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Introductory Algebra

1. Simplify by factoring. Assume that all variables under radicals represent nonnegative numbers.
√(36x^6)
Answer or note that the square root is not a real number. Show all work.

2. Use the quotient rule to simplify. Provide answer using exponential notation. Show all work.
-12a^5b^6c^12 / 3a^2b^3c^9

3. The amount P of pollution varies directly with the population N of people. City A has a population of 468,000 and produces 260,000 tons of pollutants. Find out how many tons of pollution we should expect City B to produce, if we know that its population is 362,000. Do not round until the final answer. Then round to the nearest ton as needed. Show all work.

4. Find the slope of the line (simplify answer) and y-intercept of the line (simplify answer and provide ordered pair). Show all work. (or note if slope is undefined or there is no y intercept)
-2x + 7y=14

5. Find the domain of the rational function. Show all work.

R(x) = -4+3x / x^3+8x^2-9x
Select correct choice below and fill in answer box:
A. The domain is {x | x is a real number and x≠ ________}
B. The domain is {x | x is a real number}.

6. Simplify the rational expression. Show all work.
X^2-6x-27 / x+3

7. Perform the indicated operation. Simplify answer. Do not factor. Show all work.
(9x^3-5x^2-5x+11)-(12x^3-3x^2+11x-5)

8. Use the produce rule to simplify the expression. Show all work.
(-7x^3p^5)(4y^2x^2)

9. Solve the compound inequality. Show all work. Provide simplified answer in interval notation.
-3 ≤ -2x+3 / 5 ≤ 4

10. Solve for x, show all work.
(x^2-14) / ( 2x^2+7x+3) + (-1) / (x+3) = (-4) / (2x+1)

11. Find the product. Show all work.
(5x+6)^2

12. Solve for x or indicate answer is all real numbers or that there is no solution. Show all work.
6(x-6)+x = 7(x-6)+6

13. A principal of $6,000 is invested in an account paying an annual rate of 4%. Find the amount in the account after 4 years if the account is compounded semiannually, quarterly , and monthly (a, b and c). Show all work.

14. Use the quadratic formula to solve m in the equation. Show all work. Simplify answer. Enter an exact answer using radicals as needed, use a comma to separate answers as needed.
(m+3)(5m+1)=3(m-4)+7

15. Simplify and write using positive exponents only. Use integers or fractions for any numbers in the expression. Type exponential notation with positive exponents. Show all work.
2a^-5b^4 / 10a^3b^-2

16. Find the equation of the line. Write the equation using function notation.
Through (4,-5); perdpendicular to 6y=x-12
The equation of the line is f(x) =___

17. The average annual number of cigarettes smoked by an adult in some country continues to decline. For the years 1997 - 2006, the equation y=-49.7x + 1756.2 approximates this data. Here, x is the number of years after 1997 and y is the average annual number of cigarettes smoked. If this trend continues, find the year in which the average number of cigarettes smoked is zero. To do this, let y=0 and solve for x. Show all work, round down to the nearest year.

18. In 2006, the population of a country was 48.7 million people. From 2006 to 2050, the country's population is expected to decrease by 5.3%. Find the expected population of the country in 2050. Round to the nearest tenth of a million. The population in 2050 will be _____ million. (round to the nearest tenth of a million.) Show all work.

19. Simplify. Use positive exponents only. Show all work.
(2x^-8 / y^-2)^-3

20. Show all work: In 2006, the median price of an existing home in some country was approximately $230,450. In 2001, the median price of an existing home was $157,700. Let y be the median price of an existing home in the year x, where x = 0 represents 2001.
a. Write a linear equation that models the median existing home price in terms of the year x. [Hint: The line must pass through the points (0,15770) and (5, 230450)].
y = _____ (answer in slope-intercept form.)
b. Usethis equation to predict the median existing home price for the year 2010.
y=$_____
c. Interpret the slope of the equation found in part a:
A. Every year, the median price of a home decreases by $14,550
B. Every year, the median price of a home increases by $157,700
C. Every year, the median price of a home decreases by $157,700
D. Every year, the median price of a home increases by $14,550

21. Graph the following equation. Show all work
Y=-2x + 2

22. Solve or indicate there is no solution. Show all work.
√(7-x) = x-1

23. Solve and Show all work. Answer should be an integer or simplified fraction, use a comma to separate answers as needed, or indicate there is no solution.):
4t^2 / 5 = 7t / 5 + 9/10

24. Solve the equation or indicate the solution is all real numbers or that there is no solution. Show all work.
(m-22) / (4) - (6m - 37) / (7) = 1

25. A spotlight is mounted on the eaves of a house 24 feet above the ground. A flower bed runs between the house and the sidewalk, so the closest the ladder can be placed to the house is 18 feet. How long a ladder is needed so that an electrician can reach the place where the light is mounted? Show all work.

Solution Preview

See attached for detailed solutions

1. Simplify by factoring. Assume that all variables under radicals represent nonnegative numbers.
√(36x^6)
Answer or note that the square root is not a real number. Show all work.
Solution:

2. Use the quotient rule to simplify. Provide answer using exponential notation. Show all work.
-12a^5b^6c^12 / 3a^2b^3c^9
Solution:

3. The amount P of pollution varies directly with the population N of people. City A has a population of 468,000 and produces 260,000 tons of pollutants. Find out how many tons of pollution we should expect City B to produce, if we know that its population is 362,000. Do not round until the final answer. Then round to the nearest ton as needed. Show all work.

4. Find the slope of the line (simplify answer) and y-intercept of the line (simplify answer and provide ordered pair). Show all work. (or note if slope is undefined or there is no y intercept)
-2x + 7y=14
Solution. From -2x + 7y=14, we get
7y=2x+14
So, y=(2/7)x +2
So, the slope is m=2/7.

5. Find the domain of the rational function. Show all work.
R(x) = -4+3x / x^3+8x^2-9x
Select correct choice below and fill in answer box:
A. The domain is {x | x is a real number and x≠ ________}
B. The domain is {x | x is a real number}.

Solution. As the denominator cannot be zero, we have

i.e.,

So, and and
So, the domain is
A. {x | x is a real number and x≠ -9, 0, 1}

6. Simplify the rational expression. Show all work.
X^2-6x-27 / x+3
Solution. As

7. Perform the indicated operation. Simplify answer. Do not factor. Show all work.
(9x^3-5x^2-5x+11)-(12x^3-3x^2+11x-5)
=
=

8. Use the produce rule to simplify the expression. Show all ...

Solution Summary

The expert examines simplifying factoring square roots.

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