Q: Solve the rational equation $$\frac{6}{x - 2} = 3 + \frac{2}{x - 2}$$, and state any restrictions.

$$x = 4$$, restriction: $$x \neq 2$$

Question 1

Solve the absolute value inequality $$|2x + 5| \u003c 7$$ and express your answer in interval notation.

Accepted Answer

$$(-6, 1)$$

Question 2

Given the system of equations $$3x^2$$ + $$2y^2 = 13$ and 2x^2$ - y^2$ = 4$$, how many real solutions (ordered pairs) does the system have?

Accepted Answer

4

Question 3

Solve the exponential equation $$5^x = 20$$ for $$x.$$

Accepted Answer

$$x = \frac{\ln(20)}{\ln(5)}$$

Question 4

A line passes through the point $$(3, -2)$$ and has a slope of $$\frac{1}{2}. $$What is its equation in point-slope form?

Accepted Answer

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

Question 5

Solve the system of equations for $$(x, y, z)$$:

$$\begin{cases} x + y + z = 6 \ 2x - y + 3z = 14 \ -x + 4y - z = -2 \end{cases}$$

Accepted Answer

$$(2, 1, 3)$$

Question 6

Which of the following graphs represents the function $$f(x) = 3^x$$ for $$x = -2, -1, 0, 1, 2$$?

Accepted Answer

An increasing curve passing through $$(0, 1)$$, $$(1, 3)$$, $$(2, 9)$$, and approaching $$y = 0 as$$ $$x 	o -\infty$$

Question 7

A graph is a sideways-opening parabola. Does it represent $$y as a $$function of $$x$$?

Accepted Answer

No, because some $$y-$$values correspond to more than one $$x-$$value.

Question 8

Solve the radical equation $$\sqrt{2x + 15} = x + 3.$$

Accepted Answer

$$x = 0$$

Question 9

Write the equation $$x^2$$ + $$y^2 - 6x + 8y + 9 = 0$ in standard form for a circle and state its center.

Accepted Answer

$$(x - 3)^2$ + (y + 4)^2$ = 4$$; center $$(3, -4)$$

Question 10

Find the distance between the points $$(5, 7)$$ and $$(-1, 4).$$

Accepted Answer

$$\sqrt{45}$$

Question 11

Solve by factoring: $$x^2 - 9x = 0$.

Accepted Answer

$$\{0, 9\}$$

Question 12

Write $$3^5 = y$ in logarithmic form.

Accepted Answer

$$\log_3$(y) = 5$$

Question 13

Solve using the quadratic formula: $$2x^2 + 4x + 10 = 0$.

Accepted Answer

$$x = -1 \pm 2i$$

Question 14

Solve the compound inequality $$-5 \leq 2x - 3 \u003c 7$$ and write your answer in interval notation.

Accepted Answer

$$[-1, 5)$$

Question 15

Given the relation $$\{(2, 5), (3, 7), (2, 8)\}$$, is this a function? What is its domain?

Accepted Answer

Not a function; domain $$\{2, 3\}$$

Question 16

Find the inverse of $$f(x) = 2x - 7.$$

Accepted Answer

$$f^{-1}(x$$) = \frac{x + 7}{2}$$

Question 17

Write $$3 = \log_a(81) in $$exponential form.

Accepted Answer

$$a^3 = 81$

Question 18

Solve the rational equation $$\frac{6}{x - 2} = 3 + \frac{2}{x - 2}$$, and state any restrictions.

Accepted Answer

$$x = 4$$, restriction: $$x 
eq 2$$

Question 19

A chemical reaction's temperature $$T (in$$ $$^ 5cc) is $$modeled by $$T = 10e$$^{0.2t}$$, where t$ is time in hours. After how many hours will the temperature reach 40$$^ 5cc$$? (Round to two decimals.)

Accepted Answer

$$t = \frac{\ln(4)}{0.2} \approx 6.93$$

College Algebra Final Exam Study Guide: Key Concepts and Worked Examples

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