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UNIT 2 — MILESTONE 2
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1
Consider the following subset of the real number line
How can this set be expressed using inequalities?
-3 ≤ x < 1
-3 ≥ x > 1
-3 < x ≤ 1
-3 > x ≥ 1
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CONCEPT
Introduction to Inequalities
2
What is the solution set for the following inequality?
2x + 8x ≥ 15 + 14x + 9
x ≤ -6
x ≤ -0.85
x ≥ -6
x ≥ -0.85
RATIONALE
Before solving for x, make sure that each side of the inequality is fully simplified. On the right side, we will add 15 and 9.
On the right side, 15 plus 9 is 24. On the left side, we can combine 2x and 8x.
2x plus 8x is 10x. Now we can begin to solve for x by applying inverse operations to both sides of the inequality. First, subtract 14x from both
sides.
Subtracting 14x from both sides cancels the 14x term on the right side of the inequality, leaving x terms on only the left side. Finally, divide both
sides of the inequality by -4. Remember that the sign of the inequality sign changes whenever you multiply or divide by a negative number!
The solution to the inequality is x ≤ -6.
CONCEPT
Solve Linear Inequalities
3
John is driving at a constant speed of 40 miles per hour.
How long does it take him to travel 50 miles?
An hour and 48 minutes
An hour and 15 minutes5/25/2021 Sophia :: Welcome
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An hour and 8 minutes
An hour and 30 minutes
RATIONALE
To find how long it will take John to travel 50 miles, we can use the distance, rate, time formula and solve for time. Plug in 50 miles for
the distance, and 40 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: 40 miles over 1 hour.
When dividing by a fraction, we can change this into a multiplication problem and multiply by the reciprocal of 40 miles per 1 hour,
which would be 1 hour over 40 miles.
Rewrite 50 miles as a fraction over 1, and multiply this by the reciprocal of 40 mph. Next, multiply the numerators and denominators of
the fractions.
Multiplying across the numerator and denominator, produces 50 over 40. The units of miles cancel, so we are left with hours. Finally,
divide 50 by 40.
It will take John 1.25 hours. However, because we must express our answer in hours and minutes, we must convert 0.25 hours to
minutes.
Using the conversion factor, 60 minutes to 1 hour, evaluate the fractions by multiplying the numerators and denominators.
Multiplying across the numerator and denominator produces 0.25 times 60, or 15. Units of hours cancel, leaving only minutes.
It will take John 1 hour and 15 minutes to travel 50 miles at a constant speed of 40 miles per hour.
CONCEPT
Distance, Rate, and Time
4
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
20 days
4 days
8 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.
CONCEPT
Solving Multi-step Equations
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The value V (in dollars) of an airplane depends on the flight hours x as given by the formula V = 1,800,000 – 250x. After one year, the value of the plane is between
$1,200,000 and $1,300,000.
Which range for the flight hours does this correspond to?
2000 ≤ x ≤ 2400
2200 ≤ x ≤ 2500
1800 ≤ x ≤ 2100
1500 ≤ x ≤ 1800
RATIONALE
The value of the airplane is modeled by this equation. To find the range for the flight hours, construct an inequality with
the expression for the plane's value in between the lower and upper values for the plane 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 1,800,000 from all three parts of the
inequality.
Subtracting 1,800,000 from all three parts cancels out the 1,800,000 in the middle, leaving only the x term. Next, divide
all parts of the inequality by -250 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 a 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.
CONCEPT
Compound Inequalities
6
Simplify the following expression.
‐4(x – 2) + 3(x + 2)
-x – 2
7x - 6
7x + 4
-x + 14
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
7
Dan paid $498.09 (including sales tax) for a laptop computer. The sales tax in his state is 7%.5/25/2021 Sophia :: Welcome
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What was the price of the laptop before the sales tax was added?
$472.00
$469.75
$465.50
$463.25
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.
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
CONCEPT
Solving Problems involving Percents
8
Simplify the following expression.
‐2 + 5(a – 2b + 1) – 3(8 – 2a + b)
3a – b – 25
11a – 3b – 21
11a – 5b – 25
3a – 9b – 21
RATIONALE
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 5 into (a – 2b + 1).
5(a – 2b + 1) is equivalent to 5a – 10b + 5. Next we have to distribute -3 into (8 – 2a + b). Notice the negative sign in front of 3.
-3(8 – 2a + b) is equivalent to -24 + 6a – 3b. Now we have to combine like terms.
The terms 5a and 6a combine to 11a, the terms -10b and-3b combine to -13b, and the constant terms -2, 5, and -24 combine to
-21. The expression ‐2 + 5 (a – 2b + 1) – 3(8 – 2a + b) is equivalent to 11a – 3b – 21.
CONCEPT
Terms and Factors in Algebraic Expressions
9
Which of the following is equivalent to ?5/25/2021 Sophia :: Welcome
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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, .
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, .
can be written as because 4 times 2 is 8. We can simplify the square root of 4 further.
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 9 times 2 is 18. We can simplify the square root of 9 further.
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 .
CONCEPT
Adding and Subtracting Radical Expressions
10
Laura opened a deposit account. In the first month, she made an initial deposit of $2500, and plans to contribute an additional $225 every month. The account does not
pay any interest.
After how many months will she have a total of $6,775?
15 months
21 months
20 months
18 months
RATIONALE
The amount in Laura'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 $2500. Laura deposits $225 each month, which is the difference, d. We want to know how many months, n, it will
take for the balance to reach $6775, which is the value for . Next, substitute each value in for , d, and .
Once each known value is substituted, we can solve for n. First, distribute 225 into (n – 1).
225(n – 1) is equivalent to 225n – 225. Next, combine like terms on the right side.
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2500 minus 225 is equal to 2275. Subtract this value from both sides to isolate the n term.
6775 minus 2275 equals 4500. Finally divide both sides by 225.
4500 divided by 225 is equal to 20. It will take 20 months for the balance to reach $6,775.
CONCEPT
Introduction to Arithmetic Sequences
11
Solve the following absolute value inequality:
|x – 4| > 5
-1 < x < 9
x < -1 or x > 9
x < -9 or x > 1
-9 < x < 1
RATIONALE
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 – 4| > 5 is equivalent to x – 4 > 5 OR x – 4 < -5. Let's start by finding the solution to the first inequality x – 4 > 5.
To isolate x in the first inequality, add 4 to both sides.
The solution to the first inequality is x > 9. We can repeat this process with the second inequality x – 4 < -5.
To isolate x in the second inequality, add 4 to both sides.
The solution to the first inequality is x < -1.
The solution to the absolute value inequality |x – 4| > 5 is x < -1 OR x > 9.
CONCEPT
Absolute Value Inequalities
12
Rationalize the denominator and simplify the following expression:
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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 .
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!).
In the numerator, we multiplied the first terms to get 8; multiplied the outside terms to get ; multiplied the inside terms to get ; and
multiplied the last terms to get 2. The denominator is the difference of squares, or . Next, we can combine like terms in the
numerator.
In the numerator, combine the constant terms to get 10 and combine the radical terms to get . Next, evaluate the exponents in the
denominator.
In the denominator, is 4 and is just 2. Next, we can subtract these two values.
4 minus 2 is 2. Simplify the fraction by dividing each term in the numerator by 2.
The expression is equivalent to .
CONCEPT
Rationalizing the Denominator
13
Use the FOIL method to evaluate the expression:
RATIONALE
FOIL stands for First, Outside, Inside, Last, and helps us remember how to multiply two binomials. Begin by multiplying the first two terms,
and 4.
times 4 equals . Next, multiply the two outside terms, and .
times equals -5 because when you multiply two terms with the same number under the radical symbol, the result is just that number.
Next, multiply the two inside terms, 2 and 4.
2 times 4 equals 8. Next, multiply the last two terms, 2and .
equals . Now we can combine all four parts.
After using FOIL, is equal to . Next, identify any like terms and combine.
We can combine the two constant terms, -5 and 8, to get 3. We can also combine the two radical terms, and , to get . The
expression is equal to .
CONCEPT
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Multiplying Radical Expressions
14
If a vehicle is going 45 mph, how many feet does it travel in one second?
73 feet
66 feet
28 feet
31 feet
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 45 miles into feet, multiply by
. Notice that this will
cancel the units of miles, and the result is in feet per hour.
45 times 5280 is 237600. This means that 45 miles per hour is the same as 237600 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 237600 feet. In the denominator, we have 60 times 60, which is equal to 3600. The units in
the denominator are now in seconds. Finally divide 237600 feet by 3600 seconds.
45 miles per hour is the same as 66 feet per second.
CONCEPT
Converting Unit Rates
15
For the arithmetic sequence beginning with the terms {-2, 0, 2, 4, 6, 8...}, what is the sum of the first 18 terms?
304
238
270
340
RATIONALE
To find the sum of the first 18 terms in this sequence, we first need to find the 18th term.
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 -2 and we want to find the sum of 18 terms, so n is 18. Next, we can find the common difference,
which is the difference between each term in the sequence.
We can see that each term increases by from one to the next, so plug in for d. To find the value of the 18th term, evaluate the parentheses.
fraction numerator 5280 space f e e t over denominator 1 space m i l e end fraction
a subscript n equals a subscript 1 plus d left parenthesis n minus 1 right parenthesis
a subscript 1 a subscript n
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18 minus 1 is 17. Next, multiply 2 by 17.
2 times 17 is 34. Then, add -2 and 34.
The 18th term is equal to 32. Now that we know the value of the 18th term, we can use the formula for finding the sum.
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 = 18.
Once these values are plugged into the sum formula, we can start by evaluating the parentheses.
-2 plus 32 is 30. Next, divide 18 by 2.
18 divided by 2 is 9. Finally, multiply this value by 30.
9 times 30 is 270. The sum of the first 18 terms of this sequence is 270.
CONCEPT
Finding the Sum of an Arithmetic Sequence
16
Ashley mixes two types of soft drinks with different types of concentration: one soft drink has 20% sugar and the other drink has 45% sugar.
Each can has 250 milliliters of soda.
What is the sugar concentration of the mixed soft drink?
62.5%
32.5%
38%
25%
RATIONALE
To find the concentration of the mixed soft drink, set up an equation using weighted averages. In the numerator, we will multiply the
concentration and quantity of each soft drink and then add them together. In the denominator, we will add the quantities of each soft drink.
The quantity of the first soft drink is 250 milliliters with a concentration of 20% sugar. Substitute 0.20 for C₁ and 250 for Q₁. The quantity of
the second soft drink is 250 milliliters with a concentration of 45% sugar. Substitute 0.45 for C₂ and 250 for Q₂.
Once the values are substituted into the formula, evaluate the denominator.
In this case, the denominator simplifies to 500. Next, multiply 0.20 and 250.
When multiplying the concentration and quantities of the first soft drink, we get 0.20 times 250, which is equal to 50. Next, multiply 0.45
and 250.
When multiplying the concentration and quantities of the first soft drink, we get 0.45 times 250, which is equal to 112.5. Then, add these
two values in the numerator together.
50 plus 112.5 is equal to 162.5. Finally, divide by 500.
162.5 divided by 500 is equal to 0.325. This value needs to be written as a percent.
0.325 as a percent is 32.5%. The mixed soft drink is 32.5% sugar.
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CONCEPT
Solving Mixture Problems using Weighted Average
17
Find the solution for x if |7x + 5| = 19.
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 7x + 5 = 19.
To solve for x, start by subtracting 5 from both sides.
On the left, we have only 7x. On the right, 19 minus 5 is 14. Then divide both sides by 7 to isolate x.
The solution for the first equation is x = 2. The same process can be applied to the second equation 7x + 5 = -19.
To solve, start by subtracting 5 from both sides.
On the left, we have only 7x. On the right -19 minus 5 is -24. Then divide both sides by 7 to isolate x.
The solution for second equation is
The two solutions to |7x + 5| = 19 are x = 2 and .
CONCEPT
Absolute Value Equations
18
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.
-10 degrees Fahrenheit
-7 degrees Fahrenheit
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-2 degrees Fahrenheit
RATIONALE
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.
CONCEPT
Isolating Variables
19
Suppose 4y – 1 = 15 and y = 5x – 3.
Which of the following equations is equivalent to 4y – 1 = 15, but written only in terms of x?
20x – 12 = 16
5x – 12 = 16
5x – 12 = 14
20x – 12 = 14
RATIONALE
To find an equivalent equation written in terms of x we can use the second equation y = 5x – 3. This equation indicates that y is equivalent to 5x – 3
so we can substitute (5x – 3) in for y in the first equation 4y – 1 = 15.
Now we have an equation in terms of x. To simplify it, first apply the Distributive Property and multiply 4 by the terms in the parentheses.
4 times 5x equals 20x, and 4 times -3 equals -12. Next, eliminate -1 from the left side of the equation by adding 1 to both sides.
15 plus 1 equals 16. This equation is equivalent to the original equation, but is written in terms of x.
CONCEPT
Substitution in multi-step linear equations
20
For the equation 1.75n = 7, what is the value of n?
0.25
4 6
5.25
RATIONALE
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.5/25/2021 Sophia :: Welcome
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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.
CONCEPT
Solving single-step equations
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