Q: Simplify the following expression: $$\dfrac{2x^2$ - 8}{x^2$ - 4}$$

$$\dfrac{2(x - 2)}{x - 2}$$,\ x \neq 2,\ x \neq -2$

Question 1

A chemist is preparing a solution by mixing two solutions: one with 30% acid and another with 50% acid. If the chemist wants to make 10 liters of a 42% acid solution, how many liters of the 50% acid solution should be used? (Let $$x be $$the amount of 50% solution.)

Accepted Answer

$$x = 6$$ liters

Question 2

Solve the system of equations for $$x$$ and $$y$$:

$$\begin{cases} 2x + 3y = 7 \ 4x - y = 5 \end{cases}$$

Accepted Answer

$$x = 2,\ y = 1$$

Question 3

Which of the following is the correct factorization of $$x^2 - 5x + 6$?

Accepted Answer

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

Question 4

A ball is thrown upward from the ground with an initial velocity of $$20 m/s. $$Its height $$h (in $$meters) after $$t$$ seconds is given by $$h = -5t^2$ + 20t. $$After how many seconds does the ball hit the ground again?

Accepted Answer

$$t = 4$$ seconds

Question 5

Simplify the following expression:

$$\dfrac{2x^2$ - 8}{x^2$ - 4}$$

Accepted Answer

$$\dfrac{2(x - 2)}{x - 2}$$,\ x 
eq 2,\ x 
eq -2$

Question 6

Solve for $$x$$:

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

Accepted Answer

$$x = 2$$ and $$x = 5 ($$but only $$x = 2 is $$valid)

Question 7

Which of the following is the solution set for the inequality $$2x - 7 \u003c 3x + 5$$?

Accepted Answer

$$x \u003e -12$$

Question 8

A rectangular garden has a length that is 3 meters more than twice its width. If the area of the garden is $$99$$ square meters, what is the width of the garden?

Accepted Answer

$$6$$ meters

Question 9

Which of the following is equivalent to $$\dfrac{3x^2$y^{-1}$$}{$$6x^{-1}y^3$}$$?

Accepted Answer

$$\dfrac{x^3$}{2y^4$}$$

Question 10

Solve the quadratic equation $$x^2 + 4x + 1 = 0$ using the quadratic formula.

Accepted Answer

$$x = -2 \pm \sqrt{3}$$

Question 11

A company’s revenue $$R (in $$thousands of dollars) is modeled by $$R = 50x - x^2$, where x$ is the number of units sold (in thousands). What number of units sold maximizes the revenue?

Accepted Answer

$$x = 25$$

Question 12

Which of the following is the solution to the equation $$\dfrac{2}{x} + 3 = 5$$?

Accepted Answer

$$x = 1$$

Question 13

If $$f(x) = 2x^2$ - 3x + 1$$, what is $$f(-2)$$?

Accepted Answer

$$f(-2) = 15$$

Question 14

Solve for $$x in $$the equation $$x^2 - 9 = 0$.

Accepted Answer

$$x = 3 or$$ $$x = -3$$

Question 15

Which of the following is the correct solution to the inequality $$|2x - 5| \leq 7$$?

Accepted Answer

$$-1 \leq x \leq 6$$

Question 16

A biologist is studying a bacteria population that doubles every 3 hours. If the initial population is 500, what will the population be after 9 hours?

Accepted Answer

$$4000$$

Question 17

Which of the following is the correct factorization of $$x^2 - 16$?

Accepted Answer

$$(x - 4)(x + 4)$$

Question 18

Solve for $$x$$:

$$\dfrac{x + 2}{x - 1} = 3$$

Accepted Answer

$$x = 2.5$$

Question 19

Which of the following is the solution to the equation $$2^{x} = 16$?

Accepted Answer

$$x = 4$$

Question 20

A car rental company charges a flat fee of $$\20$ plus $$\$$0.15$$ per mile driven. If a customer paid $$\50$, how many miles did they drive?

Accepted Answer

$$200$$ miles

Core Concepts in College Algebra: Equations, Factoring, and Problem Solving

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