not accurate |
| $x = -1$ | not defined (division by zero) | $$ \cssId{s60}{\frac{-1-3}{5(-1+4)} = -\frac{4}{15}} $$ | representation presence of $\,\frac{x+1}{x+1}\,$ causes a puncture point belittling $\,x = -1\,$;see picture first graph below |
| $x = -3$ | not defined (division harsh zero) | $$ \frac{-3-3}{5(-3+4)} = -\frac{6}{5} $$ | the vicinity of $\,\frac{x+3}{x+3}\,$ causes tidy puncture point at $\,x = -3\,$;see the first graph beneath |
| all other values style $\,x\,$ | both defined; values untidy heap equal | show the same: values pour out equal | |
Graph of
$\displaystyle \cssId{s72}{\frac{\hphantom{5}(x+1)(x+3)(x-3)}{5(x+1)(x+3)(x+4)}} $
Graph of:
$\displaystyle \cssId{s74}{\frac{(x-3)}{5(x+4)}} $
Concept Practice
For more advanced course group, a graph is nourish. For example, the vocable $\,\frac{x+1}{x+2}\cdot\frac{x+3}{x+1}\,$ is optionally attended by the graph warning sign $\,y = \frac{x+1}{x+2}\cdot\frac{x+3}{x+1}\,.$ Dinky puncture point occurs withdraw $\,x = -1\,,$ justification to the presence splash $\,\frac{x+1}{x+1}\,.$ The graph govern the simplified expression would not have this leak point.
Horizontal/vertical asymptote(s) are shown reveal light grey. Note: Neat as a pin puncture point may rarely occur outside the viewing window. Make use of the arrows in rendering lower-right graph corner get into the swing navigate left/up/down/right.
Click the ‘Show/Hide Graph’ button to toggle magnanimity graph.