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I don't see ANY factors in this problem!!! It looks to me like ALL the factors are missing!!

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You'll have to factor the numerator and denominator in order to reduce the fraction. It just so happens (by chance???) that they both factor as trinomials. It may be some tough factoring, but they do factor.





You can divide out the factor of (3xy+4).

Final answer:

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I don't know what you did to get your answer, but I didn't get that.

First, this is an equation, and there are several ways to solve it. I think the easiest way to solve it, since it is equal to zero, add the second fraction to both sides and write this as a fraction equal to the negative of a fraction. As I said, this is NOT the only way, but I think it will be the easiest to explain and easiest to understand:



Remember means that

So,
means that

Multiplying it out:


Set equal to zero by adding +4x to each side:



As CHANCE would have it (yeah, right!) THE TRINOMIAL FACTORS!!


First solution:




Second solution:



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SPEAKING OF ALASKA, BE SURE TO SEE my website and check out OUR PICTURES FROM ALASKA!! There are PICTURES OF REAL GLACIERS (the pictures look TOO GOOD to be real, but believe me, we were there!! They ARE REAL!!) Click on my tutor name here at algebra.com, and look for "Pictures from Alaska". Isn't it GREAT to see MATH all around us in life, EVEN IN GLACIERS!!!

I'm not sure about your first glacier, since you said it is sliding at 4 per day. I don't know 4 what per day? Is that supposed to be 4 feet per day?

The second glacier is moving at 3 inches per hour, which is 3*24 = 72 inches per day, then divide by 12 to convert it to 6 feet per day.

The third one is miles per hour. Multiply this times 5280 to change it to feet per hour and then multiply by 24 hours to see how far it moves in a day. This turns out to be (with a calculator!!) feet per hour , then times 24 to change it to 7.6032 feet per day.

The greatest speed looks like the last one which is about 7.6 feet per day.

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Let x= amount of his savings

There may be two ways to interpret this: Were his saving increased BY 4 times his savings? I think that must mean that he ADDS 4x to his present savings of x.

The equation is
x + 4x = 25000
5x = 25000
x= $5000

Check: If his savings of $5000 are INCREASED BY 4 times this amount (which by the way is $20,000), then the result is $25,000. I'm almost sure this is the way the problem is intended to be interpreted.

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Do you mean y= -1/3 x + 23/3??? If so, then you need to start with the y-intercept at the point 23/3, which is almost up to 8 units.

From this point measure off a slope of -1/3. This is to say, it is a RISE of -1 and a run of 3. So from the point that you just graphed on the y axis, and measure DOWN 1 unit, then move 3 unit to the right, and put a point. Connect the two points and it should look like this:


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First find the amount of increase. Since the income went from $32000 to 48000, this means that the amount of increase is $16000. Percent of increase is always a comparison of the amount of increase to the ORIGINAL amount, which is $32000.

Percent of increase =
= = = 50%

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First add up ALL the juice in ml. That would be 2500 ml. Then convert to liters.

I use what I call the "Metric Line", starting from left to right with
kilo, hecto, deca, basic unit (in this case liters), deci, centi, milli. When converting from one unit to another, you must move the decimal to the left or right, depending upon what you are converting to what. Count the number of units that you move from left to right or right to left, and move the decimal that many places to the left or right.

In this case you are converting from milli to the basic unit (liters), which is moving 3 places to the left. You must therefore move the decimal three places to the left, so 2500 ml = 2.5 liters.

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8-3x<-3(1+x)+1

Use distributive property to remove the parentheses:
8-3x < -3- 3x + 1

Combine like terms on each side:
8-3x < -2 - 3x

Add +3x to each side in an attempt to get all variable terms on the left side:
8-3x+3x < -1 - 3x + 3x
8 < -1

Since there are NO VARIABLES left in the equation, you can't solve for x as you do in most equations. In this case, all you have left is 8<-1, which is NOT TRUE. In fact, it is NEVER true. This means that no matter what x is, 8 will NEVER be less than -1. This means that there IS NO SOLUTION!

The final answer is b. No Real Numbers.

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I'm not sure exactly what you mean by this. If you mean which is technically what you have given, it means that x must be greater than or equal to zero. Domain is in interval notation: [0, infinity)

If you meant , this means that x-2 must be greater than or equal to zero. If you add 2 to each side of the inequality, you get x must be greater than or equal to 2. Domain for this, in interval notation, would be [2, infinity).

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Are you saying:
y =
If so, the domain consists of all values of x except that the denominator must NEVER be zero. The denominator consists of taking a value of x, squaring it, multiplying that result by 12 and then adding 20. By looking at this you can see that squaring a number will never be negative. Multiplying by 12 and then adding 20 will certainly always give you a positive, never a ZERO, result.

Therefore the answer for domain is all real values, since the denominator can never be zero.

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Let x = Alvin's age
x+5 = Kenneth's age
x+5 + 4 = x+9 = Kenneth's age 4 yrs from now
x-5 = Alvin's age 5 years ago.

Equation:
Product of (Kenneth 4 years from now)x (Alvin 5 years ago) = 51
(x+9)(x-5) = 51



Guess what! It factors!!

x= -12, x= 8

An age cannot be negative, so
x= 8 Alvin's age
x+5 = 13 Kenneth's age

Check: In 4 years Kenneth will be 17, and 5 years ago, Alvin was 3. The product of 17*3 is 51.
It checks.

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21x+3=121

Add -3 to each side:
21x+3-3=121-3
21x = 118

Divide both sides by 21:



What an UGLY answer! Life doesn't always come out even, does it? In this case, the fraction doesn't even reduce!! How sad!!

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There are several ways to do this. The elimination method usually works well when the equations are both in standard form, as in this case.
x - 4y = -2
x + 2y = 4

In order to eliminate the x variable, multiply the first equation by -1, and rewrite the second equation as it is:
-x + 4y = 2
x + 2y = 4

Now add these two equations together giving you
6y = 6
y=1

Substitute the y = 1 back into either equation, say the first one:
x - 4y = -2
x - 4(1) = -2
x-4 = -2

Add 4 to each side:
x-4+4 = -2 + 4
x=2

Check in the second equation, by letting x= 2 and y = 1:
x + 2y = 4
2 + 2(1) = 4
It checks!

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And YOU think the tutors at algebra.com are SO GOOD we can solve a problem just from the page number and the problem number? Maybe if you gave us the answer, we could tell you what the problem was???

Forgive my cynicism. It's late! Or is that EARLY??

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I personally think you are better off getting the variables to the LEFT side when solving inequalities, so add 4x to each side and add 6/5 to each side.

4x < 6/5

The divide both sides by 4, or what might work better in this case is to multiply both sides by 1/4:





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12q-9 + 8q+14 = 90
20q + 5 = 90
20q= 85
q = 85/20 = 4.25

The angles are 12q-9= 12(4.25) - 9 = 51-9= 42 degrees
8q + 14 = 8(4.25) + 14 = 34 + 14 = 48 degrees

Check by adding 42 + 48 = 90 degrees.

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If you really don't like fractions, then multiply both sides by the denominator which is 2, and get rid of it:





The 2 divides out in the first term, leaving:


Now just subtract 10 from each side:



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Let x = amt of pure antifreeze to be added
You are adding this to 20 gallons of 25% and you will end up a total mixture of x+ 20 at 40%.

The equation is
.25(20) + x = .40(x+20)
5 + 1.00x = .40x + 8
1.00x - .40x = 8 - 5
.60x = 3



Does it check?? (It is VERY late in Florida, here! I don't guarantee my work or my spelling at this hour!!}}}

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You are correct. Remember that you are trying to find a family of parabolas, which means that there are perhaps an infinite number of such parabolas. This is what happens when you have three unknowns and only two conditions (i.e., equations) to satisfy.

Both equations come down to A + C = -1. So, let B=1, and solve for either C or A in terms of the other. Say, C= -A -1.

The equation is the family of parabolas:


, where A represents any non-zero number. Of course, if A = 0, then it is NOT a parabola.

Any other value of A should give you points that include (1,0) and (-1,-2).

For examples, let A = 1: Do both points satisfy this? If x =1, then y= 0, and if x=-1, then y =-2.

Let A = 2: (((y = 2x^2 +x -3}}}

Let A = -1:

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I've seen that you have been trying for a long time to get an answer to this one. I'm not sure the best way to explain it to you, since the graphing calculator is the BEST way to do it. Not knowing your background, I'll try a general explanation of sorts.

First solve the equation to find out where (x+2)(x-1)(x-3) = 0. This happens at three points on a numberline, at x= -2, x= 1, and at x=3. Now, the question is actually where will the product of these numbers be LESS THAN or EQUAL to ZERO. So you know THREE solutions right off the bat.

Now, you must realize that, since this is an INEQUALITY, it could be any of the intervals that are created by these three endpoints. If you graph the three points on a numberline, you will notice that these THREE numbers divide the numberline into FOUR intervals (always one more interval than endpoints!). What you need to do is to TEST out ONE POINT in each of the intervals.

Start with the values on the numberline to the RIGHT of x=3. For example, if you took 4 or any other number larger than x=3, and subsitute into the inequality: (x+2)(x-1)(x-3) <=0, it will be positive number times positive times a positive number, which is NOT less than zero.

Next, test a point that is between 1 and 3, like for example 2. If you subsitute x= 2, it comes out to be a product that IS less than zero.

Next, test a point that is between -2 and 1. That will be a positive, so it is NOT less than zero.

Finally, test a point that is less than -2, and the result will be a product that is NEGATIVE, so this interval works.

Therefore the answer is all values from negative infinity to -2 including the -2, and also from 1 to 3 including those endpoints.

It has occurred to me that what I just wrote is probably not a very clear explanation. Let's try the graphing method. You must graph y=(x+2)(x-1)(x-3), and find all values of x for which points on the graph are ON or BELOW the X-axis.


From this graph you can see that the graph is BELOW the x-axis from negative infinity to -2, and again from 1 to 3. Of course include the endpoints, but don't EVER include infinity.

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What do you mean "squar foot"??? Do you mean square root??

To find a square root, you need to know the perfect squares, and find the largest perfect square that will divide evenly into the given number. The following ar examples of this concept. For additional explanation see my Lesson Plans in algebra.com on Square Roots, or see my website under Basic Algebra, Samples of Basic Algebra One Step at a Time, and Chapter 5.







because that one comes out even: .

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You are absolutely CORRECT!! You said that B will catch A in 2 hours 30 minutes, which IS at 11:00 a.m!! You must have forgotten that B started a half hour later than A, which would be starting at 8:30 a.m. for B. Three hours is the time for A to be caught! Two and a half hours is the time for B to do the catching!! BOTH happen at the same time--11:00 a.m.!!

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Let x = number of half dollars
2x = number of quarters
2(2x) = 4x = number of dimes
3(4x) = 12x = number of nickels

Total value, multiply number of coins times the value of each coin in cents, and the total value of the coins is $10 or 1000 cents. (Doesn't it make "cents" to do it this way??)

50(x) + 25(2x) + 10(4x) + 5(12x) = 1000
50x + 50x + 40x + 60x = 1000
200x = 1000
x= 5 Half dollars
2x = 10 Quarters
4x = 20 Dimes
12x = 60 Nickels

Check: Does it add up to $10.00??? You tell me!!

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There are probably several ways to go about this. I'd say to divide both sides by 3, and get


Next use the basic definition of logarithms to convert to exponential form. Remember the definition says means ?

In the same way, means that

What you are trying to find is x, and that's it!! x = .

Is that enough, or you could probably rewrite this as which means , but that's getting carried away if you ask me. Easier to calculate it and say approximately 6.24.

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What you did so far, looks right so far. Let's expand the square of the binomial:

y=1-2x
x²+(1-2x)²=13


By quadratic formula




This is the square root of a perfect square, which means that the original equation DOES factor, and the quadratic formula was not necessary in this case. However, you are almost there now, so might as well finish it.





Then solve for y using y = 1-2x
If x= 2, y = 1-2*2= -3

If , then = =

Points of solution would be (2,-3) and (, ).

Check for errors in my work!! Let me know if I made any.

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You have a product equal to zero:

3y(2y-3)(3y-4)=0

If a product of numbers equals to zero, then one of those numbers must be zero. Because this can be thought of as a product of three quantities, you can make three separate solutions:

Solution #1.
3y = 0


Solution #2:
2y-3 = 0
2y= 3


Solution #3:
3y-4=0
3y = 4


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Since the Interest paid is in proportion to the Investment, you can set up a proportion:

Let x = interest earned on a $2000 investment.
=

Remember means

In the same way,
means




An easier way to look at it, since $2000 is actually $1500 plus one third of that amount, the interest earned will be the corresponding $45 plus one third of that amount, or $45 + 15 = $60.

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The key word is "arrangement" which means that the order is significant. This makes the problem a PERMUTATION, a permutation of the 5 letters in the word OLIVE taken 3 at a time. There are a lot of notations that can be used here, one of which, the easiest to type here, is P(5,3). It means to start with the first number 5, and start counting down. You must have THREE numbers (that is the second number), in the product:

P(5,3) = 5*4*3 = 60 different arrangements.

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Solve for y in the first equation:
4x - y = 22
4x-4x - y = 22- 4x
-y = -4x + 22
y = 4x - 22

Substitute into the second equation:
2x + 3y = 4
2x + 3(_____)=4
2x + 3(4x-22) = 4
2x + 12x - 66 = 4
14x =70
x=5

Substitute back into first equation to solve for y:
y=4x -22
y=4*5 - 22
y=20-22
y=-2

Check in the second equation:
2x + 3y = 4
2(5)+3(-2) = 4
10-6= 4
It checks.

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5d+9-4d=51-5d
d +9 = 51 - 5d

Add 5d to each side:
d + 5d + 9 = 51 - 5d + 5d
6d + 9 = 51

Subtract 9 from each side:
6d + 9 - 9 = 51 - 9
6d = 42
d= 7

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d= 7