91Ó°ÊÓ

Algebraically. Be sure to label what the variable represents. Lynette can run at the rate of \(10 \mathrm{kph}\) and bike at the rate of \(24 \mathrm{kph}\). If she runs for one-half hour longer than she bikes and she covers a total of \(56 \mathrm{km},\) how long was she biking?

Perform the indicated operations and simplify. $$\frac{3}{2 x}+\frac{5}{x+2}$$

Reduce to lowest terms. $$\frac{6+\sqrt{12}}{8}$$

The period of a pendulum is the time it takes for the pendulum to sweep out one complete arc. The formula (in seconds) for the period \(T\) of a pendulum of length \(L\) feet is $$T=2 \pi \sqrt{\frac{L}{32}}$$ To the nearest tenth, find the period of a pendulum of length 2 feet.

Reduce to lowest terms. $$\frac{12-\sqrt{20}}{10}$$

Perform the indicated operations and simplify. $$\text { Solve for } x, \quad \frac{6}{x-6}=\frac{2}{x-2}$$

Reduce to lowest terms. $$\frac{10-\sqrt{24}}{12}$$

Solve the following problems algebraically. Be sure to label what the variable represents. Xavier made three investments at \(6.5 \%, 7.6 \%,\) and \(9.2 \% .\) The amount invested at \(7.6 \%\) is \(\$ 1000\) less than the amount invested at \(9.2 \%,\) and the amount invested at \(6.5 \%\) is twice the amount invested at \(7.6 \% .\) If the annual income from the three investments is \(\$ 837,\) how much is invested altogether?

Verify that \(2+\sqrt{10}\) is a solution to the equation \(x^{2}-4 x-6=0\).

Determine whether \(3-\sqrt{13}\) is a solution to the equation \(x^{2}-6 x=3\).