Q: Given $$y = \dfrac{x}{2x-6}$$, simplify the expression and state the domain of $$y.$$

$$y = \dfrac{1}{2x-6}$$, \ x \neq 3$

Question 1

Simplify the following expression and eliminate any negative exponents: $$\left( \dfrac{a^5$}{b^4$} \$$right)^3$$ \left( \dfrac{b^2$}{a^7$} \$$right)^2$$

Accepted Answer

$$\dfrac{a$$^{1}$$}{b$$^{8}$$}$$

Question 2

Given $$y = \dfrac{x}{2x-6}$$, simplify the expression and state the domain of $$y.$$

Accepted Answer

$$y = \dfrac{1}{2x-6}$$, \ x 
eq 3$

Question 3

Rationalize the denominator of the expression $$\dfrac{7}{\sqrt{1 - x}}$$ and simplify the result.

Accepted Answer

$$\dfrac{7\sqrt{1-x}}{1-x}$$

Question 4

Rationalize the numerator of $$\dfrac{\sqrt{5} - 2}{\sqrt{5} + 3}$$ and simplify.

Accepted Answer

$$\dfrac{(\sqrt{5} - 2)(\sqrt{5} - 3)}{(\sqrt{5} + 3)(\sqrt{5} - 3)}$$

Question 5

Write the point-slope equation for the line through the point $$P(2, 5)$$ with slope $$m = -2.$$

Accepted Answer

$$y - 5 = -2(x - 2)$$

Question 6

Find an equation for the line through the two points $$(3, 2)$$ and $$(1, -4).$$

Accepted Answer

$$y = 3x - 7$$

Question 7

Find the slope and $$y-$$intercept of the line $$2x - 5y = 7.$$

Accepted Answer

Slope: $$\dfrac{2}{5}$$, $$y-$$intercept: $$-\dfrac{7}{5}$$

Question 8

Suppose the initial amount of a drug in the body is 1 mg. Find a formula for $$A$$, the amount in mg, at time $$t (in $$minutes) after it is taken, if $$A is (i) $$increasing by $$0.3 mg $$per minute, (ii) decreasing by $$0.3 mg $$per minute.

Accepted Answer

(i) $$A = 1 + 0.3t$$; (ii) $$A = 1 - 0.3t$$

Question 9

For the quadratic function $$f(x) = x^2$ + 6x + 8$$, what are the $$x-$$intercepts?

Accepted Answer

$$x = -4, -2$$

Question 10

Factor the quadratic function $$x^2 - 5x + 6$.

Accepted Answer

$$(x - 2)(x - 3)$$

Question 11

Given the quadratic $$g(x) = x^2$ - 12x + 4$$, what is the vertex of the parabola?

Accepted Answer

$$(6, -32)$$

Question 12

If $$f(x) = 2x^2$ + 6x + 8$$, what is the $$y-$$intercept?

Accepted Answer

$$8$$

Question 13

Which of the following is the correct factorization of $$12x^2 + 11x - 15$?

Accepted Answer

$$(3x - 5)(4x + 3)$$

Question 14

Given $$f(x) = x^2$ + 6x + 8$$, sketch the graph and determine the axis of symmetry.

Accepted Answer

$$x = -3$$

Question 15

Simplify $$\sqrt{75x^2$} \sqrt{12x^4$}.$$

Accepted Answer

$$6x^3$$ \sqrt{6}$$

Question 16

If $$f(x) = x^2$ + 6x + 8$$, what is the value of $$f(-1)$$?

Accepted Answer

$$1$$

Question 17

Given $$f(x) = x^2$ + 6x + 8$$, what is the sum of the roots?

Accepted Answer

$$-6$$

Question 18

For the quadratic $$f(x) = x^2$ + 6x + 8$$, what is the product of the roots?

Accepted Answer

$$8$$

Question 19

Given the equation $$2x - 5y = 7$$, what is the $$y-$$intercept?

Accepted Answer

$$-\dfrac{7}{5}$$

Algebraic Manipulation and Linear Equations: Foundations for Calculus

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