Solving Rational Equations

1. $\rhd$ Equations with one rational expression (3:11) $\frac{14x+4}{-3x-2} = 8.$
Solution: $x = -\frac{-10}{19}$
2. $\star$ Solve for $r$ $r$: $\frac{4r-6}{r-7} = \frac{1}{2}$
3. $\rhd$ Solving equations with one rational expression (advanced)   Solve: $x^2 - \frac{x^2 - 4}{x-2} = 4$ with $x \neq 2$.
This produces the equation $x^2 - x - 6 = 0$, so $x = 3$ or $x = -2$.  Then he correctly verifies — doesn’t assume the conclusion! (Says: $f(-2) = 4$ after defining $f(x)$ to be the given rational expression.)
4. $\star$ Solving equations with one rational expression  Solve $\frac{-3k - 38}{k^2 - 16} = -1$
5. $\rhd$ Equations with two rational expressions (4:16)  Solve and find excluded values ( $p \neq 1, -3$). $\frac{4}{p-1} = \frac{5}{p+3}$  Solution: $p = 17$, verifies solution.
6. $\rhd$ Equations with two rational expressions (by finding the least common multiple) (4:07) $\frac{5}{2x} - \frac{4}{3x} = \frac{7}{18}$ (note that $x \neq 0$.  Solution: $x = 3$
7. $\rhd$ Rational equations with extraneous solutions (3:02) Solve: $\frac{x^2}{x+2} = \frac{4}{x+2}$.  The values $x = -2$ and $x = 2$ satisfy the resulting equation, but $x = -2$ is extraneous.
8. $\star$ Equations with two rational expressions $\frac{k+5}{k^2 -5k +6} = \frac{k-9}{k-2}$