Question 1

Simplify $$\frac{1+7i}{8+2i}$$ and write your answer in standard form.

Accepted Answer

$$\frac{1+7i}{8+2i} = \frac{1+7i}{8+2i} \cdot \frac{8-2i}{8-2i} = \frac{22+54i}{68} = \frac{11}{34} + \frac{27}{34}i$$

Question 2

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

Accepted Answer

$$(-\infty, -5) \cup (-5, 4]$$

Question 3

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

Accepted Answer

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

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

y-values: $$-1, 3, 7$$

Question 5

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

Accepted Answer

Graph A (shows a square root function starting at $$x = -6)$$

Question 6

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

Accepted Answer

$$(f-g)(1) = f(1) - g(1) = 12 - 5 = 7$$

Question 7

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 8

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

Accepted Answer

Not a function, because $$x=0 is $$paired with two different $$y$$ values.

Question 9

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

Accepted Answer

Vertex: $$(5, 0)$$

Question 10

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

Accepted Answer

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

Question 11

Convert the logarithmic function $$2y = -3 + \log_8$(x) to an $$exponential function.

Accepted Answer

Question 12

Evaluate the exponential function $$g(x) = 3$$^{-x+2}$$ at x = 1$.

Accepted Answer

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

Question 13

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

Accepted Answer

Vertex: $$(5, -1)$$, opens downward

Question 14

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

Accepted Answer

$$x = 10 or$$ $$x = -4$$

Question 15

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

Accepted Answer

Odd

Question 16

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

Accepted Answer

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

Question 17

Solve the logarithmic equation $$y = \log_6$(216). $$What is the value of $$y$$?

Accepted Answer

$$y = 3$$

Question 18

For the quadratic equation $$f(x) = 2x^2$ + 4$$, what is the y-intercept?

Accepted Answer

$$y = 4$$

Question 19

Use synthetic division to divide $$2x^5$$ + $$x^4$$ - $$10x^3$$ + $$2x^2 + x - 16$ by (x+2$$)$$. What is the remainder?

Accepted Answer

Remainder: $$-2$$

Guidance for Rational Inequality and Complex Number Division

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