Algebra Computational Problems & Mathematics Questions

Lab Section #:Name:
Final Exam, 2/19/2022
You have until 12:00pm to complete your work for the exam. You then have 12:00pm – 12:45pm to upload your work to Gradescope. All work must be your own. No notes, calculators, or resources of any kind are allowed. Follow all guidelines in the syllabus and previous emails/announcements.
You must work on physical paper. We recommend you print out this template and work directly on it. You can
also work on blank paper as long as it is organized in a similar way as the template (you do not have to rewrite the questions). There are 35 questions, each worth 3 points. All answers must be justified in order to recieve credit.
If you need extra space to show your work, use blank paper and clearly indicate on the template that your work continues on another sheet. Also remember when submitting in Gradescope to tag a question to multiple pages if you use another page to show your work.
1. Find the value of the following expression using the
rules for order of operations.
2. Solve the following equation. Express your answer as
an integer, simplified fraction, or decimal rounded to two
decimal places.
11 + (5)2 ÷ (2 + 3) · 4
1.6w + 5.4 = 1.1w + 6.4
3. Solve the absolute value equation and write your answer
in set notation.
3|y − 1| − 12 = 0
4. Solve the following formula for the indicated variable.
2c =
1
7a
;
4b + 5
solve for b.
5. Consider the following inequality.
−2x − 6 < 3x + 6 (a) (2 pts.) Write the solution using interval notation. (b) (1 pt.) Graph the solution on a number line. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 6. Consider the following inequality. |8 − 2y| ≤ −5 Solve the absolute value inequality and express the solution in interval notation. If there is no solution, write “no solution”. 7. Consider the following inequality. |2y + 7| − 3 > 0
(a) (2 pts.) Solve the absolute value inequality and express the solution in interval notation.
(b) (1 pt.) Graph the solution set on a number line.
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
2
1
2
3
4
5
6
7
8
9
10
8. Consider the following linear equation.
y = −4x − 2
(a) (2 pts.) Determine the slope and y-intercept (written as an ordered pair) of the equation above. Reduce all fractions to lowest terms.
Slope:
y-intercept:
(b) (1 pt.) Graph the line.
10
8
6
4
2
-10
-8
-6
-4
-2
2
4
6
8
10
-2
-4
-6
-8
-10
9. Find the equation of the line passing through the points
with the given coordinates. Write your answer in slopeintercept form.
(1, 2), (4, 0)
10. Simplify the expression using the properties of exponents. Only include positive exponents in your answer.

3
x2 y −1
x−2 y 2
3 
x−1 y −2
x4 y 3
−3
11. 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 + 5)2
12. Divide the polynomial in the numerator by the binomial in the denominator and simplify.
13. Factor the polynomial by grouping, if possible. If the
polynomial cannot be factored, write “not factorable”.
3×4 − x3 − 9x + 4
x+2
20u3 + 35u2 − 4u − 7
14. Completely factor the given polynomial, if possible. If
the polynomial cannot be factored, write “not factorable”.
15. Completely factor the given polynomial, if possible. If
the polynomial cannot be factored, write “not factorable”.
8y 2 − 2×2
2×2 − 16x + 24
4
16. Simplify the following expression. Assume that each
variable is positive.
√
3
16×5
17. Simplify the expression by by combining the radical
terms using the indicated operations. Assume all variables
are positive.
p
p
p
7x 72xy + 8x 2xy + 7x 18xy
18. Rationalize the denominator and simplify, if possible.
19. Consider the following quadratic equation.
3
√
9− 7
x2 − 6x + 1 = −8
(a) (2 pts.) 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.
(b) (1 pt.) Solve the quadratic equation by factoring.
5
20. Solve the quadratic equation using the square root
property. Write the radicals in simplest form.
21. Solve the following quadratic equation by completing
the square. Simplify the solutions and rationalize denominators, if necessary.
(3x − 12)2 = 27
x2 + 6x − 12 = 0
22. Solve the following quadratic equation using the
quadratic formula.
y 2 = −8y + 1
23. Consider the following quadratic function.
y = x2 − 8x + 20
Rewrite the quadratic function in the form
y = a(x − h)2 + k.
6
24. Consider the quadratic function
y = (x − 2)2 + 3
(a) (1 pt.) Identify the vertex. Write your answer as an ordered pair.
(b) (1 pt.) Since a = 1 (see Question 23), we can identify two points on the parabola by finding a point 1 unit to
the right of the vertex and a point 1 unit to the left of the vertex. Find the point on the parabola 1 unit to the
right of the vertex. Write your answer as an ordered pair with an x-value of 3. Find the point on the parabola 1
unit to the left of the vertex. Write your answer as an ordered paid with an x-value of 1.
(c) (1 pt.) Graph the parabola in the space below.
10
8
6
4
2
-10
-8
-6
-4
-2
2
4
6
8
10
-2
-4
-6
-8
-10
25. Consider the function
f (x) =
√
26. Determine the domain of the given relation or function.
√
g(x) = 2 − 9x
5 + 4x.
(a) (1 pt.) Find the value of f (5).
(b) (2 pts.) Find the value of f (a − 2).
7
27. Perform the indicated operation of multiplication or
division on the rational expressions and simplify.
28. Perform the indicated operation of multiplication or
division on the rational expressions and simplify.
3
22
÷
3
y − 3 3(y − 3y 2 )
4
y 2 − 25
÷
(y + 5)2
20
29. Perform the indicated operation of addition on the
two rational expressions and reduce your answer to lowest
terms.
3x + 1
6
+
x2 + 2x + 1 x + 1
30. Simplify the following complex fraction.
5
x2
1
x2
8
−
−
10
x3
4
x4
31. Consider the following equation.
x
2
3x
=
−
x+4
x + 1 x2 + 5x + 4
(a) (1 pt.) State any restriction(s) on the variable, if they exist.
(b) (2 pts.) Solve the equation, if possible. If there is a solution, express your answer as either an integer or a simplified fraction.
33. Find the inverse of the following function.
32. Given two functions f (x) and g(x):
f (x) = 3×2 + 4,
g(x) = 7x − 5,
g(x) =
form their composition (g ◦ f )(x).
9
6
+2
x
34. Acid solution A has a 68% acid concentration, and acid solution B has a 20% acid concentration. How many ounces
each of solution A and solution B must be mixed to produce 100 ounces of an acid solution with a 32% acid concentration?
35. Working together, Candace and Donovan can clean the snow from their driveway in 9 minutes. It would have taken
Donovan, working alone, 36 minutes. How long would it have taken Candace alone?
10