Question 1

Simplify the following expression and write your answer in standard form: $$rac{1+7i}{8+2i}$$

Accepted Answer

$$\frac{1+7i}{8+2i} = \frac{1+7i}{8+2i} \cdot \frac{8-2i}{8-2i} = \frac{(1+7i)(8-2i)}{(8+2i)(8-2i)} = \frac{8-2i+56i-14i^2$}{64-4i^2$} = \frac{8+54i+14}{64+4} = \frac{22+54i}{68} = \frac{11}{34} + \frac{27}{34}i$$

Question 2

Solve the rational inequality and write the solution set in interval notation: $$\frac{x-4}{x+5} \leq 0$$

Accepted Answer

$$(-5, 4]$$

Question 3

Given the points $$A(-1,3)$$ and $$B(3,-2)$$, what is the distance between them?

Accepted Answer

$$\sqrt{(3-(-1))^2$ + (-2-3)^2$} = \sqrt{4^2$ + (-5)^2$} = \sqrt{16+25} = \sqrt{41}$$

Question 4

For the equation $$y = 2x + 3$$, complete the table of ordered pairs that are solutions of the equation for $$x = -2, 0, 2.$$

Accepted Answer

When $$x = -2$$, $$y = 2(-2) + 3 = -4 + 3 = -1$$; $$x = 0$$, $$y = 2(0) + 3 = 3$$; $$x = 2$$, $$y = 2(2) + 3 = 4 + 3 = 7. So $$the table is: $$(-2, -1), (0, 3), (2, 7)$$

Question 5

Which graph corresponds to the function $$f(x) = \sqrt{x+6}$$?

Accepted Answer

Graph 1 (shows a square root function starting at $$x = -6$$ and increasing)

Question 6

Let $$f(x) = x^2$ + 11$$ and $$g(x) = x + 4. $$Find $$(f-g)(1).$$

Accepted Answer

$$(f-g)(1) = f(1) - g(1) = (1^2$ + 11) - (1 + 4) = (1 + 11) - 5 = 12 - 5 = 7$$

Question 7

Determine whether the relation $$\{(0,3),(-2,6),(5,1),(0,8)\} is a $$function and if it is one-to-one.

Accepted Answer

It is not a function because the input $$0 is $$paired with two different outputs ($$3$$ and $$8).$$

Question 8

Find the slope of the line through the points $$(0,2)$$ and $$(13,3).$$

Accepted Answer

$$m = \frac{3-2}{13-0} = \frac{1}{13}$$

Question 9

For the quadratic equation $$h(x) = x^2$ - 10x + 25$$, what are the coordinates of the vertex?

Accepted Answer

The vertex is at $$(5, 0)$$

Question 10

Convert the logarithmic equation $$y = \log_6$(216) to $$its exponential form.

Accepted Answer

$$6^y = 216$$

Question 11

Solve the exponential equation $$(13)^{x+5} = 2202$. What is the value of x$?

Accepted Answer

$$x = \log_{13}(2202) - 5$$

Question 12

For the function $$g(x) = (3$$)^{-x+2}$$, evaluate g(1$$)$$.

Accepted Answer

$$g(1) = 3$$^{-1+2}$$ = 3$$^{1}$$ = 3$$

Question 13

Which of the following equations represents a circle? $$x^2$$ + $$y^2 = 25$

Accepted Answer

Yes, it is a circle with center $$(0,0)$$ and radius $$5$$

Question 14

Determine if the function $$g(x) = 4x^3$ - x is $$odd, even, or neither.

Accepted Answer

Odd

Question 15

Solve the absolute value equation $$|2x - 5| = 7.$$

Accepted Answer

$$2x - 5 = 7 or$$ $$2x - 5 = -7$$; $$x = 6 or$$ $$x = -1$$

Question 16

Find the center-radius form of the equation of a circle with center $$(3, -2)$$ and radius $$5.$$

Accepted Answer

$$(x-3)^2$$ + $$(y+2)^2 = 25$

Question 17

For the quadratic equation $$f(x) = -3(x-5)^2$ - 1$$, does the parabola open upward or downward?

Accepted Answer

Downward

Guidance for Rational Inequality Problem

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