Q: Simplify the rational expression and state the excluded values: $$\frac{x^2$ + x - 6}{x^2$ - 49}.$$

$$\frac{x+3}{x+7},\ x \neq -7,\ 7$$

Q: Simplify the expression $$\frac{x^2$ + 2x}{x^2$ - 4}$$ and state the values of $$x$$ that must be excluded.

$$\frac{x(x+2)}{(x-2)(x+2)},\ x \neq 2,\ -2$$

Question 1

Convert the radical $$\sqrt[3]{9x-1} to $$rational exponent notation.

Accepted Answer

$$(9x-1)^{1/3}$$

Question 2

Which of the following is the correct radical notation for $$x^{3/4}$$?

Accepted Answer

$$\sqrt[4]{x^3$}$$

Question 3

Rationalize the denominator and simplify: $$\frac{4}{\sqrt{6}}.$$

Accepted Answer

$$\frac{2\sqrt{6}}{3}$$

Question 4

Solve the equation $$-2(6-x) = 12(x+1).$$

Accepted Answer

$$x = -\frac{5}{2}$$

Question 5

Solve the inequality $$-11x + 3(3x-1) \geq 5$$ and write your answer in interval notation.

Accepted Answer

$$(-\infty, -4]$$

Question 6

Factor the polynomial $$121x^2 - 36$ completely.

Accepted Answer

$$(11x-6)(11x+6)$$

Question 7

Solve the quadratic equation $$x^2 - 7x - 18 = 0$.

Accepted Answer

$$x = 9,\ -2$$

Question 8

Solve the radical equation $$\sqrt{10-5x} = x+8. $$Which value is a solution?

Accepted Answer

$$x = -3$$

Question 9

Solve by completing the square: $$x^2 - 14x - 2 = 0$.

Accepted Answer

$$x = 7 \pm \sqrt{51}$$

Question 10

Use the quadratic formula to solve $$2x^2 - 5x - 9 = 0$.

Accepted Answer

$$x = \frac{5 \pm \sqrt{97}}{4}$$

Question 11

Simplify the rational expression and state the excluded values: $$\frac{x^2$ + x - 6}{x^2$ - 49}.$$

Accepted Answer

$$\frac{x+3}{x+7},\ x 
eq -7,\ 7$$

Question 12

Which of the following is the correct simplification of the complex rational expression $$\frac{2x}{x} + \frac{3}{x}$$?

Accepted Answer

$$\frac{2x+3}{x}$$

Question 13

A biologist is studying the growth of a bacteria population modeled by $$P(t) = 5t^2$ + 4t = 12. $$Solve for $$t.$$

Accepted Answer

$$t = -2,\ 1.2$$

Question 14

Given the equation $$x^4$$ - $$6x^2 + 5 = 0$, which of the following is the complete set of real solutions?

Accepted Answer

$$x = -1,\ 1,\ -\sqrt{5},\ \sqrt{5}$$

Question 15

Simplify the expression $$\frac{x^2$ + 2x}{x^2$ - 4}$$ and state the values of $$x$$ that must be excluded.

Accepted Answer

$$\frac{x(x+2)}{(x-2)(x+2)},\ x 
eq 2,\ -2$$

Question 16

A chemist needs to prepare a solution by mixing two liquids. The equation $$\sqrt{2x+33} = x-1$$ models the required concentration. What is the value of $$x$$?

Accepted Answer

$$x = 8$$

Question 17

Which of the following is the correct radical notation for $$(3x-5)^{2/7}$$?

Accepted Answer

$$\sqrt[7]{(3x-5)^2$}$$

Question 18

A projectile's height is modeled by $$h(t) = -16t^2$ + 47t = 0. $$Find all real solutions for $$t.$$

Accepted Answer

$$t = 0,\ 1,\ 47$$

Question 19

Simplify the complex rational expression $$\frac{2x}{x} + \frac{3}{x}$$ and choose the correct answer.

Accepted Answer

$$\frac{2x+3}{x}$$

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