Algebraic Proficiency Final Assessment

1. Consider the following algebraic expression: -6s -4b

Step 1. Identify the first term of the algebraic expression. Indicate whether the term is a variable term or a constant term. For a variable term, identify the variable and the coefficient of the term.

Step 2. Identify the second term of the algebraic expression. Indicate whether the term is a variable term or a constant term. For a variable term, identify the variable and the coefficient of the term.

2. Simplify the algebraic expression by combining the like (or similar) terms. -2y + 4a² - 4 + 3a² - 3y + 1

3. Evaluate the following algebraic expression at x = -3, y = 3, z = -2 and simplify your answer. -9x + y/-2z

4. Translate the following phrase into an algebraic expression. Do not simplify. Use the variable names "" or "" to describe the unknowns. four divided by a number

5. Translate the following phrase into an algebraic expression. Do not simplify. Use the variable names "" or "" to describe the unknowns. four divided by a number

6. Solve the following linear equation using equivalent equations to isolate the variable. Express your answer as an integer, as a simplified fraction, or as a decimal number rounded to two places. 4/3t = 6

One of the tallest clock towers in the world is the Abraj Al Bait Towers in Saudi Arabia, which stands 1972 feet tall. Mendocino trees in California are among the tallest in the world. If 5 of these trees were stacked on top of each other, they would still be 169.5 feet shorter than the Abraj Al Bait Towers in Saudi Arabia. Solve the following equation to determine the height of the tree: 5x + 169.5 = 1972. Round your answer to the nearest tenth.

8. Solve the following linear equation and simplify your answer. -2/5(y + 1/2) = -1/5(y - 2)

9. A classic car is now selling for $1000 more than two times its original price. If the selling price is now $18,000, what was the car's original price?

10. Change the following fraction/mixed number to a percent. Round the percent to the nearest hundredth.2(1/4)

11. After 10 years, Hamid's account earned $600 in interest. If the interest rate (in decimal form) is 0.12, how much did Hamid initially invest? Step 1. Choose the correct formula.

1. C=2πr
2. P=a+b+c
3. I=Prt
4. A=bh
5. P=4s
6. P=2l+2w
7. A=1/2 h(b+c)
8. C=5/9(F-32)

Step 2. Without substitution, solve the formula chosen in the previous step for the unknown variable in terms of the known variable(s).

Step 3. Solve the problem by substituting in the appropriate value(s) of the known variable(s) in the formula you found in the previous step. Round your answer to two decimal places, if necessary.

12. A particular style of watch costs the retailer $70 per watch. At what price should the retailer mark them so he can sell them at a 10% discount off the original price and still make 35% profit on his cost?

13. Consider the following word problem: It took Eric 9 hours to drive to a football game. On the way home, he was able to increase his average speed by 22 mph and make the return drive in only 6 hours. Find his average speed on the return drive.

Step 1. Complete the following table by entering the missing values.

Step 2. Use the table from part 1 to help write a linear equation that represents the given word problem. (Use x to represent the unknown quantity.)

Step 3. Solve the equation found in part 2 for x. Use this information to answer the given word problem. Enter your answer in the box below.

14. Given the equation F = 9/5C + 32 where C is the temperature in degrees Celsius and F is the corresponding temperature in degrees Fahrenheit, and the following ordered pairs: (-10,F₁ ), (10,F₂ )

Step 1. Compute the missing y values so that each ordered pair will satisfy the given equation.

Step 2. Plot the resulting set of ordered pairs using your answers from Step 1.

15. Find the y-intercept and x-intercept of the following linear equation, state as ordered pairs. Graph the line and clearly indicate these two points on the graph. Tip: Always compute at least one more point as a double check, if your points are not all on a straight line then go back and check your arithmetic. -4x + 6y = -12

Answer: y-intercept: x-intercept:

16. Graph the following linear equation. 4x + 2y = 8
Step 1. Determine the slope and y-intercept (state undefined if applicable).
Step 2. Plot the y-intercept of the linear equation on the graph.
Step 3. Plot a second point on the graph (other than the y-intercept) such that the slope of the line connecting the point and the y-intercept is equal to the slope of the given equation.

17. A local cinema found that if the price of admission was $17, the attendance was about 2050 customers per week. When the price of admission was dropped to $12, attendance increased to about 2100 per week. Write a linear equation for the attendance in terms of the price, p. ( A = mp + b )

18. Find an equation for the line perpendicular to the line 5x - 6y = 10 having the same y-intercept as 3x - 4y = 2.

19. Consider the following inequality: -4z + 3 ≥ 4z + 7
Step 1. Solve the linear inequality for the given variable. Simplify and express your answer in algebraic notation.

Step 2. Graph the solution to the given inequality.

20. State the domain of the following function in interval notation, include your explanation in a sentence as well. y = -9x + 1 Explain what this interval represents:

21. Simplify the expression using the properties of exponents. Expand any numerical portion of your answer and only include positive exponents. 10y⁴/2y²

22. Simplify the expression using the properties of exponents. Expand any numerical portion of your answer and only include positive exponents. (2² m² n)/n² )^-2

23. Consider the following polynomial: 7x² + 3x
Step 1. Determine if the polynomial is a monomial, binomial, or trinomial (circle or clearly state).
A) Monomial B) Binomial C) Trinomial
Step 2. Determine the degree and the leading coefficient of the polynomial.

24. Perform the indicated operation by removing the parentheses and combining like terms. (-3x² - 9x) - (7x² + 2x)

25. Multiply the monomials using the product rule of exponents. (5x⁴y)(〖5x² )

26. A square is 10 inches on each side. A small square, x inches on each side, is cut out from each corner of the original square. Represent the area of the remaining portion of the square in the form of a polynomial function A(x).

Explain your approach to the solution process you used, there is more than one right way to approach his problem:

27. Find the product of the binomials using the appropriate special product (difference of two squares, square of a binomial sum, or square of a binomial difference). (x - 3)(x - 3)

28. Divide the polynomial in the numerator by the monomial in the denominator. Simplify your answer.6x + 2/-2

29. Find the GCF (greatest common factor) of the following terms.{12xy²,36,24xy}

30. Factor the given polynomial by finding the greatest common monomial factor (or the negative of the greatest common monomial factor) and rewrite the expression.-39x + 15x² - 15y

31. Factor each expression by factoring out the common binomial. b³ (a + 5) + 4(a + 5)

32. The area (in square inches) of a rectangle is given by the polynomial function A(p) = p² + 9p + 20. If the length of the rectangle is (p + 5) inches, what is the width?

33. Completely factor the trinomial, if possible. 5x² + 22x + 8 Explain what factoring process you used and why it was applicable:

34. Completely factor the trinomial, if possible. 8b² + 6b + 1 Explain what factoring process you used and why it was applicable:

35. Completely factor the polynomial, if possible. 16 + 24x + 9x² Explain what factoring process you used and why it was applicable:

36. Completely factor the expression, if possible. x³ - 27y³ Explain what factoring process you used and why it was applicable:

37. Consider the following quadratic equation: x² + 7x + 1 = -9

Step 1. Using the standard form ax² + bx + c = 0 of the given quadratic equation, factor the left hand side of the equation into two linear factors.

Step 2. Solve the quadratic equation by factoring. State the answer in set builder notation (refer to OpenStax Chapter 3 Section 2 for help with formatting set builder notation)

38. The product of two consecutive even integers is 48. Find the integers using an algebraic method, not a guess and check method, show the algebraic set-up and solution steps.

39. Use your knowledge of square roots and cube roots to determine whether the following number is rational, irrational, or nonreal. √169

40. Simplify the following expression. Assume that each variable is positive. √36y⁶

41. Change the following expression to an equivalent expression in radical notation. Assume all variables are positive. x^1/8

42. Perform the indicated operations on the following radicals. 2√9x + 4√x - 8√5

43. Multiply the following radicals and simplify your answer. -5√2 . 4√10

44. Rationalize the denominator and simplify, if possible. Assume all variables are positive. ∛10z⁴)/∛z))

45. Solve the following radical equation. Enter your answer as an integer. If there is no solution indicate "No Solution".√(t + 3) = √(3t + 7)

46. Consider the following functions. f (x) = x + 3 and g(x) = x - 4

Step 1. Find ( f + g)(x). Simplify your answer.
Step 2. Find ( f - g)(x). Simplify your answer.
Step 3. Find ( f . g)(x). Simplify your answer.
Step 4. Find (f/g)(x). Simplify your answer.

47. Find the inverse of the following function: f ( x ) = 4x - 4

48. The compound interest formula states that if P dollars are invested at an annual interest rate of r, compounded n times per year, then A, the amount of money present after t years, is given by A = P (1 + r/n)^nt. If $14,000 is invested at 9% compounded annually, how much will this investment be worth in 15 years? Round your answer to two decimal places.

49. Solve the following logarithmic equation in terms of the variable x. log⁶( x ) = -3

50. Evaluate the following expression. log⁵( 5^1/3)

51. Use a calculator to evaluate the following logarithm accurate to 5 decimal places. If it is undefined, then indicate "Undefined". ln( -47.82)

52. Solve the following equation for x. 3x² - 8x - 3) = 3²x² - 2x + 2)

53. Find the restricted values of x for the following rational expression. If there are no restricted values of x, indicate "No Restrictions". -8/x² + 8)))

54. Perform the indicated operation of multiplication or division on the rational expressions and simplify. b/a + 3 ÷ ab + 3b/a² + 6a + 9

55. Consider the following equation: x - 5/x - 8 = 4/5

Step 1. State any restriction(s) on the variable, if they exist.

Explain the rationale for stating restrictions on the domain, why is this checking step needed?

Step 2. Solve the equation, if possible. If there is a solution, express your answer as either an integer or a simplified fraction.

56. The length that a hanging spring stretches varies directly with the weight placed at the end of the spring. If a weight of 27 lb stretches a certain spring 12 in., how far will the spring stretch if the weight is increased to 44 lb? (Leave the variation constant in fraction form. Round off your final answer to the nearest inch.)

57. Solve the following quadratic equation by using the definition of a square root and write the solutions in simplified radical form. 3x² = 36

58. Solve the following quadratic equation using the quadratic formula. -x² + 3x + 5 = 0

59. Consider the following quadratic function. y = -x² - 4

Step 1. Find the vertex of this function.
Step 2. Find the line of symmetry of this function.
Step 3. Find the x-intercept(s), if any. Express your answer as ordered pair(s).
Step 4. Determine two points on the graph of the parabola other than the vertex and the x-intercepts.
Step 5. Graph the quadratic function.

60. Write 1 × 10^-3 in decimal form.

Explain the general process for converting both small and large values into scientific notation:

61. Write 1,770,000 in scientific notation.

62. Simplify the expression 392,000 × 0.0002 using scientific notation and express your answer in scientific notation.