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Substitute y for f(x) to get:
.
y = -3x - 2
.
This equation is in the slope-intercept form:
.
y = mx + b
.
in which m (the multiplier of x) is the slope of the graph and b is the value on the y-axis
where the graph crosses.
.
So for a start, you can tell by comparing the given equation to the slope-intercept form
that b (the crossing point on the y-axis) is -2. On your coordinate system put a dot at
-2 on the y-axis.
.
Next, again by comparing the given equation with the slope-intercept form, you can tell
that the slope is -3. The minus sign tells you that the graph slants downward as you move
to the right along the x-axis. For every 1 unit you move horizontally to the right, the
graph goes down 3 units. So put your pencil tip on the one point you know, that point being
-2 on the y-axis. Then move your pencil tip 1 unit horizontally to the right and stop. From
that location move your pencil point vertically down 3 units in the y-direction. Stop there
and mark that point. It is on the graph. (As a check, the point should be (1, -5).
.
As another check, you can go back to the given equation and set y equal to zero. The corresponding
value of x will be on the x-axis because points along the x-axis all have zero as their
y-value. When you set y = 0 in the given equation, it becomes:
.
0 = -3x - 2
.
Add 2 to both sides to get rid of the -2 on the right side. When you do, the equation
becomes:
.
2 = -3x
.
Finally divide both sides by -3 to solve for x. When you do that division you get:
.
2/(-3) = x
.
which is the same as:
.
x = -2/3
.
So when x is -2/3 then y is zero. So another point on the graph is (-2/3, 0). You can
plot it and see if it lies on the graph. You should be able to take a straight edge and
lay it along all three dots you have found and be able to extend a line through all three
of the dots. That line is the graph of y = -3x - 2.
.
Note that you can assign other values to x, and use the given equation to find corresponding
values of y if you want to get more points for your graph. For example, set x equal to
-2. The equation becomes:
.
y = -3(-2) - 2 = 6 - 2 = 4
.
So another point on your graph is (-2, 4)
.
When you get done your graph should look like:
.

.
Hope this helps you understand graphs for linear equations a little more.

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To solve these two problems you need only the voltage amplification equation:
.

.
and a knowledge of a few rules of Logarithms.
.
The dB equation defines decibels. It uses the base 10 Logarithms. In this equation,
is the output voltage, and is the input voltage.
.
Here are a couple of rules of Logarithms that you should be familiar with. Assume that
the base of the Logarithm is b:
.
Rule I. ….
.
Rule II. …. –
.
Rule III. …. +
.
Rule IV. …. Log base b of x = y is equivalent to saying
.
Note that in decibel calculations b (the base) is 10. This translates Rule IV to:
.
Log (x) = y is equivalent to saying
.
If you need a further explanation of these rules, you can get it from an Algebra II text,
from elsewhere on this site, or by doing a Google on Logarithms.
.
Time to work your first problem by using the voltage form of the decibel equation.
Given the amplifier has a power gain of 26 dB, and an input voltage of 1 volt. Substitute
those two values into the dB equation (26 for dB and 1 for the input voltage) and you get:
.

.
The plan is to work down to , so start by dividing both sides of this equation by
20 to get rid of the 20 on the right side. At the same time note that
is just . These simplifications lead to the equation becoming:
.

.
And after dividing 26 by 20 on the left side the equation reduces to:
.

.
Time to apply Rule IV. When you do you get:
.

.
Calculator time. Your calculator probably works this way: enter 1.3 then press the
key and you find the result to be 19.95262315 volts. Close enough for electrical work.
Call it 20 volts.
.
Now to your second problem. If the answer is to be 10, then you probably meant that the
input voltage was 10 millivolts instead of 10 microvolts .
So let’s use 10 millivolts.
.
For the given gain of 60 dB and an input of volts the equation becomes:
.

.
Just as was done in the last problem, divide both sides by 20 to reduce the problem to:
.

.
The left side divides out to a quotient of 3 and the equation becomes:
.

.
On the right side apply Rule II (division) to split the term into two separate logarithms:
.
–
.
Notice that 10*10^-3 = 10^(-2). Substitute this into the right hand logarithm and get:
.
–
.
Apply Rule I (exponent rule) to and get - . But by Rule IV
you can see that Log base 10 of 10 = 1. (Think which means y must be 1.)
So - equals -2*1 equals -2. Substitute this result for –
and the equation becomes:
.
–
.
And this simplifies to
.
Subtract 2 from both sides and the equation becomes:
.

.
Apply Rule IV to this equation and you get:
.

.
This agrees with the answer you had for this problem, but recall that 10 millivolts
was used in place of 10 microvolts to get the answer.
.
Hope this helps you get a handle on decibel equations.

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(3square root of 3)(square root of 7)
.
Am I reading your problem correctly? I interpret as:
.

.
If this is, in fact the problem, the answer can be obtained under one radical sign as:
.

.
and this simplifies to:
.

.
But this is not even close to any of the four answers you listed. Are you sure you
stated the problem correctly? Or have you listed the four answers for some other
problem?
.
If you use a calculator and compute the value of the four given answers you get:
.
A.
.
B
.
C.
.
D.
.
The way I interpreted what you said you can work your problem as:
.

.
Using a calculator, you can determine that and
.
Substituting these values into your problem for the appropriate square roots results in:
.

.
And multiplying this out results in the value of your problem being:
.

.
Note that if you use your calculator to find you get the same answer.
.
Notice also that this is not very close to the values we got above for each of the four
answers that you listed..
.
Sorry that I can't help you much more than that. If you do find your discrepancy,
please re-post the problem with the correction ... either in the problem itself or in
the answer list.
.
Hope that in this exercise there is something that helps you to learn a little more about
working with square roots.

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Given:
.

.
I used LCD and got:
.
<===
.
Then multiplied both sides by <=== should be + between terms, not -
.
You should now have:
.

.
Multiply out the right side and you get:
.

.
Combine the two terms on the left side that both contain x to get:
.

.
Add x + 4 to both sides to eliminate the terms on the left side. The result is:
.

.
Combine the two terms on the right side that contain x:
.

.
Transpose (switch sides):
.

.
This equation is now in standard quadratic form. It can be solved by graphing, by factoring,
or by using the quadratic formula. The more general approach is to use the quadratic
formula, but in this case factoring works. The left side of this equation factors to:
.

.
Notice that this equation will be true if either of the factors on the left side equals zero
because zero times anything is zero. So, one at a time, set the two factors equal to
zero and solve for x.
.

.
Subtract 4 from both sides:
.

.
Divide by 3:
.

.
That's one solution for x. Now set the other factor equal to zero:
.

.
Subtract 1 from both sides:
.

.
That's the second solution for x.
.
In summary, the two solutions for x are and
.
You can check both of these by substituting them (one at a time) for x in the original
given equation and making sure that the equation still balances on both sides.
.
Hope this helps you to understand the problem and also to correct your minor mistake.
.
Cheers ...

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Given:
.

.
Square the left side and the equation becomes:
.

.
Then get this equation into standard quadratic form by eliminating the 5 on the right side
so that the right side becomes 0. Do this by subtracting 5 from both sides. When you do
that subtraction the equation becomes:
.

.
The left side of this equation does not factor nicely. So use the quadratic formula. This
formula says that for a quadratic equation of the standard form:
.

.
The values of x that satisfy this equation are given by the equation:
.

.
By comparing our equation with the standard form you can see that a = 1, b = -2, and c = -4.
.
By substituting these values into the equation that defines the values of x you get:
.

.
This simplifies to:
.

.
Note that
.
Substituting for reduces the equation for x to:
.

.
So this problem has two solutions for x ... and
.
Hope this helps you to understand the problem a little better.
.

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For this problem you can use the slope-intercept equation. This equation has the form:
.

.
In this equation m represents the slope and b is the value on the y-axis where the graph
intersects the y-axis.
.
In this problem you are told that the slope is -2. And you are told that the graph goes through
the origin. This means that the graph intersects the y-axis at a y value of 0. This translates
to m = -2 and b = 0. Substituting these values into the slope-intercept equation results in
the equation becoming"
.

.
You can then simplify it by dropping the 0 to get the equation:
.

.
The fact that the slope is negative tells you that the graph goes down as you move to
the right.
.
You can now get some points on the graph by assigning values to x and computing the corresponding
values of y. For example, let x = -5. Plug that value in for x and the equation becomes:
.

.
So when x is -5, then y = +10. This means the point (-5, +10) is on the graph.
.
The problem tells you that (0, 0) is on the graph.
.
Then suppose that we let x = +5. Substituting this value into the equation results in:
.

.
This means that when x equals +5, the corresponding value of y is -10. So the point
(+5, -10) is on the graph. If you plot these three points you should see that they
lie on a straight line. Take a straight edge and line it on these points. Then extend a
line through all three points. When you do you should have a graph that looks like:
.

.
Hope this helps you to understand the problem and see how to get an answer.
.
Cheers
.

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It helps to visualize this one if you replace n(x) with y to get:
.
y = 0
.
Now think of it this way: since there is no x in the function, y is always zero regardless
of what value you assign to x. Is (-10, 0) a solution to this equation? Yes it is because
y equals zero. So is (0,0) a solution set because although x is zero, the critical part is
that y = 0 and that satisfies the equation. How about the point (50, 0)? Same thing.
.
If you plot all of these points, you begin to see that the graph of y = 0 (or its equivalent
n(x) = 0) is the x-axis, because the x-axis is the line of all points having a y value of zero.
.
Similarly, y = -3 is a horizontal line intersecting the y-axis at -3. No matter what value
you assign to x, the corresponding value of y must always be -3.
.
Hope this is one way of looking at the problem that will help you to make some sense out
of it.

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You can work problems of this type just as you would solve an equation with the exception that
if you divide or multiply by a negative quantity, then you must reverse the direction
of the inequality sign.
.
So let's start with the given expression:
.

.
Notice that so we can replace it by 4 to get:
.

.
Get rid of the 4 on the left side by subtracting 4 from both sides of the inequality
to get:
.

.
Now reduce the left side to just "+a" by dividing both sides of the inequality by +3
to get:
.

.
which simplifies to
.
The original inequality should be satisfied as long as the value of "a" is less than 2.
.
Let's build our self confidence by trying some values for "a". Suppose we let "a" equal
zero. That value is obviously less than +2. If we substitute zero for a in the original
inequality we get:
.
which becomes . That works. Similarly, if we let a = +1
the original inequality becomes:
.
and this further simplifies to . That works also.
.
Now let's set a = +3. That is outside the limit we found since +3 is not less than +2.
.
With a = +3 the original inequality becomes:
.
. This simplifies to , and this obviously is not true.
.
From these spot checks, it seems as though our answer is correct.
.
Hope this helps you to understand the problem.
.
Cheers.

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Given:
.
The sum of a number and four times its reciprocal is -5
.
The "number" is the unknown, so let's represent it by x.
.
The reciprocal of the number by definition is 1 divided by the number, and 4 times the reciprocal
is, therefore:
.

.
So the sum of the number and 4 times its reciprocal is:
.

.
and the problem tells you that this is -5. So set it equal to -5:
.

.
Now let's solve for x. We can do so by multiplying every term (both sides) of this
equation by x to eliminate the denominator. Do that and the equation becomes:
.

.
Add 5x to both sides to eliminate the -5x on the right side and get the equation into the
standard quadratic form of:
.

.
There are several ways that this can be solved (graphing; completing the square or its
equivalent, using the quadratic formula; but in this case factoring is probably the
easiest.) The equation factors to:
.

.
This equation will be true if either of the two factors is equal to zero. So set each factor
equal to zero and solve for the value of x that will make that happen:
.

.
Subtract 1 from both sides of the equation and the result is:
.

.
Then set the second factor equal to zero:
.

.
and subtract 4 from both sides to get:
.

.
So there are two possible values for x that will work ... -1 and -4
.
Check them out by evaluating each in the original problem.
.
If x = -1, will ? Substitute -1 for x and you get:
.

.
This value of -1 works. Now let's try the second value, x = -4. Substitute for x and you
get:
.

.
This also works. Therefore, your problem has two solutions ... x = -1 and x = -4.
.
Hope this helps you to understand the problem and how you can work it to a solution.
.
Cheers

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Given the equation:
.
(3w+4)(2w-7)=0
.
Solve for w.
.
Notice that if at least one of the two factors on the left side of the equation is zero, then
the equation will be true because zero times anything is zero.
.
Therefore, the equation will be true if either 3w + 4 = 0 or if 2w - 7 = 0 because that will
make the left side equal the right side.
.
Let's first say that (3w + 4) equals zero.
.
3w + 4 = 0
.
Solve for w by first subtracting 4 from both sides to get:
.
3w = -4
.
Then divide both sides by 3 (which is the multiplier of w) to get:
.
w = -4/3
.
So that's one value for w that will make the equation true.
.
Next let's look at the second factor ... (2w - 7) and set it equal to zero:
.
2w - 7 = 0
.
Get rid of the -7 on the left side by adding 7 to both sides to get:
.
2w = 7
.
Solve for w by dividing both sides by 2 and the result is:
.
w = 7/2
.
That's the second value for w that will make the equation true.
.
In summary, the values of w that will make the equation true are:
.
w = -4/3 and
w = 7/2
.
Hope this helps you to understand why setting each of the factors equal to zero will give
you two values for w that make the equation work.

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Given:
.

.
The problem is to factor this expression.
.
The given expression contains the difference of two perfect squares. As such it falls under
the factoring rule:
.

.
By comparing this rule with the given expression you can see that:
.

.
and this leads to
.
Similarly:
.

.
and taking the square root of both sides of this results in:
.

.
Now return to the factoring rule and substitute for a and b using the above expressions.
When you do you get:
.

.
and the right side of this equation is the answer you are looking for.
.
Hope this helps you become familiar with the factoring rule for the difference between
squares. It comes up reasonably often in book problems that you need to be aware of it, and
the more you see and use it, the more you will tend to remember it.
.
Cheers ...

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Let's check your answer. One vagabond is going at a rate of 50 miles per hour. The time
at this rate is 4 hours (2 pm to 6 pm). In that 4 hour period at 50 miles per hour, that
vagabond travels the full 200 miles by itself. If that doesn't seem right, it means that
your answer is incorrect because the second vagabond wasn't even taken into consideration.
.
Here's the way to do the problem.
.
Use the distance formula for the vagabond that travels at the unknown rate R. When you do you
get that the distance (D1) for the first vagabond is:
.
D1 = R*T
.
At this point we know that T is 4 hours. So we can substitute 4 for T and the equation
then becomes:
.
D1 = R*4 = 4R
.
Now let's do the same thing for the second vagabond. It has a rate equal to the rate of
the first vagabond plus 10 miles per hour, so its rate is R+10. It also travels for 4 hours.
If the distance traveled by this second vagabond is D2 then its distance equation is:
.
D2 = (R+10)*4
.
Multiply out the right side and you get:
.
D2 = 4R + 40
.
Now recognize that when the two vagabonds meet, the total distance (D1 + D2) that they
travel is 200 miles. So lets add our two distance equations:
.
D1 = 4R
D2 = 4R + 40
.
When you add these vertically you get:
.
D1 + D2 = 8R + 40
.
But we already noted that D1 + D2 is 200 miles. So we can substitute 200 for D1 + D2 and
we get:
.
200 = 8R + 40
.
Get rid of the 40 on the right side by subtracting 40 from both sides. When you do that the
equation becomes:
.
160 = 8R
.
Finally, solve for R by dividing both sides by 8 and you get:
.
R = 160/8 = 20 miles per hour
.
So the rate of the first vagabond is 20 miles per hour and the rate of the second vagabond
is 10 miles per hour faster or 30 miles per hour.
.
Let's check these answers.
.
At 20 miles per hour, in 4 hours the first vagabond goes 80 miles toward the second vagabond.
.
Meanwhile at 30 miles per hour, in 4 hours the second vagabond goes 120 miles toward the
first vagabond.
.
This combination means that in the 4 hours the two together cover 80 + 120 miles and when
that is done they meet having covered the 200 miles between them when they started out.
.
Hope this helps you to understand the problem and how your teacher said you could use
the distance formula to solve it.

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Given:
.

.
Begin by treating this as just an ordinary equation:
.

.
You can easily find a couple of points on this line. Do this by first setting x equal to
zero. If you do that, all that is left is . Now you know that the point (0,4)
is on the graph. (This point is on the y-axis at +4). Next set y equal to zero and you get:
.
. Solve for x by dividing both sides by 4 and the result is:
.

.
You now know that the point (1,0) is also on the line. (This point is on the x-axis
at +1.
.
Since you have two points plotted on the coordinate system, you can use a straight
edge to extend a line through these two points.
.
This is the graph of the line . It should look like this:
.

.
Now let's return to the original inequality:
.

.
Let's get this into the slope intercept form. Recall that the slope intercept form is
where b is the value of y at the point where the graph crosses the y-axis.
And m is the slope of the line. We can get the inequality into this form by subtracting
4x from both sides. When you do that the equation becomes:
.

.
but recall from above that we found +4 was the value on the y-axis where the graph
intersected. Substitute this value for b to get:
.

.
This form tells you that the line that is the graph has a slope of -4 and crosses the
y-axis at +4. Note that this slope intercept equation tells you that for each point on
the line, y must have a value equal to or greater than the value of y associated with that
point on the line.
.
Therefore, you can shade in the area of the graph above that line and also on that line.
Y can have values anywhere in that shaded area. And all the (x, y) points in that shaded
area (including the line) will satisfy the original inequality.
.
Let's try a point in the shaded area, and see if it works. Pick a point on the y-axis
above +4 ... say the point (0, 6). That point is also on the y-axis and has the values
x = 0, y = 6. It is in the shaded area above the line. Return to the original inequality
and substitute 0 for x and 6 for y.
.

.
Substituting 0 for x and 6 for y results in:
.

.
This simplifies to:
.

.
This is obviously true ... 6 is greater than 4. And since the point (0, 6) is in the shaded
area, it helps to convince us that we have identified the shaded region correctly.
.
You can try points above the line, on the line, and below the line to check out the solution
we got. You will see that points that are on the line and above the line make the original
inequality true. But points below the line will not make the inequality correct.
.
Hope this helps you to understand the solution to inequalities of this type, how to find
them and how to represent them by shading on the entire graph.
.

You can put this solution on YOUR website!
Given:
.

.
find (a) f (0), (b) f (5), and (c) f (-2)
.
Problem (a) tells you to go to the given equation, replace every x with a zero, and then
simplify the result.
.
Replace every x with zero to get:
.

.
So for (a) you have
.
Problem (b) ... same thing only replace every x with 5 to get:
.

.
So for (b) you have .
.
For problem (c) ... same thing again only this time replace every x with -2 to get:
.

.
So for (c) you have
.
Hope this helps you to understand what the problem was asking you to do and how you should
go about doing problems like these.
.

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What is the decibel level of a normal conversation,(square root of 10)- 3.2 x watt per meter^2?
.
I'm not sure what your problem is supposed to be. The opening statement asks what is the
decibel level of a normal conversation. After that you have written
.
(square root of 10)- 3.2 watts per meter^2?
.
That is not a decibel level because the decibel level has units of decibels, not watts/m^2.
.
The units of watts/m^2 sort of implies that you are to take the decibel level of normal
conversation and find the sound energy in that level of normal conversation.
.
Normal conversation is about 65 decibels.
.
The equation that defines the decibel level of sound is:
.

.
in which D is the number of decibels; Log is Log operator to the base 10; is
the sound energy, in this problem it is the energy of the sound in normal conversation;
and is the energy of sound that is the lowest sound energy that can be heard.
The internationally accepted standard for is watts/m^2 ... the
least amount of sound energy that is audible.
.
If you are to find , the sound energy of normal conversation, you can do so if
you say that the decibel level of normal conversation is 65 decibels. All you have to
do is use the equation that defines decibels and substitute the values for D and .
As was noted, D = 65 and . Substituting those two values into the
equation results in:
.

.
Recall that by the rules of Logarithms, if you take the Logarithm of a fractional
term, an equivalent is the Logarithm of the numerator minus the Logarithm of the denominator.
Applying this rule to the equation above results in:
.

.
Also recall that by the rules of Logarithms, if you have the Logarithm of a quantity that
has an exponent, an equivalent is the exponent times the Log of just the quantity.
Apply this rule to the last term on the right side of the equation and you get:
.

.
But, the Log to the base 10 of 10 is just 1. So, substitute 1 for Log(10) to get:
.

.
Note that -(-12)*1 is just +12. Substitute this and the equation becomes:
.

.
Multiplying out the right side results in:
.

.
Subtract 120 from both sides to get:
.

.
Divide both sides by 10 to find that:
.

.
Convert this to exponential form and you have:
.

.
Calculator time. Raise 10 to the -5.5 power and the answer is in scientific notation is:
.

.
That means that:
.
watts/m^2
.
Hope that I guess correctly what this problem was asking you to do. If you have a value
for the decibels for normal conversation and you need to find the sound energy of normal
conversation, just use this method only substitute your db value in place of 65 and work
it out. You'll probably need to re-post this problem if I guessed wrong. If you do re-post,
clarify what you are looking for, what that last term involving watts/m^2 is, and any
other information you want considered.
.
Hope this helps you to understand the problem a little better too.
.
Cheers!

You can put this solution on YOUR website!
Given:
.

.
What is the solution set?
.
Work this as two separate problems. For the first problem, replace the absolute value signs
with parentheses that are preceded by a + sign. For the second problem, replace the absolute
value signs with parentheses that are preceded by a - sign. Let's do it.
.
First problem. Replacing the absolute value signs by parentheses preceded by a + sign results
in:
.
+(3x + 18) = 12
.
Since the leading sign preceding the parentheses is + you can just remove the parentheses
and you get:
.
3x + 18 = 12
.
Solve by subtracting 18 from both sides and you have:
.
3x = -6
.
Divide by 3 and you get:
.
x = -6/3 = -2
.
Second problem. Replace the absolute value signs by parentheses preceded by a minus sign
and you get:
.
-(3x + 18) = 12
.
This time when you remove the parentheses, the leading minus sign requires that you change
the sign of all the terms inside the parentheses. When you remove the parentheses
you get:
.
-3x - 18 = 12
.
Get rid of the - 18 on the left side by adding + 18 to both sides to get:
.
-3x = 30
.
Solve by dividing both sides by -3 and you get:
.
x = 30/(-3) = -10
.
Check your two answers by putting them into the given absolute value equation. First check
the -2 answer.
.

.
the 3*(-2) equals -6 and substituting this -6 results in:
.

.
combining the terms in the absolute value signs gives you:
.

.
and since is +12, this equation works.
.
Next substitute -10 for x and you get:
.

.
Do the multiplication in the parentheses to get:
.

.
combine the terms in the absolute value signs to get:
.

.
And the absolute value of -12 is +12 so the equation becomes:
.

.
So this answer checks also.
.
Another way you could have considered doing this problem is to substitute each of the
given answers into the original equation to see if it works. For example, Answer A is 2.
Substitute 2 for x in the original problem and you get:
.

.
this simplifies to:
.

.
and this further simplifies to:
.

.
This is obviously wrong so A cannot be the answer. And if you do a similar process for
answers B and D you will see they do not work either. Sometimes on the ACT and SAT exams
this method will save time. Just check the answers until you find the one that works.
.
Hope this helps you to understand a method for doing absolute value problems.
.

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Call the first number F and the second number S
.
The sum of the first number and the second number equals 51. In equation form this is:
.
F + S = 51
.
Then the problem says that twice the first (2*F) plus four times the second (4*S) equals 128.
In equation form this is:
.
2F + 4S = 128
.
So we now have a set of two equations:
.
F + S = 51
2F + 4S = 128
.
We can solve this set of equations by the process of variable elimination. One way is
to multiply the to equation (all terms on both sides) by -2 to get:
.
-2F - 2S = -102
2F + 4S = 128
.
Now add the two equations vertically. When you do, the -2F in the top equation cancels the
2F in the bottom equation. Continuing with the vertical addition you get:
.
2S = 26
.
Dividing both sides of this equation by 2 results in:
.
S = 13
.
Now return to the original first equation which said that the sum of the first and second
numbers is 51 ...
.
F + S = 51
.
Substitute 13 for S and this equation becomes:
.
F + 13 = 51
.
Solve for F by subtracting 13 from both sides:
.
F = 38
.
In summary, the two numbers are 13 and 38
.
We already know that 13 plus 38 equals 51 as required by the problem.
.
Then 2 times the first is 2 * 38 and that equals 76. Add to that 4 times 13 which is 52.
The result of 76 + 52 is 128 as was also required by the problem. The answer checks.
.
Hope this helps you to understand the problem.
.

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For this equation you use the formula:
.
D = R*T
.
where D = the distance traveled, R = the rate or speed, and T = the time
.
The rate and the time are the two unknowns. For the 200 miles we can write the equation for
the actual trip as:
.
200 = R * T <--- call this the first equation
.
For the proposed trip you know that the new rate equals the old rate plus 10 mph. You can
write this new rate as (R + 10).
.
And you also know that the new time is an hour less than the old time. You can write this
new time as (T - 1)
.
Therefore you can write the equation for the new trip as:
.
200 = (R + 10)*(T - 1)
.
Multiply the right side out to get:
.
200 = R*T - R + 10*T - 10 <--- call this the second equation
.
Since you need to solve this second equation for R, solve the first equation for T in terms of
R and then substitute that into this second equation.
.
Solving the first equation for T in terms of R you divide both sides by R and you get:
.
200 = T * R
.
200/R = (T*R)/R
.
200/R = T
.
Now substitute the left side of this equation into the second equation to get:
.
200 = R*(200/R) - R + 10*(200/R) - 10
.
Simplify this by multiplying out the right side:
.
200 = 200 - R + (2000/R) - 10
.
Subtract 200 from both sides and the equation becomes:
.
0 = - R + 2000/R - 10
.
Multiply both sides by -R and you get:
.
0 = R^2 - 2000R/R + 10R
.
The middle term on the right side simplifies to -2000 and the equation becomes:
.
0 = R^2 - 2000 + 10R
.
Transpose this equation (switch sides) and rearrange terms so it is in the more conventional
form of:
.
R^2 + 10R - 2000 = 0
.
This equation factors into:
.
(R + 50)*(R - 40) = 0
.
This equation will be true if either factor on the left side equals zero. So set each equal
to zero and solve for R:
.
R + 50 = 0
.
Subtract 50 from both sides and R becomes:
.
R = -50 mph
.
Then set the second factor equal to zero:
.
R - 40 = 0
.
Add 40 to both sides to get:
.
R = 40 mph
.
Ignore the first answer of -50 mph because a negative speed doesn't really make sense.
.
So the answer is 40 mph as the speed.
.
Check this out. At 40 mph you drive the 200 miles in 5 hours.
.
Now increase the speed by 10 mph to 50 mph. If you drive 200 miles at 50 mph it will
take 4 hours. The answer checks ... increasing the speed to 50 mph reduces the time it
takes to drive 200 miles by 1 hour.
.
So the answer to the original rate is 40 mph.
.
Hope this helps you to understand the problem and a way that you can solve it.
.

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Given:
.

.
in which y equals the hourly wage in dollars and x is the number of units produced
per hour.
.
You are asked to find the hourly wage of a worker who produces 13 units per hour.
.
All you have to do is to refer to the equation and substitute 13 for x. If you do that, you
get:
.

.
Do the multiplication on the right side and the equation becomes:
.

.
Now add the two terms on the right side of the equation to get:
.

.
That's the answer. A worker who produces 13 units per hour earns a wage of $14.15 per hour.
.
Hope this helps you to understand the problem a little better.
.

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Call n the number of cubic yards that are ordered. Then to find the cost of just the gravel
you need to multiply the number of cubic yards (n) by $30 per yard. Therefore, that cost
is . To that cost the dealer will add $50 for delivery. This will make the total
cost of gravel plus delivery [call it C(n)]:
.
dollars.
.
This is one of the answers you were asked for in the problem.
.
Using this equation you can calculate the the cost of 12 delivered cubic yards by replacing
n with 12. This results in:
.

.
Multiply 30 by 12 and the equation becomes:
.

.
So 12 cubic yards of delivered gravel costs $410.
.
Hope this helps you to understand the problem a little more.
.

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Given:
.
1n = 1n + 1n
.
Collect the logarithms that contain x on one side of the equation. You can do that by
subtracting ln from both sides of the equation. When you do that subtraction
you get:
.
ln - ln = ln
.
Now apply the rule that the difference in two logarithms is the same as the log of their division.
In equation form that rule says:
.
ln - ln = ln
.
With that rule the equation becomes:
.
ln = ln
.
Look at this carefully ... comparing the left side to the right side. For this to be
true the two logarithms must be equal, and this means that:
.

.
To get rid of the denominator, multiply both sides by the denominator of to
get:
.

.
On the left side the multiplier cancels with the denominator. And multiplying the right
side, gives:
.

.
and it reduces to:
.

.
Solve this as you would any equation. You need to gather all the terms that contain x
on one side of the equation and everything else on the other side. You can do this by
subtracting 6x from both sides to get:
.

.
combine the terms on the left side and the result is:
.

.
solve for x by dividing both sides by -5 and the outcome is:
.

.
and that's how to get the book answer. Hope this helps to familiarize you with logarithms.

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Given the expression:
.
-2 + ln e^3
.
Simplify this expression.
.
You can just about do this in your head by applying two rules of logarithms. The first rule
is that if you take the logarithm of a quantity that with an exponent, you can make an equivalent
term by multiplying the logarithm of the quantity times the exponent. If you apply this
rule to the given expression, you convert the expression to:
.
-2 + 3*ln e
.
The second rule (actually a definition) is that you can convert a logarithm to exponential
form by raising the base to the exponent on the other side and setting that equal to
the quantity you are taking the logarithm of. Easier to do than to say. Let's use this
rule/definition to find ln e.
.
Let's set y equal to ln e. We know that the base of the natural logarithms is e. If
we raise that base to the exponent y it will equal the quantity that the ln function
is operating on. So
.
ln x = y means (base)^y = x
.
and since the (base) is e we can say e^y = x
.
Now notice that for this problem x is e. Substituting e for x gives us
.
e^y = e
.
If you look at this carefully, you can see that to make the left side equal to the
right side, the exponent y has to be 1 ... making the equation e^1 = e.
.
So we now know that y = 1, but recall that we had said y = ln e. This tells us that
1 = ln e
.
Note - you can also do this on a scientific calculator. Fine the function key for e^x.
Enter 1 for x and then press the e^x key. You should get 2.718281828 as the value of
e^1 which is the same as e. Then press the ln key to take the natural logarithm of
2.718281828. You should get 1 as the answer. This tells you that the natural logarithm
of e (or 2.718281828) is 1 ... just as we found in the previous paragraph.
.
Anyhow, if we go back to the expression in the form:
.
-2 + 3*ln e
.
and we substitute 1 for ln e, we get:
.
-2 + 3*1 = -2 + 3 = +1
.
And that's the book answer.
.
Hope this helps you with your understanding of logarithms in general and natural logarithms
in particular.

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Given:
.

.
Note that I presume you meant and not which is what you wrote
if the rules of algebraic convention are followed.
.
There are four steps to finding an inverse of the function. These steps are:
.
1. Replace f(x) with y
2. Then replace y by x and also change x to y
3. Solve for y
4. Finally replace y by
.
Let's do it by following these 4 steps. Start with the given:
.

.
Replace f(x) by y to get:
.

.
Step 1 is done. Next do step 2 by changing y to x and also changing x to y. When you
do that the equation from step 1 becomes:
.

.
Step 2 is now done. On to step 3 which tells you to solve for this equation for y.
You can isolate the term that contains y. Do that by adding to both sides of
the equation to get:
.

.
Notice that the terms and cancel each other on the right side so
the equation is reduced to:
.

.
You can get rid of the denominator 5 on the right side by multiplying all the terms on
both sides of this equation by 5. Multiplying everything by 5 results in:
.

.
Now get rid of the denominator of 3 on the right side by multiplying everything on both
sides by 3 to get:
.

.
Now solve for y by dividing both sides by 10 and the result is:
.

.
Step 3 is done. Now replace y with and the answer becomes:
.

.
Presuming that I interpreted your problem correctly, this is the answer you were asked to
find. I'm not sure where you got the answer that you did. Even if I interpreted it incorrectly,
the four step process is correct, and you can use this same method to solve any problem with
inverting a function.
.

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You are looking for a number ... so let's call the missing number X.
.
When "twice a number (or 2X) is subtracted from 24" is a statement that translates to:
.
24 - 2X
.
The problem says that the result equals 6 times the number (or 6X)
.
Set this up as an equation:
.
24 - 2X = 6X
.
Solve this equation by first getting all the terms containing X on one side of the equal sign
and all the plain numbers on the other side. You can do this by adding 2X to the left
side to cancel out the -2X. But if you add 2X to the left side, you must also add 2X to the
right side to keep the equation in balance. Adding 2X to both sides results in:
.
24 - 2X + 2X = 6X + 2X
.
combining the -2X and +2X on the left side results in them canceling out and this
makes the equation become just:
.
24 = 6X + 2X
.
Combining the 6X and +2X on the right side makes the equation:
.
24 = 8X
.
Solve by dividing both sides of this equation by 8 and you get:
.
24/8 = X
.
and after doing the division on the left side you get the answer of:
.
X = 3
.
as being the number you are trying to find.
.
Check ... from 24 subtract twice the number ... this is 24 - (2*3) = 24 - 6 = 18. Does
that equal 6 times the number or 6 times 3? 6 times 3 also equals 18. So the answer checks.
.
Hope this helps you to see how you just start breaking down a problem and converting
the given information into algebraic terms. Then solve the equation that results from
relating the algebraic terms to each other.
.

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Given:
.

.
Find the inverse through the following process.
.
First replace f(x) with y to get:
.

.
Now replace x by y and y by x to get:
.

.
Solve this equation for y just as you would an ordinary equation. This can be done by
first adding 6 to both sides of the equation to eliminate the -6 on the right side to
get:
.

.
Then solve for y by dividing both sides by -3
.

.
Simplifying this results in:
.

.
Transpose sides and the result is:
.

.
Apply the negative sign in front of the right side to each of the terms in the numerator
and you get:
.

.
Finally replace y by to get:
.

.
Other than reversing the order of the two terms in the numerator this is the
answer you
gave in the problem.
.
Hope this helps you understand the four basic steps in finding the inverse of a function:
.
1. Replace with y
2. Interchange x and y
3. Solve for y in terms of x and
4. Replace y with
.

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Let N equal the number of pounds of $9 per pound nuts.
.
Let T equal the number of pounds of $12 per pound nuts.
.
From the problem you can infer that when you mix these two amounts together, the total
weight of the mixture will be 100 lbs. In equation form this becomes:
.
N + T = 100
.
If you multiply $9 times N you get the total dollar amount of N lbs of nuts in the mixture.
.
If you then multiply $12 times T you get the total dollar amount of the T lbs of nuts in
the mixture.
.
The price of the mixture is given as $11.25 a pound and there are 100 lbs of the mixture.
So the mixture is worth $11.25 times the 100 lbs of mixture ... a total of $1125.
.
So the dollar amount for each type of nut must add up to be $1125. In equation form this
is:
.
$9*N + $12*T = $1125.
.
So we have two equations:
.
N + T = 100 and
9N + 12T = 1125
.
Let's solve these two equations by elimination of the variable N. Multiply all the terms
in the top equation by 9 so that the term involving N in the top equation equals the term
that involves N in the bottom equation. The multiplication of the top equation by 9
results in:
.
9N + 9T = 900
9N + 12T = 1125
.
Now subtract the bottom equation from the top equation and you get:
.
0*N - 3T = -225
.
and the term 0*N = zero so the equation is just
.
-3T = - 225
.
Solve for T by dividing both sides by -3 to get:
.
(-3T)/(-3) = -225/-3
.
The division leads to:
.
T = 75
.
So the mixture contains 75 lbs of nuts that cost $12 per pound.
.
Since there are 100 lbs of the mixture, and we have accounted for 75 lbs of it, the
remaining 25 lbs must be the nuts that cost $9 per lb.
.
This is how you do this problem. You could have used other methods to solve the pair of
equations (methods such as substitution or determinants) but variable elimination
used above works just as well. Hope this all makes sense to you and you can see how you
need to find two equations to solve this problem.

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Given:
.
r = 1250/A
.
(1) Does this follow the Principle of Continuity which states that the velocity (r) of a liquid
flowing through a pipe increases as the cross-sectional area of the pipe decreases,
and decreases as the cross sectional area of the pipe increases.
.
The answer is that it does. Try it yourself. Suppose the area is 1 square inch. What is
the velocity (r)? Substitute 1 for A and you find that r = 1250/1 = 1250 gal per min.
.
Next increase the Area to 2 square inches. Substitute 2 for A and you find that r, the
velocity is now: r = 1250/2 = 625 gal per min. So as the Area got bigger (increased),
the velocity got smaller (decreased) ... exactly what the Principle of Continuity
said it should do.
.
(2) Given that the hose for the morning fire had a cross-sectional area of 5 square
inches, what was the velocity of the water? Just substitute 5 for A and you get:
.
r = 1250/5 = 250 gallons per minute
.
(3) During the evening fire, the velocity of the water was 100 gallons per minute.
Substitute this value for r in the equation and solve for A, the cross-sectional area:
.
100 = 1250/A
.
Multiply both sides of the equation by A to eliminate the denominator of A that appears
on the right side:
.
100A = 1250
.
Next divide both sides of the equation by 100 to find A:
.
A = 1250/100 = 12.5 square inches
.
Hope this helps you to understand the problem a little better.

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False
.
If x and y vary directly, then as one of them increases, the other must increase also.
(or as one of them decreases, the other must decrease also).
.
This means that x cannot be in the denominator unless y is also in a denominator.
.
Think about this. Let's say that x = 1. Then the given equation becomes:
.
y = 2/x = 2/1 = 2
.
Now let's increase x by saying that x is going to be 2. If y varied directly as x then
because we increased x, y should also increase. But when we substitute 2 for x, the
equation becomes:
.
y = 2/x = 2/2 = 1
.
Notice that when x went from 1 to 2, the value of y went down from 2 to 1. So y does not
vary directly as x. It varies inversely ... the bigger x gets the smaller y gets.
.
Hope this helps you to understand the problem a little better. And sometimes it helps
just to plug numbers in and see what happens to one of the variables if you increase
(or decrease) the other one ... just as we did above.

You can put this solution on YOUR website!

.
Let's just assume that the original amount is 1. Then to decay by half the amount
the resulting value of would be . Substitute these two values and you
get:
.

.
and after the multiplication by 1, the right side becomes just:
.

.
Because an exponent contains the variable we are to solve for, that's a clue that we can use
logarithms and the rules of logarithms to solve. And because e is also involved, let's
take the natural log of both sides to get:
.

.
A rule of logarithms that can be applied is that if the quantity that the logarithm
acts on has an exponent, you can make that exponent a multiplier of the logarithm.
Applying this rule to our problem results in:
.

.
But the natural logarithm of e is equal to 1. So substitute 1 for to get:
.

.
Simplify the right side by doing the multiplication and get:
.

.
Now you can use a calculator to find the natural logarithm of or its equivalent 0.5.
This logarithm is -0.69314718 and when this is substituted for the equation
becomes:
.

.
Finally solve for t by dividing both sides by -0.058 to get:
.

.
Therefore, the closest answer in your list of possible answers is 12 days. Hope this helps
you to see a way to solve problems such as these.

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.
You can solve this by using the same basic rules as you would in solving an equation,
with the exception that if you multiply or divide by a negative number, you must reverse
the direction of the inequality signs. In particular, whatever operation you do on one
section of the trichotomy, you must do the same operation on both of the other sections.
.
Don't forget that we are trying to solve for +x. Therefore, we need to reduce the center
section of this inequality down to +x.
.
So let's get rid of the +1 in the center section by subtracting +1 from all three sections.
.
When we do that subtraction we get:
.

.
Now to isolate +x in the center section of the inequality, let's divide all three sections
by -3. And when you divide by that negative number, remember that you have to reverse
the direction of the inequality arrows.
.

.
In a little more standard use of signs this becomes:
.

.
That's the answer ... on a number line x must lie to the left of but must be
to the right of . Let's try a number in that span and see if it works.
.
How about letting x equal -3. That number is between and so it
should satisfy the inequality. Start with:
.

.
and substitute -3 for x to get:
.

.
do the multiplication in the center section to get:
.

.
do the addition in the center section and you find:
.

.
As you can see, by letting x = -3 the inequality is satisfied.
.
Try some other values outside the span that we found to see that they don't work.
For example, try letting x = -2 and then x = -4. These are outside the span that we
found in solving the equation.
.
Hope this helps you to understand how to solve a trichotomy similar to this one.

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Since the integers are consecutive, if the smaller integer is x then the next integer
has to be one greater than x. So we can represent the two integers as x and x+1. Since
the sum of the two integers is 41 we can write:
.
x + (x + 1) = 41
.
Remove the parentheses on the left side to make the equation:
.
x + x + 1 = 41
.
Combine the two x's to get:
.2x + 1 = 41
.
Subtract 1 from both sides to eliminate the 1 on the left side. This results in:
.
2x = 40
.
Divide both sides by 2 to solve for x and the result is:
.
x = 40/2 = 20
.
Since the two integers are x and x + 1, the integers are 20 and 20 + 1 or 20 and 21.
.
If you add 20 and 21, you find that the sum of these two consecutive integers equals 41,
just as the problem required.
.
Hope this helps you to understand the problem a little better ...