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College Algebra (MAC-1105C) - Pre-Assessment Exam

Which number is irrational?        

  

-5
Answer:

Explanation:

An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q.
Irrational numbers have decimal expansions that neither terminate nor become periodic.
Every transcendental number is irrational.

Which graph corresponds to the set of real numbers:
                                                                                                    {− 4.25, − π, 7, 2.75}?

 

 

What is the temperature in Fahrenheit if the current temperature is −5°C?

23.0°F

Explanation:

 ·  =  = -9

32 – 9 = 23

Driver A travels west from Boston at a constant speed of 60 mph.

15 hours

Explanation:

First, we write the formula:
60(3 + x) = 72x
Next, we simplify and solve for x.
180 + 60x = 72x

-60x    -60x

That leaves us with
180 = 12x
Now we divide 180 by 12.
180 / 12 = 15
x = 15                                         

What is the solution to x + 4 < 4x - 5 expressed in interval notation?

(3, ∞)

Explanation:
Let's solve your inequality step-by-step.
x + 4 < 4x − 5
Subtract 4x from both sides. x + 4 − 4 x < 4 x − 5 − 4x
−3x + 4 < −5
Subtract 4 from both sides. −3x + 4 – 4 < −5 − 4
−3x < −9
Divide both sides by -3.
−3x −3 < −9 −3
x > 3

(3, ∞)

Which graph represents the following equation? 

y = 1/2x - 4

 

Explanation

Rewrite in slope-intercept form.
The slope-intercept form is y = mx + b where m is the slope and b is the y-intercept.
y = mx + b
Reorder terms. y = 1/2x − 4
Use the slope-intercept form to find the slope and y-intercept.
Slope: 1/2 y-intercept: (0, −4)
Any line can be graphed using two points. Select two x values,
and plug them into the equation to find the corresponding y values.
Reorder terms. y = 1/2x − 4
Create a table of the x and y values.
x    y
0   4
2   -3

Graph the line using the slope and the y-intercept, or the points.
Slope: 12      y-intercept: (0, −4)
x     y
0   -4
2   -3

What is the slope of the line that passes through the points (0, – 8) and (1/2, 4)?

24

Slope is equal to the change in y over the change in x or rise over run.
m =

The change in x is equal to the difference in x-coordinates (also called run), and the change in y
is equal to the difference in y-coordinates (also called rise).
m =    plug in the numbers m =

Now we apply the distributive property and solve for m.
(please note, 0.05 was converted from ½)
m =  =

m = 24

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What is the equation of the line that passes through the points (–3, 7) and (5, –1)?

y = −x + 4

Explanation:
Use y = mx + b to calculate the equation of the line,
where m represents the slope and b represents the y-intercept.

 =  =  = -1, so m = -1

 

 

d = –625p + 112,500

 

 

 = -625

A stockbroker’s monthly salary is $1,848 plus a sales commission of 5.2%
of the broker’s total sales for the month.
Which equation represents the stockbroker’s total monthly income (I)
based on monthly sales of x dollars?

I = 0.052x + 1,848

 

   

A store is creating a 30 lb mixture of chocolates and nuts.
Chocolates cost $5.00 per pound, and nuts cost $2.50 per pound.
The store wants to make a mixture that costs, on average, $3.00 per pound.
How many pounds of nuts are needed for this mixture?

24 lb.

Explanation:
The two equations we need are:

x + y = 30
5x + 2.5y = 3x(x + y)

Now we solve this equation for y and then plug the value of into the other equation.
We subtract x from both sides so y is alone.
x + y = 30 =               y = 30 – x
Now we plug in the value of y into the second equation.
5x + 2.5(30 – x) = 2(30)
Now we simplify.                          5x + 75 – 2.5x = 60
Subtract 75 from both sides.     5x – 1.5x = -15
Simplify the left side.                   2.5x = -15
Divide 2.5 on both sides.            -15 / 2.5 = -6
x = -6
The weight of the chocolate = 6
Now we subtract 6 from the 30 lb. mixture.
30 – 6 = 24

 

           

A college student works two part-time jobs.
The student works as a tutor for $40 per hour and at a coffeehouse for
$12 per hour. In one week the student works 22 hours and earns $404.
How many hours did the student work at the coffeehouse that week?

17

Explanation:

This is the equation we need!
12t + 40(22 - t) = 404
We will solve for t.
12t + 880 - 40t = 404
Simplify the left     12t – 40t = -28t
-28t + 880 = 404
We subtract 880 from both sides.
-28t – (880 = 880) = 404 – 808
We are left with     -28t = -476
Now we divide both sides by -28.
-28t / -28 = t
-476 / -28 = 17
t = 17

A dog breeder sells purebred puppies for $1,200 each and adult dogs for $800 each.
The breeder sells 8 dogs at a dog show for $8,400.
How many puppies did the breeder sell?

5

Explanation:
This is the equation we need to write and solve for p.
1200p + 800(8 - p) = 8400
1200p + 6400 – 800p = 8400
400p + 6400 = 8400
400p = 2000
p = 5

A child finds 30 nickels and dimes between sofa cushions.
How many dimes did the child find if the total value of the coins is $1.90?

8

Explanation:
This is the formula we need to write:
10d + 5(30 - d) = 190
Factor 5(30 – d)
10d + 150 – 5d = 190
Simplify
5d + 150 = 190
Subtract 190 - 150
5d = 40
Divide 40 / 5
d = 8

A bookstore has a clearance shelf with $3 books and $5 books.
On Saturday, the bookstore sold 96 of the clearance books for a total of $400.
How many of the $3 books were sold?

40

Explanation:
This is a problem that involves a slope.
Normally, we would need to solve for the y = mx + b format,
However, we are only looking for one value, so we can take a shortcut!
You can look and match my equation to the values from the problem.
It would be a lot longer however I like to take as many shortcuts as I can.
I would suggest the same for you. The equation makes perfect sense, so look it over it carefully.

This is the equation we need to solve.           3x + 5(96) -5x = 400
We simplify, combine, and solve.
3x – 5x + 5(96) = 400 =
-2x + 480 = 400
Now we subtract 400 – 480 = -80
-2x = -80
Now we divide      -80 / -2
x = 40

What is the result when (A³(A⁷) is simplified?

A¹⁰

Explanation:
Simply add the exponents
because we add when we multiply a variable with exponents.
A^{3 + 7} = A¹⁰

What is the simplification of the following expression?
          (4x⁴ – 3x² – 2x – 1) – (5x³ + 2x² + x - 4)

4x⁴ – 5x³ – 5x² – 3x + 3

First, we need to factor the right side, since this is a Simplification.
WE ARE NOT SUBTRACTING YET!
– (5x³ + 2x² + x - 4) is the same as – 1(5x³ + 2x² + x - 4)
So, we need to multiply each number on the right side by -1
Now we can line the equations up and simplify.
When we multiply, we need to separate like terms.
4x⁴ comes first, and so on. I have lined them up.

-5x³     – 2x²   – x      + 4

 4x⁴               – 3x²   – 2x    – 1

= 4x⁴ – 5x³   -5x²      -3x      + 3

What is the product of the following expression?

–2x²(3x² – x – 1)

–6x⁴ + 2x³ + 2x²

Explanation:
This is straight-forward.
We simply multiply the value of the left side by the values in parenthesis.
Then we combine like terms and simplify.
–2x² · 3x² =   6x⁴
–2x² · -x =     2x³
–2x² · -1 =     2x²
Now we put everything in proper order of operation.
–6x⁴ + 2x³ + 2x²

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Factor: 10xy + 16x – 45y – 72.

(5y + 8) (2x – 9)

Explanation:
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
(10xy + 16x) − 45y − 72
Factor out the greatest common factor (GCF) from each group.
2x (5y + 8)                −9(5y + 8)
Factor the polynomial by factoring out the greatest common factor, 5y + 8.
(5y + 8) (2x − 9)

Factor the following trinomial:   x² – x – 6.

(x – 3) (x + 2)

Explanation:
First, we consider the form x² + bx + c.
We then find a pair of integers whose product is c and whose sum is b.
In this case, whose product is −6 and whose sum is −1.
−3, 2
Write the factored form using these integers.
(x − 3) (x + 2)

Factor the following trinomial:
            3x² + x − 10

 (3x – 5) (x + 2)

Explanation:

Rewrite 1 as −5 plus 6                  3x²+(−5 + 6) x – 10
Apply the distributive property     3x² − 5x + 6x – 10
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.    (3x² − 5x) + 6x − 10
Factor out the greatest common factor (GCF) from each group.    x (3x − 5) + 2(3x − 5)
Factor the polynomial by factoring out the greatest common factor, 3x−5.
(3x−5) (x+2)

What is the complete factorization of the following trinomial?
                                        6x ² + 26x + 20

2 (x + 1) (3x + 10)

Explanation:
First, we factor out the GCF of 2 from 6x²+26x+20.
Factor out the GCF of 2 from each term in the polynomial.
Factor out the GCF of 2 from the expression 6x².
2(3x²) + 26x + 20
Factor out the GCF of 2 from the expression 26x.              2(3x²) + 2(13x) + 20
Factor out the GCF of 2 from the expression 20.                2(3x2) + 2(13x) + 2(10)
Since all the terms share a common factor of 2, it can be factored out of each term.
2(3x²+13x+10)
Factor by grouping. For a polynomial of the form ax² + bx + c rewrite the middle term as a sum of two terms whose product is a ⋅ c = 3 ⋅ 10 = 30 and whose sum is b = 13
2(3x² + 3x + 10x + 10)
Factor out the greatest common factor from each group. Group the first two terms and the last two terms.
2((3x² + 3x) + 10x + 10)
Factor out the greatest common factor (GCF) from each group.                          2(3x(x + 1) + 10(x + 1))
Factor the polynomial by factoring out the greatest common factor, x + 1.       2((x + 1) (3x + 10))

What is the complete factorization of the following binomial?
                                            9x² – 324

9(x – 6) (x + 6)

Explanation:
Factor out the GCF of 9 from 9x²−324.
Factor out the GCF of 9 from each term in the polynomial.
Factor out the GCF of 9 from the expression 9x²                9(x²) − 324
Factor out the GCF of 9 from the expression −324.           9(x²) + 9(−36)
Since all the terms share a common factor of 9, it can be factored out of each term.
9(x² − 36)                  Rewrite 36 as 6²                 9(x² − 6²)
Since both terms are perfect squares, factor using the difference of squares formula
a² − b² = (a + b) (a − b) where a = x and b = 6
9(x+6) (x−6)

What are the solutions to the following equation?
                5x² + 14x – 3 = 0

− 3, 1/5

This is a quadratic equation. We need to solve it.

                =         

 
Explanation:
Next, we raise 14 to the power of 2.

Next, we simplify the right side           196 – 4(5
So, we are left with                       
Next, we can rewrite  by squaring 256 and drop the radical.
The square root of   = 16².

Since it is a positive real number we can drop the radical.
We are left with     so we can simplify by dividing the top by  2.

We are left with

If we say  we will get  which equals            -3

If we say  we will get  which equals                 

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The incline of a roller coaster is 2 times as long as its elevation, and the horizontal length of the
roller coaster is 11m more than the elevation. What is the length of the incline in meters?

30

Explanation:
This is the equation we need to solve.
x² + (x + 11)² = (2x)²
Now we simplify.
(x + 11) (x + 11)
x² + 11x + 11x + 121
x² + x² + 11x + 11x + 121 = 4x²
2x² + 22x + 121 = 4x²
Now we subtract 4x² from both sides.
-2x² + 22x + 121 = 0
We need to solve the quadratic equation.

a = -2, b = 22, c = 121
The answer is x = 15.02627944.
We round that to 15.
The length of the incline is 2 times the elevation,
So, we multiply 15(2) = 30

Two teams of interns are wrapping donated gifts at a hospital. If the teams work alone, Team A can wrap
all of the gifts in 7 hours, and Team B can wrap all of the gifts in 5 hours.
How many hours will it take the interns to wrap all of the gifts if both teams work together?
Round the answer to the nearest half hour.

3.0

Explanation:
Simplest way to do this is like so and we do not need to solve for x.
I see this around the internet, and it was never solved correctly.
I did this in my head when I took the exam, however, I will spell it out now!
This is how I did it!
IF team A had another team that could wrap gifts at the same speed,
the two team A’ s would get it done in half the time! 7 hours / 2
IF team B had another team that could wrap gifts at the same speed,
the two team B’ s would get it done in half the time! 5 hours / 2
So, this is how we wrap this problem up!
I put x in front but there is no need to because all we do is add and simplify!

 +  / 2

 Add the fractions and simplify.

 / 2

So, we have 12 / 2 / 2.
12 / 2 = 6      and     6 / 2 = 3

What is the solution to the following equation?

 =

2 answers.   , 2

Explanation:
Multiply the numerator of the first fraction by the denominator of the second fraction.
Set this equal to the product of the denominator of the first fraction
and the numerator of the second fraction.

(x + 2) ⋅ 4 = 2x (3x − 2)

Solve the equation for x - Since x is on the right side of the equation,
switch the sides so it is on the left side of the equation.
2x (3x − 2) = (x + 2) ⋅ 4      Simplify        2x (3x − 2)
Rewrite        0 + 0 + 2x (3x − 2) = (x + 2) ⋅ 4
Simplify by multiplying through           2 ⋅ 3x ⋅ x − 4x = (x + 2) ⋅ 4
Simplify each term.          6x² − 4x = (x + 2) ⋅ 4           Simplify (x + 2) ⋅ 4              6x² − 4x = 4x + 8
Move all terms containing x to the left side of the equation.                6x² −8x = 8
Subtract 8 from both sides of the equation. 6x² − 8x – 8 = 0
Factor the left side of the equation.               2 (3x + 2) (x − 2) = 0
If any individual factor on the left side of the equation is equal to 0,
the entire expression will be equal to 0.       3x + 2 = 0                  x – 2 = 0
Set 3x + 2 equal to 0 and solve for x                Set 3x + 2 equal to 0                     3x + 2 = 0
Solve 3x + 2 = 0 for x.                                        x =
Set x − 2 equal to 0 and solve for x                x = 2

x^4/15

Explanation:
Since we will be dividing variables with exponents in a fraction,
We would simply multiply:

1x^2/3 ⋅ 1x^2/5

What is the value of x in the equation 3 + = 10?

24

Explanation:
3 +  = 10
We subtract 3 from both sides = 7
Now we square both sides and remove the radical

 )²= 7², so the radical drops out and we are left with:

2x + 1 = 7²    Now we can simplify 7²
We are left with 2x + 1 = 49
So, now we subtract 1 from both sides.
2x 1 – 1 = 49 – 1     =          2x = 48
Now we divide both sides by 2.
x = 24

What is the solution to the equation 15 - = 10?

15

Explanation:
This is the same steps as above.
15 - = 10
Subtract 15 from both sides.     - = -5
Square both sides (- )² = -5²
Drop the radical and simplify -5²

*Note the negative (– ) on the left gets dropped as well)

So, we are left with          2x - 5 = 25
Now we add 5 to both sides.    2x = 30
Divide by 2
x = 15

What is the solution to the equation (x + 2)² = –12

−2 2i

Explanation:
This is another quadratic problem. First, we simplify to quadratic form.
First, we move all terms to the left side of the equation and simplify.
Add 12 to both sides of the equation.
(x + 2)² + 12 = 0
Simplify (x + 2)² + 12
x² + 4x + 4 + 12 = 0
Add 4 and 12
We are left with:
x² + 4x + 16 = 0

a = 1, b = 4, c = 16

What is the result when (P⁸(P⁷) is simplified?

P¹⁵

Explanation:
Simply add the exponents
because we add when we multiply a variable with exponents.

P^{8 + 7} = P¹⁵

What is the solution to the equation 2x² – 4x + 4 = 0?

x = 1 - 1i
x = 1 + 1i

Explanation:
You can do this the long way or use my quadratic equation solver!
a = 2, b = -4, c = 4
http://www.coursesavior.com/quadsolver

 

What is the solution to the equation x² + 2x + 2 = 0?

x = 1 - 1i
x = 1 + 1i

Explanation:
You can do this the long way or use my quadratic equation solver!
a = 1, b = 2, c = 2
http://www.coursesavior.com/quadsolver

 

A rectangle has an area of 256 cm².

It has a width that is 48 cm shorter than 4 times its length.
What is the rectangle's length?

16 cm²

Explanation:
A = LW
x (4x - 48) = 256
4x² - 48x – 256 = 0

a = 4, b = -48, c = -256
You can do this the long way or use my quadratic equation solver!

http://www.coursesavior.com/quadsolver

A rectangle has an area of 240 cm². It has a width that is 43 cm longer than its length.
What is the rectangle’s length?

5 cm

Explanation:
This is the formula…
x (x - 43) = 240
x² - 43x = 240
x²- 43x – 240 = 0
We are left with a quadratic equation.

a = 1, b = -43, c = -240

You can do this the long way or use my quadratic equation solver!
http://www.coursesavior.com/quadsolver

A rectangle has an area of 150 cm². It has a length that is 5 cm longer than its width.
What is the rectangle's length?

15 cm

Explanation:
150 = x² - 5x
Factor x² − 5x − 150 using the AC method.
Consider the form x²+bx+c.
We need to find a pair of integers whose product is c and whose sum is b.
In this case, whose product is −150 and whose sum is −5
−15, 10
Write the factored form using these integers.
(x − 15) (x + 10) = 0
If any individual factor on the left side of the equation is equal to 0,
the entire expression will be equal to 0.
x – 15 = 0
x + 10 = 0
Set x – 15 equals to 0 and solve for x
x = 15              Set x+10 equal to 0 and solve for x
x = −10
The final solution is all the values that make (x − 15) (x + 10) = 0 true.

x = 15, −10

 Which graph represents the function y = (x – 2)² + 1?

 

I solved a similar problem above!
Use the same steps to solve or use a graphing calculator.

Which graph represents the function y = (x– 4) ² + 5?

 

I solved a similar problem above!
Use the same steps to solve or use a graphing calculator.

Explanation:

Simply place g(x) 6x – 1 over f(x) 3x – 2

The population of a town is 10,000 people.
The population is growing according to the function P(t) = 10,000 2
where t is the number of years from the present and P is the population.
What is the estimated population of the town after 3 years?

10,644

Explanation:

This is the formula we need to use for this problem.
P(t) = 10000(2)^0.03t
We are given t = 3 so we plug it in for the value of t.
10000(2)^0.03(3)
Now we need to simplify a little to solve with a calculator.
Enter it in the calculator like so:

10000 · 2^0.09

 

What is the solution to the equation _log64x = 2/3

16

 Explanation:

        64^2/3 = 16

An insurance annuity has an initial principal of $3,500 and an annual interest rate of 7%.
The interest is compounded continuously.
How many years will it take the account to earn $475 in interest?
Round to the nearest tenth.

1.8 years

Explanation:
First, convert R as a percent to r as a decimal
r = R / 100
r = 7 / 100
r = 0.07 per year,
Then, solve the equation for t
t = ln (A / P) / r
t = ln (3,975.00 / 3,500.00) / 0.07
t = 1.818025 years

A savings account has a beginning principal of $12,500 and an annual interest rate of 1.75%.
The interest is compounded continuously.
How many years will it take the account to earn $1,125 in interest?
Round to the nearest whole number.

5 years

Explanation:
First, we convert R as a percent to r as a decimal.
r = R / 100
r = 1.75 / 100
r = 0.0175 per year
Then, solve the equation for t
t = ln (A / P) / r
t = ln (13,625.00 / 12,500.00) / 0.0175
t = 4.92444 years
We round to the nearest whole number = 5

What is the value of x in the exponential equation 4^{5 – 9x} =   ^{x – 2}?

Explanation:
Apply the product rule to         4^{5 – 9x} =   
One to any power is one             4^{5 – 9x} = 

Move 8^{x – 2} to the numerator using the negative exponent rule ()
4^{5 – 9x} = 8^{-(x – 2)}
Create equivalent expressions in the equation that all have equal bases.
2^2(5-9x) = 2^{3(-(x – 2))}
Since the bases are the same,
then two expressions are only equal if the exponents are also equal.

2(5 − 9x) = 3(−(x − 2))

Solve for x    Simplify 2(5 − 9x)               10−18x = 3(−(x − 2))
Simplify 3(−(x−2))             10 − 18x = −3x + 6
Move all terms containing x to the left side of the equation.    10 − 15x = 6
Move all terms not containing x to the right side of the equation.      −15x = −4
Divide each term in −15x = −4 by −15 and simplify.

College Algebra (MAC-2233) Lessons:        Pre-Assessment Exam                  Final Exam (Proctored)            Page 1    2    3

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