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Your problem is:

The negative exponent means that the term raised to that negative value can be changed
into a fraction having 1 as the numerator and the term with a positive exponent as the
denominator. Easier to show by an example than to explain it.
Think "I can change into by making the denominator
of a fraction that has 1 as its numerator." When you do that your problem becomes:

Then you can eliminate the denominator if you multiply both sides by
If you do that multiplication on the left side you have a fraction that has
in both the numerator and denominator. This is equivalent to 1. On the right side the multiplication results in and the problem becomes:


Dividing both sides by 9 gives:

Now just take the square root of both sides. Notice that because
. Also notice that both the positive and negative forms
of are needed because when either is squared they produce a positive value
of
The two answers to this problem are:
and
You can check this by putting each of these answers (positive and negative) one at a time
into the equation:

and see that with these values the left side of the equation equals 9.

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If I understand your problem correctly it is:

One of the ways to solve this is to take the square root of both sides to get:
and
Then add + 7 to both sides and the result is:

If you calculate this you will find the two answers are about 9.645751311 and 4.354248689
Another way to solve it is to square out the left side of the equation by multiplying . If you do that, the equation becomes:

Subtract 7 from both sides to make the equation:

This is in the form for applying the quadratic formula that says if a quadractic equation is in the form:
then the solution for x is in the form:

For this problem a = 1, b = -14, and c = 42
Substituting the values of a, b, and c for this equation results in:

If you work this out you will get the same answers as you got previously
Hope this helps.

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Think about this ... x can be any value and y will always equal 3.5.
If x is 0, then y = 3.5
If x is 1, then y = 3.5
If x is 5, then y = 3.5
If x is -4, then y = 3.5
If x is -10, then y = 3.5
If you plot all these points on a graph you should begin to see that the graph is a horizontal line that intercepts the y-axis at +3.5 and continues on forever in both plus and minus directions.
Any time that there is no x term in the equation, the graph is a horizontal line that crosses the y-axis at the given value of y. For example, the graph of y= -4 would be a horizontal line that crosses the y-axis at the point -4 and continues on forever in both directions (to the left and to the right).

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You are given:

Notice that 5x + 10 can be factored by removing a 5 to make it 5(x+2). If you do this the equation becomes:

Next you need to make all the denominators the same. The common denominator is 5(x+2) so you don't need to do anything to the first term. It already has the common denominator. The second term lacks a 5 in the denominator. Therefore, multiply this term by . This is equivalent to multiplying the second term by 1 because is 1. When you do this the equation becomes:

Finally, on the right side of the equation the denominator lacks the . Therefore, multiply the right side by [effectively multiplying it by 1. When you do that the equation becomes:

Now you can get rid of the common denominator entirely by multiplying all the terms in the equation by . This cancels out the denominator and leaves you with the equation:

Its all downhill from here. Just do the distributive multiplication on both sides to get:

Add the two terms containing x on the left side to get:

Add 15 to both sides:

Subtract 7x from both sides:

Multiply both sides by -1:

If you substitute this value for x in the original equation, you should find that it makes the left side of the equation equal to
Hope this helps.

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Tell the answer to, or how to put, y=-3/2x+4 -into standard form
The slope-intercept form is in which m and b are constants, in which m is the slope of the graph and b is the value where the graph intercepts the y-axis.
So what you need to do for the first problem is to move the terms around and solve for y to get the equation to match the slope intercept form.

Move the 4x term to the right side of the equation, by subtracting 4x from both sides. This subtraction cancels out the 4x on the left side and makes a -4x appear on the right side as follows:

Now divide all terms on both sides of the equal sign by -5 to get:

When you divide -4 by -5 the answer becomes positive. And dividing +10 by -5 results in -2. So the answer becomes:

The next problem involves putting an equation into standard form. The standard form is:

where a, b, and c are all constants. So what you need to do is take the equation you are given, an equation in slope-intercept form, and rearrange things to get it into standard form. You are given:

Get rid of the denominator of 2 by multiplying all the terms in this equation by 2 to get:

Next, move the -3x term to the other side of the equal sign by adding 3x to both sides. Adding +3x to the right side cancels the -3x. And adding 3x on the left side makes a +3x appear on the left side. The result is:

This is the standard form that you are looking for.
Hope this helps.

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It probably is more educational to think about this problem rather than to worry about how to get it into a calculator. [What model TI calculator would make a difference in the approach.]
At the end of the year if you want to know what your return on investment is ... how would you do it. You just need to understand that the rate of return is the total dollar amount of interest you made divided by the total dollars you had invested.
You had three investments as follows: $165,000 + $85,000 + $235,000. The total of these invested amounts is $485,000.
Because of the different rates of return on each investment, you have to calculate the interest you get for each individual investment.
Investment 1 of $165,000 has a rate of return of 11.5% of the investment. This return is ($165,000 * 0.115) and it equals $18,975 of interest.
Investment 2 of $85,000 has a rate of return of 22.75% of the investment. This return is ($85,000 * 0.2275) and it equals $19,337.50 of interest.
Investment 3 of $235,000 has a rate of return of 10.25% of the investment. This return is ($235,000 * 0.1025) and it equals $24,087.50 of interest.
This means your total return for the year is the sum of $18,975.00 + $19,337.50 + 24,087.50 and this is equal to $62,400.
As we said previously the rate of return for the entire year is the total interest earned ($62,400) divided by the total invested ($485,000). If you do the division you find that the answer is 0.12865979 which converts to 12.866% and rounds further to 12.9%. Answer B is the correct answer.
Hope this helps you grasp the fundamentals of the problem.

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Let's check your answer by multiplying your factors together using the FOIL method (first terms, outside terms, inside terms, and last terms)
Your answer was:

Multiplying the first terms in each factor results in:

Multiplying the outside terms in each factor results in:

Multiplying the inside terms in each factor results in:

And finally, multiplying the last terms in each factor:

Algebraically adding all these terms in descending powers of x results in:

That is exactly the expression you started with. So your answer is correct. Excellent job. Keep up the good work!

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The easiest way to do it is to it is to punch it in to a scientific calculator. These calculators have sin, cos, and tan keys and cost in the range of $5 to $10. Be sure that the calculator is set for the degrees mode, enter 135 and press the tan key to find that the answer is -1.
Such a calculator is almost a necessity if you are dealing with uncommon angles. By uncommon I mean angles such as 8 degrees, 139.2 degrees, 273 degrees, and so forth. If you don't have a calculator to handle angles such as these, you probably need to resort to a table of trigonometric functions such as the "old-time" mathematicians used when they were struggling along.
Now, as to the particular problem you presented. As you read the following, it probably will be more understandable if you sketch the following thoughts onto a Cartesian coordinate system so you have a better picture of what is going on.
135 degrees is a fairly common angle. If you draw it onto a Cartesian coordinate system, it is an angle that is located in the second quadrant -- the upper left hand quarter of the graph. This angle is represented by a line that originates at the origin and slopes up and to the left such that the angle between this line and the negative x-axis is 45 degrees.
Notice that in addition to being at 45 degrees with the negative x-axis, the angle between this line and the positive y-axis if 45 degrees. If you start an angle measurement from the positive x-axis to the sloped line, measuring in the counter-clockwise direction, you will see that the angle is 135 degrees (90 degrees to the +y-axis plus 45 degrees from the +y-axis to the sloped line).
Recall from geometry that a right triangle (the kind most commonly used in trigonometry) that contains a 45 degree angle has the other acute angle of 45 degrees. (45 + 45 + 90 = 180 degrees for the triangle.) Also recall from geometry that the sides of such a 45 degree triangle are in the ratio of 1 to 1 to sqrt(2). These dimensions come into play during the next couple of steps.

Let's examine the characteristics of the line you drew from the origin and sloped it up and to the left at an angle of 45 degrees with the negative x-axis can be described this way: go to the left on the negative x-axis 1 unit from the origin. From that point (-1, 0) go vertically up one unit. The point that you are at now is (-1, +1) and it lies on the sloped line from the origin. [The distance from the origin to the point (-1, +1) is sqrt(2).]
[Note you could also multiply everything by 10. Go to the left on the negative x-axis 10 units from the origin to the point (-10, 0). From that point go vertically upward ten units to the point (-10, + 10). That point is on the line from the origin and the distance from the origin to the point (-10, +10) is equal to 10 times the square root of 2. This whole exercise just makes the sketch a little bigger.]
Looking on your sketch you should see that the three points (0,0), (-1,0) and (-1, +1) form a right triangle. [The same can be said for the scaled up points of (0,0), (-10,0) and (-10, +10)]
Now recall the definition of the tangent. It is the length of the side opposite to the angle divided by the length of the side adjacent to the angle. The angle is always measured between the sloped line and the x-axis. So in this case the side opposite the 45 degree angle is +1, the distance upward in the y direction, and the length of the side adjacent is -1, the distance from the origin to the left. Therefore:

Just for grins, how about finding sin(135). From the definition of sine you know that it equals the side opposite divided by the hypotenuse. Just as before, the side opposite is the length from the point (-1,0) to the point (-1, +1), a distance of +1 unit. The hypotenuse is the length from the origin to the point (-1, +1) which is the square root of 2. The hypotenuse is defined as always being a positive number. So sin(135):

which can be found by approximating the square root of 2 as 1.4142136 to get:

and this divides out to be Sin(135) = +0.707107
This has probably been somewhat confusing, but hang with it and it will eventually get clear to you. If you have further questions about it, post a follow up message.

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You apparently have done everything correctly so far. If you have written the problem correctly, there is a chance that the book is wrong.
You have correctly identified PQ as 11 units long. Therefore, you correctly said that side SR must also be 11 units long. And since SR equals 7x-3, you correctly set 7x - 3 equal to 11 units and solved this equation to find that x=2 units.
You did not explain much further, but QR is given as 3x-3. Since you now know that x=2, you can substitute to find that:

This simplifies to:

and this tells you that QR = 3
But the opposite sides of a rectangle are equal in length. Since side PS is opposite to side QR, its length must also be 3.
In summary you have PQ = 11, QR = 3, SR = 11, and PS = 3. If you add these four lengths you find that the perimeter is 28 units long.
If this doesn't agree with the book answer, you either provided some wrong information on the problem or the book answer is wrong. Go through the above work and see if it tracks exactly with the problem information from the book. If it does, then the publisher of the book made an error. [Believe me, it wouldn't be the first time a publisher made a mistake.]

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This is a classic "If ... then" relationship where the "if" identifies the hypothesis (something supposed to be true for the sake of argument) and the "then" identifies the conclusion (the conclusion is the outcome of the hypothesis).
Therefore, "if you get all a's on your geometry tests" is a hypothesis (something we suppose to be true)
and
"then everyone will want to sit near you in class" is the conclusion (outcome that will happen if the hypothesis is true).
Notice that the hypothesis may never happen, but if it does, the conclusion could be wrong.
For example, you may get all a's but maybe people don't like you ... or maybe they don't want the temptation of copying from you, so they may not want to sit by you anyway.

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To find the perimeter (P) of a rectangle you need to add 2 times the length (L) to 2 times the width (W) of the rectangle. In equation form this is written:

You are given that P is 47mm and L is 18.5mm. Substitute these values into the equation to get:

Multiply out the 2*18.5 and the equation becomes:

Subtract 37 from both sides to remove it from the left side of the equation. The result is:

Finally, divide both sides by 2 and you find that:
W = 5 mm.
Then recognize that the area of a rectangle is found by multiplying its length times its width. You were given that the length was 18.5 mm and you found that the width was 5 mm. Multiplying these two together results in an area (A) equal to:

which multiplies out to:

and the units of the answer are square mm

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2(x+5)-x-4=7
First do the distributive multiplication by multiplying the two terms in the parentheses by the leading 2. This results in:
2x + 10 - x - 4 = 7
Next combine the 2x with the -x to get just x. When you do the equation becomes:
x + 10 - 4 = 7
Then combine the +10 and the -4 to get +6 which makes the equation:
x + 6 = 7
Then subtract 6 from both sides so that it no longer appears on the left side:
x = +1
That is the answer. Check it by substituting +1 for x in the original problem to get:
2(1+5) - 1 - 4 = 7
Add the numbers in parentheses to get:
2(6) - 1 - 4 = 7
12 - 1 - 4 = 7
11 - 4 = 7
7 = 7
Since this is a true statement, the answer x = 1 is correct.

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The start of the series of numbers that end in 3 are:
3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 103, ....
Numbers in this series that are not prime are 33 because it factors into 3 times 11; 63 because its factors include 21*3 and 9*7; and 93 because it can be factored into 31*3. Because these are just 3 of the counter examples, the premise that all digits that end in the number 3 are prime cannot be true.

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Find the x-intercepts y = x^2 + 5x + 2
One way to do this is to graph the equation by selecting values of x, plugging them in, and finding corresponding values of y. Then plot the (x,y) points. This might help you to narrow down the places where the graph crosses the x-axis, but it also involve quite a lot of work if the answer is not a "nice" integer. [In this problem the answers are not integers.]
Think in terms of the axes of a graph. In order for a number to be on the x-axis, what does its y value have to be? Can you picture that y must be zero. For example, is the point (3, 4) on the x-axis? No, it is not because the y value of 4 tells you the point is located 4 units above the x-axis. A point such as (5, 0) is located on the x-axis because it is 0 units above the x-axis.
So to find if the equation has values on the x-axis you can set the value of y equal to zero and see if there are any values of x that will make that equation true. Setting y equal to zero results in the equation:

[Notice that I have switched the equation around -- putting the y value (zero) on the right side and the polynomial in x on the left side. No reason for it, its just a little more conventional.]
If you could factor the equation to get it in the form

then you could set each of the factors equal to 0 because if either factor is zero the equation will be true. This would tell you the values of x for which the equation is true. However, in this problem the polynomial in x is not easily factored.
A more general method to solving this equation is to use the quadratic formula which says that in a quadratic equation of the standard form:

the values of x that satisfy this equation are:

By comparing the equation from this problem to the standard form you can see that a = 1, b = +5, and c = +2. Plugging these values into the quadratic formula tells you that the two values of x that satisfy your problem are:


and this can be simplified to:

which can be further simplified to:

as an approximation you can say that the square root of 17 is about 4.1231 so that the two values for x are:

and

These simplify to x=-0.43845 and x=-4.56155
However, these are only approximately correct because the square root of 17 is not a "nice" number.
Hope this helps.

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Given:

Let's begin by simplifying the equation. Let's multiply everything by 2 to get rid of the denominator in the y-term.

Now add 24 to both sides to move the -24 to the right side of the equal sign.

Now all you have to do is to assume a value for x and find a corresponding value for y. For example, let's assume x = 1. The equation becomes:

and this becomes:

Subtract 12 from both sides to get:

and by dividing both sides by -3 you find that y = -12
In the Cartesian point form (x,y) this means that (1, -12) is an answer.
You can find other answers using the same process as above. For example, if you assume x = 2 and substitute that into the equation, you should find that y = -8 so that (2, -8) is another solution. And if you assume x = 4, you should find that y must = 0 so that (4, 0) is another solution.
Another way to look at this is graphically. Return to the equation:

First set x = 0 and the result will be y = 48/-3 = -16. Plot the point (0, -16) on the graph. Then let y = 0 and the result will be x = 48/12 = 4. Plot the point (4, 0) on the graph. Draw a line through those two points and extend the line in both directions. Any Cartesian point on that line will give you a set of x and y values that satisfies the equation. There are any number (an infinite amount) of values that will answer the problem.
Hope this helps you understand the problem a little better.

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The standard form of an equation with two variables (in this case the two variables are x and y) is:

in which A, B, and C are constants with A and B not both zero.
So what has to be done for this problem is to rearrange y = 3x -1 into the above form.
Subtract 3x from both sides. If you do, you get:

This simplifies to:

If you compare this to the standard form, you will see that it is the same form in which A = -3, B = 1, and C = -1.
Hope that helps you to see how to arrange linear equations with two variables into standard form.

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One share of A stock is worth 3 1/2 shares of B stock. If the total value of the stock is $9000, how much was invested in each company?
This is a poorly designed question because it has no firm answer.
For example, suppose a share of stock in company B is worth $200. That means that a share of company A stock is worth 3 1/2 times as much or $700. Now suppose that you own 2 shares of company A stock. That means you have $1400 invested in company A. The remaining $7600 ($9000 minus $1400) must be invested in company B and at $200 per share the number of shares you have in company B is 7600/200 = 38 shares.
Using the same value for the stocks, let's now suppose that you own 4 shares of company A stock at $700 per share for a total of $2800 invested in company A. The remaining $6200 ($9000 minus $2800) must be invested in company B and at $200 per share the number of shares you have in company B is 6200/200 = 31 shares.
And a third example. At $700 per share if you owned 8 shares of company A you would have a total of $5600 invested in company A. That means the remaining $3400 is invested in company B ... so you own 17 shares of company B.
In these three examples a share of stock in company A is worth 3 1/2 times a share of stock in company B. And also in each example the total amount invested is $9000. Those are all the limits set forth in the problem, so all three examples meet the requirements of the problem. Yet the amounts invested in each company differs for the in each example. There are many more examples you could come up with.
If there were a limit on the number of shares that could be owned for each company or if the problem said that the total amount invested in company A was 3 1/2 times the total amount invested in company B that might make the problem practical. It is not a practical problem the way it is written.
I think that what was intended in this problem was to say that the amount invested in company A (call it A) was 3 1/2 (use 3.5 in place of 3 1/2) times the amount invested in company B (call it B) and the total of the two investment amounts was $9000.
In such a case you can write:
A = 3.5B
and
A + B = $9000
Substitute 3.5B for A in the second equation
3.5B + B = $9000
4.5B = $9000
B = $9000/4.5 = $2000
And since A = 3.5B then A = 3.5*$2000 = $7000
But, again, that's not what the problem says as it is currently written.

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I presume that you are thinking of a right-circular cone ... that is, a cone whose height (h) is measured along a line perpendicular to the base that goes through the center of the circular base and through the vertex of the cone.
The formula for this surface area is:

where the variables in the equation are defined as: S is the surface area; r is the radius of the circle that forms the base; and s is the slant length of a line that goes from the perimeter of the circle forming the base to the vertex.
By the way of explanation the length of s is the hypotenuse of a right triangle whose legs are r and h where h is the length of the perpendicular between the center of the circular base and the vertex.
For your problem, you are given that the radius of the circular base is 10 feet and the height of the cone is 18 feet. From this calculate s using the Pythagorean theorem:
s =
s =
s =
s = 20.5913 ft
Next substitute the known values of the variables into the surface area formula:


When you do the multiplication you find that the surface area is 646.895 square feet.

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If the sum of two integers is 8 (even), then the integers could be 5 and 3, and the two integers are both odd.
You could do the same with the even sum of 12 being the sum of the odd integers 7 + 5 or 9 + 3.

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All you have to do is to recognize that (1,-2) is a way of saying that x = 1 and y = -2.
You are given the equation 3x - 4y = 12
(1,-2) is a solution to this equation if you substitute 1 for x and -2 for y in the equation and the result is 12 = 12.
In other words:
3x - 4y = 12
3(1) - 4(-2) = 12
3 + 8 = 12
11 = 12
Notice that the left side of the equation is not equal to the right side. Therefore, (1, -2) is not a solution to the equation.

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You are given the expression:
f(x) = 2x - 1
The notation f(-9) simply tells you that you substitute -9 for x in the expression
f(x) = 2x-1
In this case the substitution of -9 for x leads to:
f(-9) = 2(-9) - 1 = -18 - 1 = -19
so f(-9) = -19

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The cube of x is written
Raising this to the 5 power is written as
Notice that represents multiplied by itself 5 times ... or:
****
and we multiply common terms with exponents by adding the exponents to get:
**** =
which simplifies to
The shorthand way of doing this is to use the rule that when a term with an exponent is raised to another exponent, you multiply the two exponents. In this case:

is found by multiplying the 3 by the 5 to get
Hope this helps you to understand exponents a little more.

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If you assume that the first integer is N, then the next three integers are N+1, N+2, and N+3.
You need to know the sum of these four integers in a row. The sum is N+(N+1)+ (N+2)+(N+3). Since all the parentheses are preceded by a + sign, the parentheses can be removed without changing any of the terms in the parentheses. The result is:
N+N+1+N+2+N+3
and by adding like terms this sum is simplified to 4N+6
Next you need to find the sum of the remaining 8 numbers. You can get this by adding all 12 numbers and then subtracting the sum of the four consecutive numbers we found above. Probably the quickest way to find the sum of all 12 numbers is to just add 21 through 32 on a calculator. You can also find it by multiplying 12 times (21+32) and dividing that answer by 2. [This method involves the rule that to add consecutive numbers in a series, add the first number in the series to the last number in the series, multiply that sum by the number of numbers in the series, and divide that result by 2.] In either case you should find that the sum of the 12 consecutive numbers in the series whose first number is 21 and whose last number is 32, is a total of 318. If you subtract from 318 the sum of the four consecutive numbers, the remainder will be the sum of the remaining eight numbers. Since the sum of the 4 consecutive numbers is 4N+6, the subtraction is:
318-(4N+6) = 318-4N-6 = 312-4N
This is the sum of the other eight numbers
The problem tells you that the sum of the four consecutive numbers equals one-half the sum of the remaining eight numbers. In equation form this is:
4N+6 = (1/2)

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Were you told what the typical heartbeat rate is? Some sites list it as 72 beats per minute for adult males and 76 to 80 beats per minute for adult females. Another site lists the spectrum of heart beats as 60 to 110 with an average of around 80 for ages 12 through 16 and as 50 to 90 beats per minute with an average of 70 for persons greater than 16 years of age.
These rates are for persons that are calm and at rest without stress. Obviously if the person is undergoing aerobic exercises, the heartbeat rate will increase significantly.
Notice that the rates are expressed as the number of beats per minute. If you count the beats for 10 seconds, you have only counted for one-sixth of a minute. So you must multiply the number of beats in that 10 seconds by 6 times to find the number of beats per minute.
And going in the other direction, if you assume that the typical heartbeat rate is 72 beats per minute, then you can determine the number of beats in 10 seconds by dividing the 72 beats per minute by 6 because there are 6 each 10-second intervals in a minute. So the answer is 72/6 = 12 beats in 10 seconds. If you were given a different heartbeat rate, divide it by 6 to find how many beats you would count in 10 seconds.
For practice you should assume different average heartbeat rates and determine how many beats would occur in 10 seconds. For example, if the average heartbeat rate for a teenager is 80 beats per minute, how many beats would be counted in 10 seconds? The answer is about 13 beats in 10 seconds, but how do you determine that?
Or suppose that a person has a heartbeat rate of 68 beats per minute. How many beats would you count in 15 seconds? Since there are 4 periods of 15 seconds each in a minute, each period would get one-fourth of the 68 beats that occur in an entire minute. So the answer can be obtained by dividing 68 by 4.
Nurses often count the number of beats in 15 seconds and multiply that number by 4 to find the heartbeat count in beats per minute for a patient.
Hope this helps you to understand the problem a little better.

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Any time that you have a complex number in the denominator look to change it to a real number by multiplying it by its complex conjugate ... which is to say multiply it by itself except with a change in the sign between the real and imaginary portions of the complex number.
Easier to understand if we work this problem. In the denominator there is the term 1-i. Multiply it by 1+i. (This is the complex conjugate of 1-i ... same terms only change of the sign between the terms.)
Recall from algebra that: (a-b)(a+b) = a^2 - b^2. Thats exactly what we have except a = 1 and b=i. So the product of (1-i)(1+i) = 1^2 -(i^2) = 1 - i^2.
But also recall that by definition i^2 = -1. So our multiplication which resulted in 1 - i^2 becomes 1 - (-1) when we substitute -1 for i^2. And 1 - (-1) becomes 1 +1 = 2. Therefore, after multiplying the original denominator of 1-i by its conjugate of 1+i the denominator is converted to 2.
However, we can't multiply the denominator by 1+i without also multiplying the numerator by 1+i. [In effect that is equivalent to multiplying the original fraction by (1+i)/(1+i) which is to say, multiplying the original fraction by 1.]
Since the original fraction had 1+i and the numerator, we multiply that by 1+i (which is to say that we square the numerator in this case). The result is:
(1+i)*(1+i) = 1^2 + 1i + 1i + i^2. This simplifies to 1+2i+i^2, but recall that i^2 equals -1 by definition. Therefore we substitute -1 for i^2 to get 1+2i-1. The real parts cancel and we are left with just 2i in the numerator. We now divide this by the denominator of 2 to get 2i/2 = i.
The answer is just i.
Hope this helps you to think about complex number division

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Let's use a little logical reasoning.
If the owner got $120,000 in revenue by spending $2000 for ads, then we set up the ratio:
120,000/2000
but this ratio will be constant throughout the range. So we can ask if 180,000 is the sales, what amount will have to be spent on ads to get the same ratio?
We can set up a proportion:
120,000/2000 = 180,000/x
Notice that the sales are the numerators of both sides and the advertising costs are the denominators of both sides.
Multiply both sides by x to get the x into a numerator:
120,000x/2000 = 180,000
Multiply both sides by 2000 to eliminate the denominator on the left side:
120,000x = 180,000 * 2000
Divide both sides by 120,000 to solve for x which is the advertising dollars for $180,000 in sales:
x = (180,000 * 2000)/120,000
Do the multiplication and division on the right side to find that the advertising expenses associated with $180,000 in sales is $3,000.
Hope this helps.

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A way to do this problem using geometry is to apply the features of similar triangles.
Similar triangles have two major features:
(1) The measures of the corresponding angles of each triangle are equal.
(2) The lengths of the corresponding sides of each triangle are in the same ratio.
To get ready to do this problem, make a sketch of the cross-sectional area of the Washington Monument, just as was done in your textbook diagram. Then erase line DE and point D.
Next, place a straight edge along the dotted line BC and extend the dotted line upward above the monument for some distance. Then place the straight edge along along line AE and extend it further above the monument until it intersects with the extension of the dashed line. Label the point of intersection with the letter F. (Keep points A, B, C, and E just as you described them in the problem. Lengths AB, BC, and CE are the same as you described in the problem.)
Now look at two triangles: triangle ABF and triangle ECF. They are similar triangles, and we can state this as a fact if we can show that their corresponding angles are equal. Note that line EC is perpendicular to the dashed line BCF. Also line AB is perpendicular to the dashed line. And since they are both perpendicular to the same line, lines EC and AB are parallel. Line BCE is a transverse to parallel lines EC and AB. Therefore, angle ABC in triangle ABF and angle ECF in triangle ECF have equal measures because they are corresponding angles of parallel lines that are cut by a transverse.
Notice that angle F is an angle common to both triangle ECF and triangle ABF.
Since two angles of each of the two triangles have equal measures, the third angles of each triangle must have equal measures also (because the sum of the three angles of a triangle is 180 degrees.
Now we can use the fact that corresponding sides of each triangle are in the same ratio. That ratio can be found by noting that side EC in triangle ECF corresponds to side AB in triangle ABF (they are both opposite to the angle F so they are corresponding sides).
The ratio of the long side (side AB) to the short side (side EC) is:
8.4 meters/5.25 meters = 1.6
Now look at the corresponding sides formed by the dashed line. In the big triangle (triangle ABF) this side is length BC plus length CF. However the problem says that BC is 152.5 meters. So the side BF in the big triangle is 152.5 plus the length CF. Write this as (152.5 + CF). In the small triangle (triangle ECF the corresponding side is just CF. Now, set the ratio of these two sides -- big side to small side:
(152.5 + CF)/(CF)
but this ratio must be 1.6 as we determined above. Therefore, you can write:
(152.5 + CF)/CF = 1.6
Multiply both sides by the length CF to get rid of the denominator on the left side:
152.5 + CF = 1.6*CF
Subtract CF from both sides:
152.5 = 1.6CF - CF = 0.6CF
Divide both sides by 0.6 to find that CF = 152.5/0.6 = 254.1667
Recognize now that the total height BF equals BC + CF = 152.5 + 254.1667
So the total height is 406.6667 meters which rounds to 406.7 meters.
Hope this helps extend your knowledge of geometry.

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Four times s is 4s.
Therefore, the sum of 4 times s and r is (4s + r)
The parentheses are used to make it clear that you are dealing with the sum. You are asked to multiply the sum by t.
t times this sum is t*(4s+r)
This is the algebraic expression you were asked to find.
Evaluate the expression by substituting 2 for r, 5 for s, and 0.5 for t to get:
0.5 * (4*5 + 2)
Inside the parentheses first do the multiplication 4 times 5. Then add 2 to that answer. 4*5 is 20 and adding 2 results in 22. Substitute 22 into the expression in place of (4*5 + 2) to get:
0.5 * 22
And 0.5 * 22 = 11
The answer to the evaluation for the given values of r, s, and t is 11