18 questions were answered correctly.
2 questions were answered incorrectly.
1
Find the solution for x if |6x + 8|=32.
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... [Show More]
correct
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RATIONALE
To solve this equation, we will need to create two
separate equations without absolute value bars.
One equation will contain the expression exactly as it
appears within the absolute value bars. The second
equation must consider the case when the expression
has the opposite value. Next, solve each equation
separately, starting with 6x + 8=32.
To solve for x, start by subtracting 8 from both sides.CONCEPT
Absolute Value Equations
2
Suppose that 3y – 5 = 45 and y = 3x – 2.
Which of the following equations is equivalent to 3y – 5 = 45, but
written only in terms of x?
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9x – 6 = 50 correct
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9x – 6 = 40
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15x – 10 = 40
On the left, we have only 6x. On the right 32 minus 8 is
24. Then divide both sides by 6 to isolate x.
The solution for the first equation is x = 4. The same
process can be applied to the second equation 6x + 8 =
-32.
To solve, start by subtracting 8 from both sides.
On the left, we have only 6x. On the right -32 minus 8 is
-40. Then divide both sides by 6 to isolate x.
The solution for the second equation is .
The two solutions to |6x + 8| = 32 are x = 4 and
.•
15x – 10 = 50
RATIONALE
CONCEPT
Substitution in multi-step linear equations
3
Jacob opened a money-market account. In the first month,
he made an initial deposit of $500, and he plans to contribute an
additional $75 every month. The account does not pay any
interest.
After how many months will he have a total of $1,475?
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12 months
To find an equivalent equation written in terms of x we
can use the second equation y = 3x – 2. This equation
indicates that y is equivalent to 3x – 2 so we can
substitute 3x – 2 in for y in the first equation 3y – 5 = 45.
Now we have an equation in terms of x. To simplify it, first
apply the Distributive Property and multiply 3 by the
terms in the parentheses.
3 times 3x equals 9x, and 3 times -2 equals -6. Next,
eliminate -5 from the left side of the equation by adding
5 to both sides.
45 plus 5 equals 50. This equation is equivalent to the
original equation, but is written in terms of x.•
15 months
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13 months
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14 months correct
RATIONALE
Once each known value is substituted, we can solve
for n. First, distribute 75 into (n – 1).
75(n – 1) is equivalent to 75n – 75. Next, combine like
terms on the right side.
500 minus 75 is equal to 425. Subtract this value from
both sides to isolate the n term.
1475 minus 425 equals 1050. Finally divide both sides
by 75.
1050 divided by 75 is equal to 14. It will take 14
months for the balance to reach $1,475.
The amount in Jacob's deposit account can be
modeled using the formula for an arithmetic sequence.
The term is the value after n months, is the
initial balance, d is the common difference (or steady
monthly deposits), and n is the number of months.
Use the information provided to find each value.
The initial deposit, , is $500. Jacob deposits $75
each month, which is the difference, d. We want to
know how many months, n, it will take for the balance
to reach $1475, which is the value for . Next,
substitute each value in for , d, and .CONCEPT
Introduction to Arithmetic Sequences
4
Simplify the following expression.
‐4(x – 2) + 3(x + 2)
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-x – 2
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-x + 14 correct
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7x + 4
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7x - 6
RATIONALE
Notice that there is a factor of -4 multiplied by (x – 2).
Distribute the -4 by multiplying it by each individual term
in (x – 2).
-4 times x is -4x, and -4 times -2 is 8. We see that -4(x –
2) is equal to -4x + 8. Next, we must do the same with
3(x + 2).
3 times x is 3x, and 3 times 2 is 6. We see that 3(x + 2)
is equal to 3x + 6. Now we can combine like terms to
simplify.
We can combine -4x and 3x to get -x. We can also
combine 8 and 6 to get 14. The simplified expression is
-x + 14.CONCEPT
Properties in Algebraic Expressions
5
Dan paid $498.09 (including sales tax) for a laptop computer. The
sales tax in his state is 7%.
What was the price of the laptop before the sales tax was added?
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$463.25
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$469.75
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$472.00
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$465.50 correct
RATIONALE
To calculate the price of an item with tax
included, multiply the original price of the item
by (1 + r), where r is the percent tax
expressed as a decimal. The price of the
laptop with tax included is $498.09 and the
percent tax, r, is 7%, or 0.07 when written as
a decimal. Substitute these values in the
equation and solve for the original price.
Once the known values are substituted into
the formula, evaluate the parentheses by
adding 1 and 0.07.CONCEPT
Solving Problems involving Percents
6
Use the FOIL method to evaluate the expression:
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correct
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RATIONALE
1 plus 0.07 equals 1.07. Next, undo the
multiplication on the right side by dividing both
sides by 1.07.
$498.09 divided by 1.07 is $465.50. The price
of the laptop without sales tax is $465.50
FOIL stands for First, Outside, Inside, Last, and helps
us remember how to multiply two binomials. Begin by
multiplying the first two terms, 2 and .CONCEPT
Multiplying Radical Expressions
7
If a vehicle is going 60 mph, how many feet does it travel in one
second?
times equals . Now we can combine all
four parts.
2 times -7 equals -14. Next, multiply the two inside
terms, and .
We can combine the constant terms, -14 and 3, to get
-11. We can also combine the radical terms, and
, to get . The expression
is equal to .
After using FOIL, is equal to
. Next, identify any like terms and
combine.
2 times equals . Next, multiply the two outside
terms, 2 and -7.
times equals 3 because when you multiply two
terms with the same number under the radical symbol,
the result is just that number. Next, multiply the last two
terms, and -7.•
528 feet
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44 feet
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264 feet
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88 feet correct
RATIONALE
In general, we use conversion factors to
convert from one unit to another. A
conversion factor is a fraction with equal
quantities in the numerator and
denominator, but written with different
units. Here, not only do we need to convert
miles into feet, but we also need to convert
hours into seconds. We'll start with
converting miles to feet first.
There are 5280 feet in 1 mile. To convert 60
miles into feet, multiply by
. Notice that this will cancel the
units of miles, and the result is in feet per
hour.CONCEPT
Converting Unit Rates
8
For the equation 1.75n = 7, what is the value of n?
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4 correct
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6
60 times 5280 is 316800. This means that
60 miles per hour is the same as 316800
feet per hour. Next we need to convert
hours into seconds.
If you don't know how many seconds are in
an hour, you can use what you know about
minutes in an hour, and seconds in a
minute to create more conversion factors.
There are 60 minutes in 1 hour, and there
are 60 seconds in 1 minute. Notice that this
will cancel the units of hours and units of
minutes, and the result is in feet per
seconds.
In the numerator, we have 316800 feet. In
the denominator, we have 60 times 60,
which is equal to 3600. Notice that the
units in denominator are now in
seconds. Finally, divide 316800 feet by
3600 seconds.
60 miles per hour is the same as 88 feet
per second.•
0.25
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5.25
RATIONALE
CONCEPT
Solving single-step equations
9
Which of the following is equivalent to ?
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•
To find the value of n, we need to isolate it to one side of
the equation. We do this by performing inverse operations
to both sides of the equation. Since n is being multiplied by
1.75, we will divide both sides by 1.75, because division is
the inverse of multiplication.
On the left side, n will become isolated once we divide by
1.75. On the right side, we have 7 divided by 1.75.
7 divided by 1.75 is equal to 4. The solution to the
equation is n = 4.•
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correct
RATIONALE
Radical expressions can easily be added and
subtracted similar to combining like terms, if there is
the same number underneath the radical. If they are
different, use the Product Property of Radicals to
simplify if possible. Let's consider the first
term, .
can be written as because 9 times 2 is
18. We can simplify the square root of 9 further.
is already in its simplest form, since 2 is a
prime number that cannot be broken down any
further. Let's simplify the next term, .
The square root of 4 is 2, so the expression
becomes when fully evaluated. Let's
consider the next radical, .
can be written as because 4 times 2 is
8. We can simplify the square root of 4 further.CONCEPT
Adding and Subtracting Radical Expressions
10
Rationalize the denominator and simplify the following
expression:
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The square root of 9 is 3, so the expression
becomes when fully evaluated. Now we can
rewrite the original expression with the newly
evaluated terms.
can be rewritten as
. We have simplified the
expression to radical expressions with the same
number underneath the radical sign. Lastly, we can
combine all of the coefficients in front of the radical
sign.
7 minus 6 plus 3 equals 4. The expression is
equivalent to .•
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correct
RATIONALE
Use FOIL to expand numerator and denominator. Since
we used the conjugate of the denominator, you will
notice the denominator will be the difference of squares
(and it is now rational!).
When working with radicals, we want to avoid having a
radical in the denominator. The first step in the process
is to rationalize the denominator. To do this, multiply the
numerator and denominator by the conjugate of the
denominator. The conjugate is a binomial with the
opposite sign between its term, or .
In the numerator, we multiplied the first terms to get
; multiplied the outside terms to get 8; multiplied
the inside terms to get 5; and multiplied the last terms
to get . The denominator is the difference of
squares, or . Next, we can combine like
terms in the numerator.
In the numerator, combine the radical terms to get
and combine the constant terms to get 13. Next,
evaluate the exponents in the denominator.CONCEPT
Rationalizing the Denominator
11
Fred mixes a solution using two types of solutions: 0.60 liters
containing 10% alcohol and 0.40 liters containing 35% alcohol.
What is the alcohol concentration of the mixed solution?
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56%
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44%
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20%. correct
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25%
RATIONALE
5 minus 4 is 1. The expression is equivalent to
.
In the denominator, is just 5 and is 4. Next,
we subtract these two values.
To find the concentration of the mixed solution, set
up an equation using weighted averages. In the
numerator, we will multiply the concentration and
quantity of each solution and then add them
together. In the denominator, we will add the
quantities of each solution.CONCEPT
Solving Mixture Problems using Weighted Average
12
Consider the following subset of the real number line.
Once the values are substituted into the formula,
evaluate the denominator.
In this case, the denominator simplifies nicely to 1,
so we can ignore it and evaluate the numerator (but
note this will not always be the case). First multiply
0.10 and 0.60.
When multiplying the concentration and quantities
of the first solution, we get 0.10 times 0.60, which is
equal to 0.06. Next, multiply 0.35 and 0.40.
When multiplying the concentration and quantities
of the second solution, we get 0.35 times 0.40,
which is equal to 0.14. Finally, add these two
values together.
0.06 plus 0.14 is equal to 0.20. This value needs to
be written as a percent.
0.20 as a percent is 20%. The mixed solution is
20% alcohol.
The quantity of the first solution is 0.60 liters with a
concentration of 10% alcohol. Substitute 0.10 for
and 0.60 for . The quantity of the second
solution is 0.40 liters with a concentration of 35%
HCl. Substitute 0.35 for and 0.40 for .
How can this set be expressed using inequalities?
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-2 > x > 10
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-2≥x≥10
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-2≤x≤10
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-2 < x < 10 correct
RATIONALE
CONCEPT
Introduction to Inequalities
13
For the arithmetic sequence beginning with the terms {5, 6, 7, 8,
9, 10...}, what is the sum of the first 17 terms?
This number line represents a range of
values, which indicates that x will be
between the two numbers, -2 and 10. The
parentheses ( and ) around both end
points indicate that we cannot include either
the exact values of -2 and 10 in our
solution, so use the inequality sign for less
than: <.
This inequality states that x is between, but
not including, -2 and 10.•
243
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187
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200
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221 correct
RATIONALE
To find the sum of the first 17 terms in this sequence, we
first need to find the1 7th term.
We can see that each term increases by 1 from one to
the next, so plug 1 in for d. To find the value of the 17th
term, evaluate the parentheses.
To find the value of any nth term, use the formula
, where is the value of the first
term, is the value of the nth term, d is the common
difference, and n is the number of terms. There are two
variables we can substitute right away: the value of the
first term and the number of terms.
We see that the first term, , is 5 and we want to find
the sum of 17 terms, so n is 17. Next, we can find the
common difference, which is the difference between
each term in the sequence.CONCEPT
Finding the Sum of an Arithmetic Sequence
14
Solve the following absolute value inequality:
|x – 12| ≥ 7
•
x ≤ 5 or x ≥ 19 correct
17 minus 1 is 16. Next, multiply 1 by 16.
1 times 16 is 16. Then, add 5 and 16.
The 17th term is equal to 21. Now that we know the
value of the 17th term, we can use the formula for finding
the sum.
Once these values are plugged into the sum formula, we
can start by evaluating the parentheses.
5 plus 21 is 26. Next, divide 17 by 2.
17 divided by 2 is 8.5. Finally, multiply this value by 26.
8.5 times 26 is 221. The sum of the first 17 terms of this
sequence is 221.
In the formula , is the value of the
first term, is the value of the nth term, and n is the
number of terms. We know that , , and n =
17.•
x≤-19 or x≥-5
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5≤x≤19
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-19≤x≤-5
RATIONALE
CONCEPT
We can rewrite the absolute value inequality as a
compound inequality depending on the direction of
the inequality symbol. For absolute value
inequalities using the greater than symbol, write two
separate inequalities: one using greater than the
positive value and one using less than the negative
value.
The inequality |x – 12|≥7 is equivalent to x – 12≥
7 OR x – 12≤-7. Let's start by finding the solution
to the first inequality x – 12≥7.
To isolate x in the first inequality, add 12 to both
sides.
The solution to the first inequality is x≥19. We can
repeat this process with the second inequality x - 12
≤-7.
To isolate x in the second inequality, add 12 to both
sides.
The solution to the first inequality is x≤5.
The solution to the absolute value inequality |x – 12|
≥7 is x≤5 OR x≥19.Absolute Value Inequalities
15
What is the solution set for the following inequality?
4x – 3 + 6x > 5x + 7
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x > 2 correct
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x < 2
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x > 0.8
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x < 0.8
RATIONALE
Before solving for x, make sure that each side of the
inequality is fully simplified. On the left side, we will
combine like terms 4x and 6x.
On the left side, 4x plus 6x equals 10x. Now we can
begin to solve for x by applying inverse operations to
both sides of the inequality. First, subtract 5x from both
sides of the inequality.
Subtracting 5x from both sides cancels the 5x term on
the right side of the inequality, leaving x terms on only
the left side. Next, add 3 to both sides of the inequality.CONCEPT
Solve Linear Inequalities
16
Simplify the following expression.
‐3 + 4(a – 3b + 2) – 5(2 – a + 3b)
•
3a – b – 11
•
9a – 27b – 5 correct
•
-a + 3b – 5
•
3a – 9b – 5
RATIONALE
Adding 3 to both sides leaves only the x term on the left
side. Finally, divide both sides of the inequality by 5 to
isolate x.
The solution to this inequality is x > 2.
To simplify this expression, we must use the
Distributive Property to distribute the number
outside parentheses by multiplying each term
inside. There are two instances here. First, we
will distribute 4 into (a – 3b + 2).
4(a – 3b + 2) is equivalent to 4a – 12b + 8. Next
we have to distribute -5 into (2 – a + 3b). Notice
the negative sign in front of .CONCEPT
Terms and Factors in Algebraic Expressions
17
The relationship between the actual air temperature x (in degrees
Fahrenheit) and the temperature y adjusted for wind chill (in
degrees Fahrenheit, given a 30 mph wind) is given by the
following formula:
y = -26 + 1.3x
Estimate the actual temperature if the temperature adjusted for
wind chill is -35 degrees Fahrenheit.
•
-7 degrees Fahrenheit correct
•
-10 degrees Fahrenheit
•
-2 degrees Fahrenheit
•
-17 degrees Fahrenheit
RATIONALE
-5(2 – a + 3b) is equivalent to -10 + 5a – 15b.
Now we have to combine like terms.
The terms 4a and 5a combine to 9a, the
terms-12b and -15b combine to -27b, and the
constant terms -3, 8, and -10 combine to -5.
The expression‐3 + 4(a – 3b + 2) – 5(2 – a +
3b) is equivalent to 9a – 27b – 5.CONCEPT
Isolating Variables
18
Bert is driving along a country road at a constant speed of 30
miles per hour.
How long does it take him to travel 50 miles?
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An hour and 50 minutes
•
An hour and 30 minutes
•
•
An hour and 40 minutes correct
•
An hour and 15 minutes
The temperature adjusted for wind chill is given by this
equation. The wind chill is -35 degrees, so substitute this
value for y.
Once y is replaced by -35, we can solve for x. First, add
26 to both sides to undo the -26.
On the left side, we will now have -9 because -35 plus
26 equals -9. On the right side, we are left with 1.3x. To
isolate x, divide both sides by 1.3.
-9 divided by 1.3 is approximately equal to -7 The actual
temperature is about -7 degrees.•
RATIONALE
To find how long it will take Bert to travel 50 miles,
we can use the distance, rate, time formula and
solve for time. Plug in 50 miles for the distance,
and 30 miles per hour for the rate, or speed.
Once we have plugged in the values, we need to
write miles per hour as a fraction: 30 miles over 1
hour.
When dividing by a fraction, we can change this
into a multiplication problem and multiply by the
reciprocal of 30 miles per 1 hour, which would be
1 hour over 30 miles.
Rewrite 50 miles as a fraction over 1, and multiply
this by the reciprocal of 30 mph. Next, multiply
the numerators and denominators of the
fractions.
Multiplying across the numerator and
denominator, produces 50 over 30. The units of
miles cancel, so we are left with hours. Finally,
divide 50 by 30.
It will take Bert about 1.67 hours. However,
because we must express our answer in hours
and minutes, we must convert 0.67 hours to
minutes.
Using the conversion factor, 60 minutes to 1 hour,
evaluate the fractions by multiplying the
numerators and denominators.CONCEPT
Distance, Rate, and Time
19
The value V (in dollars) of a vehicle depends on the miles x that it
has been driven. This is given in the formula V = 22,000 – 0.25x .
After one year, the value of a vehicle is between $12,000 and
$15,000.
Which range of miles driven corresponds to this range of values
based on the given formula?
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18,000≤x≤19,000
•
28,000≤x≤40,000 correct
•
30,000≤x≤40,000
•
19,000≤x≤22,000
RATIONALE
Multiplying across the numerator and
denominator produces 0.67 times 60, or
approximately 40. Units of hours cancel, leaving
only minutes.
It will take Bert about 1 hour and 40 minutes to
travel 50 miles at a constant speed of 30 miles
per hour.CONCEPT
Compound Inequalities
20
The value of the vehicle is modeled by this
equation. To find the range for the miles driven,
construct an inequality with the expression for
the vehicle's value in between the lower and
upper values for the vehicle given in the
problem.
Solving these kinds of inequalities is very
similar to solving equations. Use inverse
operations to isolate the variable, but be sure to
perform the same operations in all parts of the
inequality. First, subtract 22,000 from all three
parts of the inequality.
Subtracting 22,000 from all three parts cancels
out the 22,000in the middle, leaving only the
x term. Next, divide all parts of the inequality by
-0.25. Remember that whenever you multiple
or divide an inequality by a negative number,
the sign of the inequality symbols change
direction.
We are now left with only an x term in the
middle. However, the inequality symbols are
not arranged in standard form, with the smaller
number first.
This is an equivalent statement from the
previous inequality. If you re-write the
inequalities in this way, you'll notice the
symbols change directions, too.The number of employees who will be able to perform a specific
task is given by the equation , where N is the number of
people who are able to perform the task, and t is time (in days).
How long will it take until 100 employees are able to perform the
task?
•
16 days correct
•
8 days
•
20 days
•
4 days
RATIONALE
The number of employees who can perform the task is
modeled by this equation. To find how long it will take until
100 employees can perform the task, we will substitute this
value for N and solve for t.
Once N is replaced, we can solve for t. First, divide both
sides by 25 to undo 25 multiplied by the square root of t.
On the left, 100 divided by 25 is 4. To undo the square root
of t on the right, square both sides.
On the left, 4 squared is 16. It will take 16 days for 100
employees to be able to perform the task. [Show Less]