Objective - Solving Rational Equations

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Solve rational equations that lead to linear and quadratic equations.

Link to section in online textbook.

First, watch this video to learn how solving rational functions. Since our domain can be restricted, we need to check these values!

Solve the rational equation below. Remember to check your solutions to make sure they are valid! If there is no solution, answer “NA”.

$\sage {leftTerm1} = \frac {\sage {rightTerm1Num}}{\sage {rightTerm1Denom}}$

Solution: $x = \answer [format=string]{NA}$

Solve the rational equation below. Remember to check your solutions to make sure they are valid! If there is no solution, answer “NA”.

$\sage {leftTerm2} = \frac {\sage {rightTerm2Num}}{\sage {rightTerm2Denom}}$

Solution: $x = \answer [tolerance=0.05]{\sage {solution2}}$

Solve the rational equation below. Remember to check your solutions to make sure they are valid! If there are more boxes than solutions, answer “NA”.

$\sage {leftTerm3} = \sage {thirdTerm3}$

Solutions: $x = \answer [format=string]{NA}$ and $x = \answer [format=string]{NA}$

Solve the rational equation below. Remember to check your solutions to make sure they are valid! If there are more boxes than solutions, answer “NA”.

$\sage {leftTerm4} = \sage {thirdTerm4}$

Solutions: $x = \answer [tolerance=0.05]{\sage {solution4}}$ and $x = \answer [format=string]{NA}$

Solve the rational equation below. Remember to check your solutions to make sure they are valid! If there are more boxes than solutions, answer “NA”.

$\sage {leftTerm5} = \sage {thirdTerm5}$

Solutions: $x = \answer [tolerance=0.05]{\sage {solution5a}}$ and $x = \answer [tolerance=0.05]{\sage {solution5b}}$

Main takeaway: Before looking, you should work through the previous problems. Have you finished working through the examples? $\answer [format=string]{Yes}$

To solve rational equations, we want to multiply to remove the denominators. When in doubt, multiply by the denominator of each one at a time. This may not always be the most efficient way (multiplying by the GCD would be) it will eventually get the equation into a more manageable form. Like with radical functions, we also need to check our solutions to make sure they are valid – that we are not dividing by 0.

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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