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Start with the given equation


Subtract 14 from both sides.


Divide both sides by 2



Since the equation is never true (note: remember, the absolute value of any number is always positive), there are no solutions to


============================================
Answer:

There are no solutions

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Start with the given expression.


Now let's FOIL the expression.


Remember, when you FOIL an expression, you follow this procedure:


Multiply the First terms:.


Multiply the Outer terms:.


Multiply the Inner terms:.


Multiply the Last terms:.


---------------------------------------------------
So we have the terms: , , , and


Now add every term listed above to make a single expression.


Combine like terms.


So FOILs to .


In other words, .

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Remember that if , then


In this case, since , this means that


Start with the given equation.


Rewrite as


Cross multiply


Multiply


Rearrange the equation


Take the cube root of both sides.


Take the cube root of 8 to get 2


So the solution is

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I'm assuming the expression is


Start with the given expression.


Multiply


Distribute


Rearrange the terms.


Multiply


Combine the roots.


Multiply


Evaluate the square root of 49 to get 7


Multiply


Rearrange the terms.


So

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Start with the given compound inequality.


Subtract from all sides.


Combine like terms on the left side.


Combine like terms on the right side.


So the solution is

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You are correct. Remember that if , then


In your case, since , this means

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First, let's simplify


Start with the given expression.


Factor into


Factor into


Break up the square root using the identity .


Take the square root of to get .


Take the square root of to get .


Rearrange and multiply the terms.


So simplifies to


In other words, where every variable is non-negative.


-------------------------------------------------------

So simplifies to


Ie. where


Now let's simplify


Start with the given expression.


Break up the root.


Factor to get


Break up the first root.


Take the cube root of to get


Convert to exponential notation.


Multiply the exponents.


Rewrite the first exponent as


Rearrange the terms.


Factor out


Square and simplify


Multiply


Convert back to radical notation.



==============================================

Answer:


So completely simplifies to


In other words, where

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First, let's graph


In order to graph , we need to plot a few points.


To get points in the form of (x,y), we need to find corresponding 'y' values to given 'x' values.


Let's find the y value when note: you can start at any x value.


Start with the given equation.


Plug in .


Square to get


Multiply


Combine like terms.


So if , then which gives us the point (-2,12).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Square to get


Multiply


Combine like terms.


So if , then which gives us the point (-1,5).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Square to get


Multiply


Combine like terms.


So if , then which gives us the point (0,0).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Square to get


Multiply


Combine like terms.


So if , then which gives us the point (1,-3).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Square to get


Multiply


Combine like terms.


So if , then which gives us the point (2,-4).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Square to get


Multiply


Combine like terms.


So if , then which gives us the point (3,-3).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Square to get


Multiply


Combine like terms.


So if , then which gives us the point (4,0).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Square to get


Multiply


Combine like terms.


So if , then which gives us the point (5,5).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Square to get


Multiply


Combine like terms.


So if , then which gives us the point (6,12).


----------------------------


Now let's make a table of the values we just found.

Table of Values:

x
y
-2
12
-1
5
0
0
1
-3
2
-4
3
-3
4
0
5
5
6
12

Now let's plot the points:

Graph:

Now draw a curve through all of the points to graph :


Graph of


So the graph of is




Now shade the graph green from x=2 to the right like so:





To find the inverse, we can do the following...


Start with the given equation.


Swap x and y


Take half of the y coefficient -4 to get -2. Square -2 to get 4. Add AND subtract this value on the right side.


Group the first three terms.


Factor that group


Add 4 to both sides.


Rearrange the terms.


Take the square root of both sides.


Add 2 to both sides.


or Break up the 'plus/minus'


Now because the directions state that the graph has "the limited domain x>=2", this means that we're only going to focus on the positive portion of the inverse. So we're going to ignore .


So the inverse of (over the domain ) is


I'll let you finish up and graph the inverse.

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Start with the given equation.


Add 2 to both sides.


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "x":


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


or Break up the expression.


or Reduce.


So the solutions are or

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Unfortunately, you are incorrect.














I think you're confusing the negative sign out front with negative exponents.


So you basically have the points

(0,-1), (1, -3), (2, -9), (3, -27)


Do the same with negative 'x' values to get more points so you can graph

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Well it helps to have the entire problem. What is the given function?


If you're having problems graphing in general, here's an example:


Problem: Graph


Solution:



In order to graph , we need to plot a few points.


To get points in the form of (x,y), we need to find corresponding 'y' values to given 'x' values.


Let's find the y value when note: you can start at any x value.


Start with the given equation.


Plug in .


Evaluate to get


So if , then which gives us the point (-2,0.25).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Evaluate to get


So if , then which gives us the point (-1.5,0.354).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Evaluate to get


So if , then which gives us the point (-1,0.5).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Evaluate to get


So if , then which gives us the point (-0.5,0.707).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Evaluate to get


So if , then which gives us the point (0.5,1.414).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Evaluate to get


So if , then which gives us the point (1,2).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Evaluate to get


So if , then which gives us the point (1.5,2.828).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Evaluate to get


So if , then which gives us the point (2,4).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Evaluate to get


So if , then which gives us the point (2.5,5.657).


----------------------------


Let's find the y value when


Start with the given equation.


Plug in .


Evaluate to get


So if , then which gives us the point (3,8).


----------------------------


Now let's make a table of the values we just found.

Table of Values:

x
y
-2
0.25
-1.5
0.354
-1
0.5
-0.5
0.707
0.5
1.414
1
2
1.5
2.828
2
4
2.5
5.657
3
8

Now let's plot the points:

Graph:

Now draw a curve through all of the points to graph :


Graph of

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I'll do the first two to get you going

# 1



Start with the given expression.


Rewrite as . Rewrite as .


Now factor by using the sum of cubes formula. Remember the sum of cubes formula is


Multiply

-----------------------------------
Answer:

So factors to .

In other words,

---

# 2


Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .


Now multiply the first coefficient by the last term to get .


Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?


To find these two numbers, we need to list all of the factors of (the previous product).


Factors of :
1,2,4,5,10,20
-1,-2,-4,-5,-10,-20


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to .
1*20 = 20
2*10 = 20
4*5 = 20
(-1)*(-20) = 20
(-2)*(-10) = 20
(-4)*(-5) = 20

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number	Second Number	Sum
1	20	1+20=21
2	10	2+10=12
4	5	4+5=9
-1	-20	-1+(-20)=-21
-2	-10	-2+(-10)=-12
-4	-5	-4+(-5)=-9

From the table, we can see that the two numbers and add to (the middle coefficient).


So the two numbers and both multiply to and add to


Now replace the middle term with . Remember, and add to . So this shows us that .


Replace the second term with .


Group the terms into two pairs.


Factor out the GCF from the first group.


Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


Combine like terms. Or factor out the common term


===============================================================



Answer:


So factors to .


In other words, .


Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).

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I'll do the first three to get you going. The rest will be like # 2 and # 3

# 1


Start with the given expression.


Multiply the fraction by .


Note: is the complex conjugate of . Multiplying two conjugate pairs will result in a real number.


Combine the fractions.


FOIL the numerator.


FOIL the denominator.


Multiply.


Replace with -1


Multiply


Combine like terms.


Break up the fraction.


Reduce.


So .


So the expression is now in standard form where and

---

# 2




Start with the given equation.


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "x":


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


Break up the fraction.


Reduce.


or Break up the expression.


So the solutions are or

---

# 3

Start with the given equation.


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "x":


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


or Break up the expression.


So the solutions are or

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What do you want to do here? Please post full instructions.

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Add 4ws to both sides.


Factor out the GCF 'v' on the right side


Divide both sides by to isolate 'v'.


Rearrange the equation.


So the solution is


Note: you made a mistake in thinking that you can combine 3s and 5w to get 8sw....the terms 3s and 5w are NOT like terms, so they CANNOT be combined.

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I'm assuming you want to solve for y?


Start with the given equation.


Distribute


Multiply


Subtract 4 from both sides.


Multiply the last term by


Combine like terms.


So after solving for 'y', we get

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Start with the given inequality.


Multiply both sides by the LCD 8.


Multiply.


----------------------------------------------------------------------

Answer:

So the solution is which is written as in interval notation.

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Start with the given formula


Plug in the given velocity of


Plug in the height of 0 (since the height is zero at the ground)


Rearrange the terms.


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "t":


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the solutions are or

This means that at times of 0 and 3 seconds, the ball will be at the ground.


Since we already know that the ball is at the ground at 0 seconds, the other solution is more interesting. So we're going to ignore


====================================================

Answer:


So it will take 3 seconds for the ball to return to the ground.

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Jump to Answer


Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last coefficient is .


Now multiply the first coefficient by the last coefficient to get .


Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?


To find these two numbers, we need to list all of the factors of (the previous product).


Factors of :
1,3,9
-1,-3,-9


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to .
1*9 = 9
3*3 = 9
(-1)*(-9) = 9
(-3)*(-3) = 9

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number	Second Number	Sum
1	9	1+9=10
3	3	3+3=6
-1	-9	-1+(-9)=-10
-3	-3	-3+(-3)=-6

From the table, we can see that there are no pairs of numbers which add to . So cannot be factored.


===============================================================



Answer:


So doesn't factor at all (over the rational numbers).


So is prime.


Jump to Top

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You're off to a great start. I'll start where you left off.


Start with the given equation.


Rearrange the terms.


Expand. Note:


Distribute


Multiply


Combine like terms.


So


This means that the polynomial with the roots , , and is

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First off, this isn't an equation (it doesn't have an equals sign '=')


Second, there's not much to solve here since 'x' is already isolated.


The only thing left to do is to write the solution set in interval and set-builder notation (and possibly graph it on a number line)



The solution set in interval notation is ()


Also, the solution set in set-builder notation is


Here's the graph of the solution set

Graph of the solution set

Note:
There is an open circle at which means that we're excluding this value from the solution set
Also, there is an open circle at which means that we're excluding this value from the solution set.

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The problem is either which simplifies to


Or....


The problem is which does NOT simplify at all.


So I'd make sure that you have the right problem.

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a)
Let's build the tree:



Using the tree, we get the sample space:

{b,b,b,b}
{b,b,b,g}
{b,b,g,b}
{b,b,g,g}
{b,g,b,b}
{b,g,b,g}
{b,g,g,b}
{b,g,g,g}

{g,b,b,b}
{g,b,b,g}
{g,b,g,b}
{g,b,g,g}
{g,g,b,b}
{g,g,b,g}
{g,g,g,b}
{g,g,g,g}

So for instance {b,g,g,b} means that the couple had a boy, girl, girl, and then a boy (in that order).

Note: recall, the sample space is the set of ALL possible outcomes.
-------------------------------

b)
Since we want to know the chances of the couple having "at least three boys", this means they want to know the chances of having 3 boys OR 4 boys (since at least means that figure or more).


Looking back at the list of all possible outcomes (ie the sample space) from part a), we see that we have the combinations for 3 boys:
{b,b,b,g}, {b,b,g,b}, {b,g,b,b}, and {g,b,b,b}

So there are 4 cases where the couple would have 3 boys.


So

-----------------------------

Also, since there is only ONE way to have 4 boys (of a total of 4 children), this means that






-------------------------------------------------------


Now simply add the two probabilities to find the chances of either one occurring:



















So the probability of the couple having AT LEAST 3 boys is which is 0.3125 in decimal form which gives a 31.25% chance.

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What is the '2x' for the circle? Is that the radius, diameter, or something else? If possible, please post a picture. If you need help with that, check out this lesson.

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# 1


So


# 2



So

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3[x-(2-x)-2x]=3(4-x) ... Start with the given equation.


Distribute


Distribute again.


Combine like terms on the left side.


Subtract 12 from both sides.


Combine like terms.


Divide both sides by -3 to isolate 'x'.


Reduce


Rearrange the equation


----------------------------------------------------------------------

Answer:

So the solution is





Check:

3[x-(2-x)-2x]=3(4-x) ... Start with the given equation.


3[6-(2-6)-2*6]=3(4-6) ... Plug in (ie replace each 'x' with 6)


3[6-(2-6)-12]=3(4-6) ... Multiply


3[6-(-4)-12]=3(-2) ... Combine like terms.


3[6+4-12]=3(-2) ... Rewrite -(-4) as +4


3[-2]=3(-2) ... Combine like terms.


-6=-6 ... Multiply


Since the equation is true, this verifies the solution

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Start with the given equation.


Get every term to the left side.


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "x":


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


or Break up the expression.


or Reduce


So the solutions are or

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I'm assuming that the first 'x's on each side are multiplication symbols.


Start with the given equation.


Take the log of both sides.


Break up the log using the identity


Subtract from both sides.


Subtract from both sides.


Rewrite the logs on the left side using the identity


Factor out the GCF 'x' on the left side.


Combine the logs using the identity


Divide both sides by to isolate 'x'


Approximate the answer (with a calculator)


So the exact answer is , which can be simplified to and the approximate answer is

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I'll do the first one to get you started.

# 1

Start with the given equation.


Distribute


Add 2ab to both sides.


Subtract cdx from both sides.


Factor out the GCF 'x'


Divide both sides by .

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Let C=total cost of lessons


So at first, the cost is $50. So no matter how long she stays, she'll be paying $50. So at first, the cost is


Now let's say that she now decides to do another month on top of the first month. She's already paid $50, but she now has to pay an extra $30. So for one additional month, the cost is now . If she does it for another month after that, she has to pay another $30, giving the cost of


Now let's say that she does 8 additional months. This means that she has to pay 8 $30 fees. So the cost would be (there are 8 thirty's here)

To save time and sanity, instead of adding repeated values, we can just multiply. So instead of having eight 30's out there, we can rewrite that part as .

So in general, if she stays for 'x' additional months, you're going to add on 'x' charges of $30 which will cost . Add this to the original cost of $50 to get the total charge


Since we want to hold the cost to $350 (that's her budget), we simply make and plug it in to get


Note: you can rearrange the last equation to get

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Start with the given function.


Plug in


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "x":


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


or Break up the expression and reduce.


So the solutions are or