College Algebra - Final
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1
Which of the following equations is correctly calculated?
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-2×-7 = -14
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81 ÷ -9 = -9
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-36 ÷ 4 = 9
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4
×12 = -48
RATIONALE
This is correct. The quotient of a positive and negative
number is always negative.
This is incorrect. The product of two negative numbers is
always positive. The correct product is 14.CONCEPT
Multiplying and Dividing Positive and Negative Numbers
2
Select the solution to the following system of equations:
•
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correct
•
•
RATIONALE
This is incorrect. The quotient of a positive and a
negative number is always negative. The correct quotient
is -9.
This is incorrect. The product of two positive numbers is
always positive. The correct product is 48.When we add the two equations, the term will
cancel, leaving . Now we can solve for by
dividing both sides by .
Dividing by gives a solution of .
Once we divide both sides by , we get a
solution of . We can solve for by plugging
in for in either equation.
Let's use the second equation . Once
is plugged in for , evaluate and solve for .
Here, by multiplying by , the result
contains a term, which will cancel when
combined with the second equation. Now, we can
add the two equations together.
Before you can add equations, sometimes you
must multiply entire equations by a scalar value for
the method to work as intended. For this system,
we can multiply the second equation by .
Subtracting a negative is the same as adding a
postive . Next, subtract from both sides.
On the right side, minus equals . Now we
can solve for by dividing both sides by .CONCEPT
Solving a System of Linear Equations using the Addition Method
3
Perform the following operations and write the result as a single
number.
[4 + 8×(5 – 3)] ÷ 5 + 6
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10 correct
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10.8
•
2
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1.8
RATIONALE
This is the solution to the system of equations.
Following the Order of Operations, we must first
evaluate everything in parentheses and grouping
symbols. When there are brackets or braces, evaluate
the innermost operations first. Here, we must evaluate
5 minus 3 first.
5 minus 3 is 2. There are still operations inside
grouping symbols to evaluate. Multiplication comes
before addition, so we must evaluate 8 times 2 next.CONCEPT
Introduction to Order of Operations
4
Consider the quadratic function .
What do we know about the graph of this quadratic equation,
based on its formula?
•
The vertex is and it opens upward.
•
The vertex is and it opens downward.
•
The vertex is and it opens upward.
8 times 2 is 16. Next, we add 4 and 16 to complete the
operations inside parentheses.
4 plus 16 is 20. Now there is just division and
subtraction. Division comes before subtraction in the
Order of Operations, so we divide 20 by 5 next.
20 divided by 5 is 4. Lastly, add 4 and 6.
4 plus 6 is 10.
•
The vertex is and it opens downward. correct
RATIONALE
Take note of the sign in the numerator. Evaluate the
division to get the x-coordinate of the vertex.
The sign of tells us if the parabola opens upward
or downward. If is positive, the parabola opens
upward. If is negative, the parabola opens
downward.
Compare the given equation to the general
. The values of and in particular
give us useful information about the graph.
In this case, since is negative, we know the
parabola opens downward. Next, we can use the
values of and to find the x-coordinate of the
vertex.
The values and can be plugged into this formula
to give us the x-coordinate of the vertex.
From the given equation, plug in for and for
. Simplify the denominator.CONCEPT
Introduction to Parabolas
5
Suppose and .
Find the value of .
•
This is the y-coordinate to the parabola's vertex.
Return to the original equation, but write in the
calculated x-coordinate, , for every instance of .
Then, evaluate the equation.
squared is . Next multiply this by the
coefficient, .
times is . Finally, evaluate the addition and
subtraction.
From the equation, we know that the parabola's
vertex is at and opens downward.
The x-coordinate of the vertex is . To determine
the y-coordinate of the vertex, plug this x-value into
the original equation and solve for .
times is . Next, evaluate times .•
•
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correct
RATIONALE
CONCEPT
Function of a Function
6
To evaluate this composite function, focus on the
innermost function first. Evaluate first by plugging
in for the variable in the function .
Once has been replaced with , evaluate the
expression.
The function evaluates to . To evaluate ,
use the value of , which is , as the input for the
function .
Once has been replaced with , evaluate the
expression.
This tells us that is equal to .Consider the following system of two linear equations:
Select the graph that correctly displays this system of equations
and point of intersection.
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•
•
• correct
RATIONALE
Start by graphing each equation in the
system. Then we will identify the
intersection point.On the left, is now isolated. Simplify the
right side by dividing each term by .
Re-writing each equation in the form
will help us easily identify the yintercept and the slope to graph each line.
Starting with the first equation, we want to
isolate the variable y onto one side of the
equals sign. First, subtract from both
sides.
Subtracting from both sides leaves
only on the left. Then divide both sides
by to isolate .
This graph shows a y-intercept at and
a slope , meaning we need to go down
one and to the right four to plot a second
point. Connecting these two points makes
this line. Repeat these steps with the
second equation.
The equation simplifies to .
This tells us that the y-intercept is at
and the slope is .CONCEPT
Solving a System of Linear Equations by Graphing
7
The equation simplifies to . This
tells us that the y-intercept is at and
the slope is .
Subtracting from both sides leaves only
on the left. Then divide both sides by
to isolate .
This graph shows a y-intercept at and
a slope , meaning we need to go up
three and to the right four to plot a second
point. Connecting these two points makes
this line.
Once both lines are plotted, you can see
the point of intersection at .
The goal is to isolate the variable onto
one side of the equals sign. First, subtract
from both sides.
On the left, is now isolated. Simplify the
right side by dividing by .A football field is a rectangle 80 meters wide and 110 meters
long. Coach Trevor asks his players to run from one corner to the
other corner by running diagonally across the field.
What is the distance from one corner of the field to the other
corner? Answer choices are rounded to the nearest meter.
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115 meters
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190 meters
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136 meters correct
•
185 meters
RATIONALE
We can use the Pythagorean Theorem to calculate the
length of a diagonal. The variables a and b represent the
sides of the field, and c represents the diagonal. First,
substitute 80 for a and 10 for b. (Note that we could also
substitute 110 for a and 80 for b).
Once we have the given values plugged into the
Pythagorean Theorem, we can evaluate the exponents.
80 squared is 6400, and 110 squared is 12100. Now we
can add these values together.
6400 plus 12100 is equal to 18500. Finally, we can take
the square root of both sides to find the value of c.CONCEPT
Calculating Diagonals
8
Briana monitors the number of E. coli infections reported in a
certain neighborhood in a given week. The recent numbers are
shown in this table:
According to her reports, the reported infections are growing at a
rate of 50%.
If the number of infections continues to grow exponentially, what
will the number of infections be in week 6?
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456 people correct
•
203 people
The square root of 18500 is approximately 136. The
distance from one corner of the field to the other,
rounded to the nearest meter, is 136 meters.
When we have a squared term, such as , taking the
square root of both sides will cancel this operation.
Week Number of People
0 40
1 60
2 90
3 135•
304 people
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297 people
RATIONALE
CONCEPT
Exponential Growth
9
Suppose , , and .
Find the value of the following expression.
In general, exponential growth is modeled using this
equation. We will use information from the problem to find
values to plug into this equation.
There will be 456 people infected in week 6.
to the power of is . Finally, multiply this by
.
plus is . Next, take this value to the power of .
The initial number of infections is , so this is the value
for . The infection rate is , so this is our value for
(remember to write it as a decimal). We want to know
how many infections there will be in week 6, so we will use
for the value for . We will need to solve for . Start by
simplifying what's inside the parentheses.•
•
correct
•
•
RATIONALE
This question involves several properties of
logarithms. The Quotient Property of Logs states
that division inside a logarithm can be expressed as
subtraction of individual logarithms.
This means we can express as
. Next, the Product Property of
Logs states that multiplication inside a logarithm
can be expressed as addition of individual
logarithms.CONCEPT
Applying Properties of Logarithms
10
The spread of a disease can be modeled as , where
N is the number of infected people, and t is time (in days).
This means we can express as
. Then, the Power Property of
Logs states that exponents inside a logarithm can
be expressed as outside scalar multiples of the
logarithm.
This means we can express as
and as . Finally, we can substitute
the values we were previously given for ,
, and .
Recall that , , and .
Once these given values are substituted into the
expression, simplify each term and then perform
the addition and subtraction.
times is , and times is . Finally, add
these values together.
The logarithmic expression evaluates to .How long will it take until the number of infected people reaches
1,400?
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28 days
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98 days
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49 days correct
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14 days
RATIONALE
CONCEPT
Solving Multi-step Equations
11
The spread of the disease is modeled by this equation. To
find how long it will take until 1400 people are infected, 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 200 to undo 200 multiplied by the square root of t.
On the left, 1400 divided by 200 is 7. To undo the square
root of t on the right, square both sides.
On the left, 7 squared is 49. It will take 49 days for 1400
people to become infected by the disease.The diameter of a hydrogen atom is about meters. A
protein molecule has an overall length of 3000 times (or
times) the diameter of a hydrogen atom.
What is the length of the protein molecule, in meters, if it were
written in scientific notation?
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meters
•
meters
•
meters correct
•
meters
RATIONALE
When multiplying numbers in scientific
notation, you must deal with the numbers
and 10s separately. First, multiply 1.05 and
3.
To find the length of the protein molecule,
multiply by 3000, which can
be expressed as .CONCEPT
Multiplication and Division in Scientific Notation
12
Write the following expression as a single complex number.
•
•
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correct
•
RATIONALE
1.05 times 3 equals 3.15. Now you can use
the Product Property of Exponents on the
10s and add the exponents.
-10 plus 3 equals -7, which is the exponent
for the base 10. Finally, combine the
number part and the power of 10 together.
The length of the protein molecule (in
meters), when written in scientific notation,
is .CONCEPT
Multiply Complex Numbers
13
When multiplying two complex numbers, we
can use FOIL.
The expression can be written
as a single complex number, .
and can combine to . Now we can
simplify the last term, , which contains the
imaginary unit squared. Recall that the is
equivalent to , which can be substituted in
our expression.
times is equal to . Finally, combine like
terms and .
is replaced by . Next, evaluate .
Multiply the first terms , the outside terms
, the inside terms , and the last
terms . Next, evaluate each
multiplication.
The expression multiplies to
. Next, combine like terms.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?
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20x – 12 = 14
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5x – 12 = 16
•
20x – 12 = 16 correct
•
5x – 12 = 14
RATIONALE
CONCEPT
Substitution in multi-step linear equations
14
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.The graph of a linear function passes through the points
and .
Find the slope of this function.
•
•
•
•
corerct
RATIONALE
Since we have two points from a linear function, use the
slope formula to find the slope of the line.
The slope is the difference in coordinates from the two
points divided by the difference in coordinates from the
same two points. When plugging in the values it is
important to be consistent with which coordinates are
subtracted in the calculations.CONCEPT
Determining Slope
15
The population of a small town is decreasing at a rate of 4% each
year. The following table shows a projection of the population, ,
after years.
If the population of the small town is currently 13,000 people, how
many years will it take for the population to reach 8,500 people?
Now that the numbers are plugged in, evaluate the
subtraction in both the numerator and the denominator.
In the numerator, the difference in coordinates is
minus , which is the same as plus , or . In the
denominator, the difference in coordinates is minus
, which is is the same as plus , or . The slope of
the line is .
Years Population
0 13,000
1 12,480
2 11,981
3 11,502
4 11,042•
9.4 years
•
9.8 years
•
10.8 years correct
•
10.4 years
RATIONALE
in general, exponential decay is modeled using this
equation. We will use information from the problem
to find values to plug into this equation.
We are now left with an exponential expression on
the right side. To undo the variable exponent, take
the log of both sides.
minus is . Next, divide both sides by
.
The initial population is , so this is the value
for . The rate of change is so this is our value
for (remember to write this as a decimal). We
want to know how many years until the population
is , which will be our value for . We need to
solve for the time, . Start by simplifying what's
inside the parentheses.CONCEPT
Exponential Decay
16
Geri runs a total of 3.1 miles in a cross-country race.
How many feet is this equivalent to?
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4,989 feet
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5,280 feet
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15,840 feet
Once we have taken the log of both sides, apply
the Power Property of Logs, which allows
exponents inside logs to be written as outside
factors.
It will take 10.4 years for the population to reach
8500.
is now isolated. To evaluate, use a calculator to
find the value of and .
Since is outside the log expression, divide both
sides by .
is equal to and
is equal to . Finally, divide.•
16,368 feet correct
RATIONALE
CONCEPT
Converting Units
17
Find the solution for x if |7x + 5| = 19 .
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. We want
to convert miles to feet. We know how many feet are
in 1 mile. We will use this fact to set up a conversion
factor.
Notice how the fractions are set up. The units of
miles will cancel, leaving only feet. Finally we can
evaluate the multiplication by multiplying across the
numerator and denominator.
In the numerator, 3.1 times 5280 equals 16368. 3.1
miles is equivalent to 16,368 feet.
There are 5280 feet in 1 mile. To convert 3.1 miles
into feet, multiply by the fraction
.•
•
•
correct
•
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.CONCEPT
Absolute Value Equations
18
Consider the following table.
Which graph corresponds to this table?
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
.• correct
•
•
•
RATIONALE
When given a table of points, we can match
the and values with the coordinate
points on the graph. Remember, the
value indicates the distance the point
moves left (negative) or right (positive) from
the origin, . The value indicates the
distance the point moves up (positive) or
down (negative) from the origin.CONCEPT
Graph of a Line
19
The vertical position (in feet) of a rock seconds after it was
dropped from a cliff is given by the formula .
The base of the cliff corresponds to .
After how many seconds will the rock hit the ground at the base of
the cliff?
•
5
seconds correct
For , only move five points to the right
from .
For , move one point to the left and
six points up from .
For , only move five points up from
.
For , move three points to the left
and eight points up from .
For , move four points to the right and
one point up from .•
8
seconds
•
6
seconds
•
7
seconds
RATIONALE
To determine how long it will take for the rock to
hit the ground, set the equation equal to zero.
When a quadratic equation is set equal to zero,
you can use the quadratic formula to solve for .
squared equals , and times
times equals . Then, find the difference
between and .
In this case, , , and . Once
you have substituted values in for , begin
to solve for t.
Use the coefficients in the general expression
to substitute for in the
quadratic formula.
We can start with the denominator, since it is
easily simplified by multiplying and . Next,
simplify the expression underneath the radical in
the numerator.CONCEPT
Using Quadratic Equations to Represent Motion
20
Simplify the rational expression by canceling common factors:
One equation involves subtraction, while the
other equation involves addition. Next, evaluate
the numerator in each equation.
It will take seconds for the rock to hit the
ground at the base of the cliff.
minus is equivalent to . Now
that the expression underneath the radical is
simplified, take the square root of .
In the first fraction, minus equals .
In the second fraction, plus equals .
Finally, divide each numerator by .
The square root of is . To find the
solutions, create two equations (due to the ).
The two solutions to the quadratic equation are
seconds and seconds. However, in this
context, the variable represents time, and time
cannot be negative.•
•
•
•
correct
RATIONALE
In order to cancel like terms, we have to first factor
both the numerator and denominator, and then look
for common factors in the top and bottom of the
fraction. Let's take a look at the numerator first.
Two integers that multiply to and add to are
and . This means that and make up the
integer pair for this quadratic.
To factor this quadratic, we need to identify a pair
of integers whose product is the constant term ( )
and whose sum is the coefficient of the x-term ( ).CONCEPT
Simplifying Rational Expressions
21
For the arithmetic sequence beginning with the terms {-2, 0, 2, 4,
6, 8...}, what is the sum of the first 18 terms?
The original fraction is now rewritten with both
numerator and denominator written in factored
form. To simplify, we cancel factors that appear in
both the top and the bottom.
The factor can be canceled. The rational
expression can be simplified to .
The original denominator can be written
as . Now we can rewrite the original
fraction with the new factored numerator and
denominator.
The original numerator can be written as
. Let's take a look at the denominator.
Two integers that multiply to and add to are
and . This means that and make up the
integer pair for this quadratic.
Again, we need to identify a pair of integers whose
product is the constant term ( ) and whose sum
is the coefficient of the x-term ( ).•
340
•
304
•
238
•
270 correct
RATIONALE
To find the sum of the first 18 terms in this sequence,
we first need to find the 18th term.
18 minus 1 is 17. Next, multiply 2 by 17.
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.
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.
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.CONCEPT
Finding the Sum of an Arithmetic Sequence
22
It took Todd 11 hours to travel over pack ice from one town in the
Arctic to another town 330 miles away. During the return journey,
it took him 15 hours. Assume the pack ice was drifting at a
constant rate, and that Todd’s snowmobile was traveling at a
constant speed relative to the pack ice.
What was the speed of Todd's snowmobile?
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.
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.
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.•
10 miles per hour
•
4 miles per hour
•
26 miles per hour correct
•
20 miles per hour
RATIONALE
When two rates work together, add the two
rates. When two rates work against each other,
the opposing rate is negative, which produces an
expression involving subtraction. To solve this
problem, set up a system of two equations.
It took Todd hours to travel over pack ice from
one town to another town miles away.
During this trip, he traveled in the same direction
as the drifting ice. During his return journey, it
took him hours to go the same distance. The
return trip back was made in the opposite
direction of the drifting ice. Two rates are
involved in this problem: the rate of Todd’s
snowmobile and the rate of the drift of the pack
ice.
We can identify Todd’s rate as and the rate of
the drifting ice as . Next, consider the
combined speed when the rates work together
and when they work against each other.CONCEPT
Dividing by equals . The rate of Todd’s
snowmobile, , is 26 miles per hour.
For the first trip, the distance, , is miles,
the combined rate is , and the time, , is
hours. On the trip back, the distance, , is
miles, the combined rate is , and the
time, , is hours. Next, simplify the system of
equations by dividing each equation by its
respective time.
Both equations must be written in the form of
, which states that distance is equal to
rate times time. Use the combined rates for the r
variable.
Adding the two equations together, the constant
terms on the left side, and , sum to .
The terms in each equation add to and
is eliminated. To solve for , divide both sides by
.
In the first equation, divide both sides by ,
which equals on the left side and leaves
on the right side. In the second equation,
divide both sides by , which equals on the
left side and leaves on the right side. To
solve for the rates, add the two equations.Distance, Rate, and Time in a System of Equations
23
Consider the equation .
Find the solutions to the equation by using the quadratic formula.
•
and correct
•
and
•
and
•
and
RATIONALE
Now that the equation is set equal to
zero, identify the coefficients , , and
that are found in the standard equation,
.
When we have a quadratic equation, we
can find the solutions by using the quadratic
formula. However, be sure that the equation
is set equal to zero. First, subtract from
both sides.One equation involves subtraction, while
the other equation involves addition. Next,
evaluate the numerator in each equation.
squared equals , and times times
equals . Next, find the difference
between and .
The square root of is . Next, evaluate
the denominator by multiplying and .
In the equation , , ,
and . Substitute these values for ,
, and , into the quadratic formula.
Plugging the values of , , and into the
quadratic formula will produce the solutions
to the quadratic equation. Be sure to
perform the operations in the correct order.
First, simplify what is underneath the
radical, or .
times equals To find the solutions,
create two equations (due to the ).
minus is equal to . Next, take
the square root of .CONCEPT
The Quadratic Formula
24
What is the solution set for the following inequality?
x + 4 < 3(x – 2)
•
x > 5 correct
•
x > 1
•
x < 1
•
x < 5
RATIONALE
minus is , and plus is . To
find the solutions, divide the numerators by
the denominator, .
The two solutions to the quadratic equation
are and .
Before solving for x, make sure that each side of the
inequality is fully simplified. On the right side, we will
distribute 3 into (x – 2).CONCEPT
Solve Linear Inequalities
25
Perform the multiplication and combine like terms.
•
•
correct
•
On the right side, 3(x – 2) is equivalent to 3x – 6. Now we
can begin to solve for x by applying inverse operations to
both sides of the inequality. First, subtract 3x from both
sides.
Subtracting 3x from both sides cancels the 3x term on the
right side of the inequality, leaving x terms on only the left
side. Next, subtract 4 from both sides of the inequality.
Subtracting 4 from both sides leaves only the x term on the
left side. Finally, divide both sides of the inequality by -2 to
isolate x. Remember that the sign of the inequality sign
changes whenever you multiply or divide by a negative
number!
The solution to this inequality is x > 5.•
RATIONALE
When FOILing, multiply the first terms, outside
terms, inside terms, and last terms together.
Once the binomials have been multiplied together,
evaluate the multiplication.
The will need to be multiplied by everything
inside the parentheses.
When multiplying these two terms, we will start by
distributing into .
To multiply a set of three binomials, we can choose
any two binomials to multiply using FOIL, and then
distribute the remaining binomial to get a final
product. Here, we will use FOIL to multiply
, but you can choose any two
binomials to start.
and combined is . can be
expressed as . We still need to
distribute the third binomial, .
is equivalent to . The
next step is to combine like terms, and .times equals .
This is another part of the final product. The final
step is to add these two parts together.
times equals . This is
one part of the final product. We will distribute
into as well.
This is the final product of the three binomials,
found by adding the two parts:
and .
When multiplying these two terms, we will start by
distributing into . [Show Less]