a.__ | -4x6 = -2916 | ___ | Divide both sides of the equation by -4 |
x6 = 729 |
| Take the ± 6th root of both sides | |
x = \(\pm\sqrt[6]{729}\) | Plug this into a calculator or note that 36 = 729 so we can rewrite 729 as 36 | ||
x = \(\pm\sqrt[6]{3^6}\) |
| Simplify | |
x = ± 3 | |||
| This equation has two solutions. There are two values of x which make the equation true. They are: x = 3 and x = -3 | ||
b. | 93 - 7x3 = 23 | Subtract 93 from both sides of the equation. | |
-7x3 = -70 |
| Divide both sides of the equation by -7 | |
x3 = 10 | Take the cube root of both sides. | ||
x = \(\sqrt[3]{10}\) |
| To get an approximate solution, plug \(\sqrt[3]{10}\) into a calculator. | |
x ≈ 2.154 | |||
| |||
c. | (-5p)5 = -65 | Take the fifth root of both sides. | |
-5p = \(\sqrt[5]{-65}\) |
| Divide both sides by -5 | |
p = \(\frac{\sqrt[5]{-65}}{-5}\) | SImplify | ||
p = \(\frac{\sqrt[5]{-1}\,\cdot\,\sqrt[5]{65}}{-5}\) |
| ||
p = \(\frac{-1\ \cdot\ \sqrt[5]{65}}{-5}\) | |||
p = \(\frac{\sqrt[5]{65}}{5}\) |
| ||
p ≈ 0.461 | |||
| |||
d. | \(-7+\frac{14}{x-3}\ =\ -5\) | Add 7 to both sides. | |
\(\frac{14}{x-3}\ =\ 2\) |
| Multiply both sides by (x - 3) and note x ≠ 3 | |
14 = 2(x - 3) | Divide both sides by 2 | ||
7 = x - 3 |
| Add 3 to both sides | |
10 = x | |||
x = 10 |
|
Assuming that \(f(x)\ =\ 2+\frac{20}{x-3}\) and \(g(x)\ =\ -3+\frac{32}{2x+4}\)
a.__ | |
\(y\ =\ f(x)\\~\\ y\ =\ 2+\frac{20}{x-3} \) | |
| y is undefined when x - 3 = 0 so there is a vertical asymptote at x = 3 |
\( y-2\ =\ \frac{20}{x-3}\\~\\ (x-3)(y-2)\ =\ 20\\~\\ x-3\ =\ \frac{20}{y-2}\) | |
| x is undefined when y - 2 = 0 so there is a horizontal asymptote at y = 2 |
The equations for the vertical and horizontal asymptotes of the graph of f(x) are: x = 3 and y = 2 | |
b. | |
\(y\ =\ g(x)\\~\\ y\ =\ -3+\frac{32}{2x+4}\) | |
| y is undefined when 2x + 4 = 0 so there is a vertical asymptote at x = -2 |
\(y+3\ =\ \frac{32}{2x+4}\\~\\ (2x+4)(y+3)\ =\ 32\\~\\ 2x+4\ =\ \frac{32}{y+3}\) | |
| x is undefined when y + 3 = 0 so there is a horizontal asymptote at y = -3 |
The equations for the vertical and horizontal asymptotes of the graph of g(x) are: x = -2 and y = -3 | |
c. | |
Here is a graph: https://www.desmos.com/calculator/7zrrangdvv | |
| (Note you can hide or show f(x) or g(x) and its asymptotes by clicking the circles beside the function.) |
Graph 2 belongs to f(x) and graph 1 belongs to g(x) . | |
d. | |
A possible function is: \(f(x)\ =\ -3+\frac{1}{x-4}\)_ |