Q: Which of the following systems has no solution?

$$\begin{cases} x + y = 2 \\ 2x + 2y = 5 \end{cases}$$

Question 1

A chemist is mixing two solutions. The first solution contains $$\frac{1}{x}$$ liters of acid, and the second contains $$\frac{2}{x-3}$$ liters. If the total acid must be $$1$$ liter, which rational equation should the chemist solve, and what values of $$x$$ must be excluded?

Accepted Answer

Solve $$\frac{1}{x} + \frac{2}{x-3} = 1$$; exclude $$x = 0$$ and $$x = 3.$$

Question 2

Solve the quadratic equation $$2x^2 - 4x - 6 = 0$ using the quadratic formula.

Accepted Answer

$$x = 3,\ x = -1$$

Question 3

A biologist is studying the growth of a bacteria population modeled by $$P(t) = 500e$$^{0.3t}$$. How long will it take for the population to reach 2000$?

Accepted Answer

$$t = \frac{\ln 4}{0.3}$$

Question 4

Which of the following is the correct domain for the function $$f(x) = \sqrt{5 - 2x}$$?

Accepted Answer

$$(-\infty,\ 2.5]$$

Question 5

Given $$f(x) = 2x + 1$$ and $$g(x) = x^2$, find (f$$ \circ g)(3)$$.

Accepted Answer

$$19$$

Question 6

A company’s profit $$P (in $$thousands) is modeled by $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold (in thousands). What is the maximum profit, and how many units should be sold to achieve it?

Accepted Answer

Maximum profit is $$18 at$$ $$x = 2$$ thousand units.

Question 7

Solve the absolute value equation $$|3x - 5| = 7.$$

Accepted Answer

$$x = 4,\ x = -\frac{2}{3}$$

Question 8

Which of the following is the correct interval notation for the solution to the inequality $$2x - 3 \u003c 5$$?

Accepted Answer

$$(-\infty,\ 4)$$

Question 9

Given $$f(x) = 3x - 2$$, find the inverse function $$f^{-1}(x$$)$$.

Accepted Answer

$$f^{-1}(x$$) = \frac{x + 2}{3}$$

Question 10

Expand $$\log_2$ \left( \frac{8x^3$}{y} \right)$$ using logarithm rules.

Accepted Answer

$$3\log_2$ x + 3 - \log_2$ y$$

Question 11

A patient’s medication decays according to $$A = 100e$$^{-0.2t}$$, where A$ is the amount left after t$ hours. How long until only 30$ mg remains?

Accepted Answer

$$t = \frac{\ln \left( \frac{100}{30} \right)}{0.2}$$

Question 12

Solve for $$x$$: $$\log_3$(x - 2) = 2.$$

Accepted Answer

$$x = 11$$

Question 13

Which of the following systems has no solution?

Accepted Answer

$$\begin{cases} x + y = 2 \ 2x + 2y = 5 \end{cases}$$

Question 14

If $$f(x) = |x - 4|$$, for which value of $$x is$$ $$f(x) = 0$$?

Accepted Answer

$$x = 4$$

Question 15

Given the system $$\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$$, solve for $$x$$ and $$y$$ using Cramer’s Rule.

Accepted Answer

$$x = 2,\ y = 1$$

Question 16

Which of the following is a one-to-one function?

Accepted Answer

$$f(x) = 2x + 1$$

Question 17

A savings account earns $$5\%$$ annual interest, compounded quarterly. If $$\1000$ is deposited, what is the amount after 6$ years? Use A = P$$\left(1 + \frac{r}{n}\$$right)^{nt}$.

Accepted Answer

$$A = 1000\left(1 + \frac{0.05}{4}\$$right)^{24}$$

Question 18

Condense $$2\log_5$ x + \log_5$ y - 3\log_5$ z$$ into a single logarithm.

Accepted Answer

$$\log_5$ \left( \frac{x^2$ y}{z^3$} \right)$$

Question 19

Solve the radical equation $$\sqrt{2x + 3} = 5.$$

Accepted Answer

$$x = 11$$

Question 20

Which of the following best describes the graph of $$y = -x^2$ + 4x - 1$$?

Accepted Answer

A parabola opening downward with vertex at $$\left(2, 3\right)$$

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