Question 1

Solve for $$x in $$the equation:
$$
\frac{5}{6}\left(x - \frac{3}{4}\right) - \frac{2}{3}x = \frac{7}{8}\left(x + \frac{5}{6}\right) - \frac{1}{4}
$$

Accepted Answer

$$x = 1$$

Question 2

Find all real solutions (zeroes) of the function:
$$
g(x) = \frac{7}{9}x - \frac{5}{4} - \frac{2}{3}x
$$

Accepted Answer

$$x = 5.4$$

Question 3

What is the domain of the function $$f(x) = \sqrt{12 - 3x - 2}$$? Express your answer in interval notation.

Accepted Answer

$$(-\infty, 5.5]$$

Question 4

A chemist is mixing two solutions. The first solution contains $$\frac{5}{6}$$ liters of acid per liter, and the second contains $$\frac{2}{3}$$ liters per liter. If $$x$$ liters of the first solution and $$x$$ liters of the second are mixed, and the total acid content must be $$2$$ liters, what is the value of $$x$$?
$$
\frac{5}{6}x + \frac{2}{3}x = 2
$$

Accepted Answer

$$x = 1.2$$

Question 5

Which of the following is the correct process to find the zeroes of $$g(x) = \frac{7}{9}x - \frac{5}{4} - \frac{2}{3}x$$?

Accepted Answer

Set $$g(x) = 0$$ and solve for $$x$$

Question 6

A function $$f(x) is $$defined as $$f(x) = \sqrt{12 - 3x - 2}. $$For which value of $$x$$ does $$f(x)$$ become undefined?

Accepted Answer

$$x = 6$$

Question 7

If $$g(x) = \frac{7}{9}x - \frac{5}{4} - \frac{2}{3}x$$, what is $$g(0)$$?

Accepted Answer

$$-\frac{5}{4}$$

Question 8

Suppose a medical dosage is modeled by $$f(x) = \sqrt{12 - 3x - 2}$$, where $$x is $$the number of hours after administration. What is the maximum number of hours after which the dosage is still defined?

Accepted Answer

$$x = 5.5$$

Question 9

Which of the following intervals represents all $$x$$ for which $$f(x) = \sqrt{12 - 3x - 2} is a $$real number?

Accepted Answer

$$(-\infty, 5.5]$$

Question 10

A company’s profit is modeled by $$g(x) = \frac{7}{9}x - \frac{5}{4} - \frac{2}{3}x. $$What is the minimum value of $$x$$ for which the profit is zero?

Accepted Answer

$$x = 5.4$$

Question 11

If $$f(x) = \sqrt{12 - 3x - 2}$$, what is $$f(0)$$?

Accepted Answer

$$\sqrt{10}$$

Question 12

Which of the following best describes the transformation from $$y = x to$$ $$y = \frac{7}{9}x - \frac{5}{4} - \frac{2}{3}x$$?

Accepted Answer

A change in slope and a downward shift

Question 13

Solve for $$x if$$ $$\frac{5}{6}(x - \frac{3}{4}) - \frac{2}{3}x = 0$$

Accepted Answer

$$x = 0.75$$

Question 14

If $$g(x) = \frac{7}{9}x - \frac{5}{4} - \frac{2}{3}x$$, what is the slope of $$g(x)$$?

Accepted Answer

$$\frac{1}{9}$$

Question 15

A real-world scenario: A tank contains $$12$$ liters of water. Water is being drained at a rate of $$3$$ liters per hour, and $$2$$ liters are lost due to evaporation. The amount of water left after $$x$$ hours is $$f(x) = \sqrt{12 - 3x - 2}. $$For how many hours will there be water left in the tank?

Accepted Answer

$$x \leq 5.5$$

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