You can put this solution on YOUR website!

Start with the given inequality


Break up the absolute value (remember, if you have , then or )

or Break up the absolute value inequality using the given rule




Now lets focus on the first inequality


Start with the given inequality


Subtract 1 from both sides


Combine like terms on the right side


Divide both sides by 2 to isolate x



Divide


Now lets focus on the second inequality


Start with the given inequality


Subtract 1 from both sides


Combine like terms on the right side


Divide both sides by 2 to isolate x



Divide



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

Answer:

So our answer is

or



So the solution in interval notation is: (] [)




Here's the graph of the solution set




Note:
There is a closed circle at which means that we're including that value from the solution set.


Also, there is a closed circle at which means that we're including that value from the solution set.

You can put this solution on YOUR website!

Start with the given equation


Break up the absolute value (remember, if you have , then or )

or Set the expression equal to the original value 10 and it's opposite -10




Now lets focus on the first equation


Add 3 to both sides


Combine like terms on the right side


Divide both sides by 5 to isolate x



Reduce

---

Now lets focus on the second equation



Add 3 to both sides


Combine like terms on the right side


Divide both sides by 5 to isolate x



Reduce





So the solutions to are:

and

You can put this solution on YOUR website!
Let x=amount of money in your pocket.


So when you have "at least $25", this means that you have 25 or more. Mathematically speaking, this means that some unknown quantity "x" is greater than or equal to 25.


So symbolically, this inequality is

You can put this solution on YOUR website!

The formula for the coordinates of the centroid is

and

note: notice how we're simply averaging the coordinates

where , , and are the coordinates of the three vertices a, b, and c respectively.


a)

Start with the formula for finding the x-coordinate of the centroid.


Plug in the x-coordinates of the given points


Add


Multiply.


So the x-coordinate of the centroid is

------

Start with the formula for finding the y-coordinate of the centroid.


Plug in the x-coordinates of the given points


Add


Simplify.


So the y-coordinate of the centroid is



So the centroid is (0,5.33)

---

b)



Start with the formula for finding the x-coordinate of the centroid.


Plug in the x-coordinates of the given points


Add


Multiply.


So the x-coordinate of the centroid is

------

(2,7),(8,1) and (14,11)
Start with the formula for finding the y-coordinate of the centroid.


Plug in the x-coordinates of the given points


Add


Simplify.


So the y-coordinate of the centroid is


So the centroid is (0,5.33)

---

c)

(a,p), (b,q) and (c,r).
Start with the formula for finding the x-coordinate of the centroid.


Plug in the x-coordinates of the given points



So the x-coordinate of the centroid is

------

Start with the formula for finding the x-coordinate of the centroid.


Plug in the x-coordinates of the given points


So the y-coordinate of the centroid is


So the centroid is (,)

You can put this solution on YOUR website!
First, let's draw the picture. Be sure you label every length you can





So we can see that the hypotenuse of the largest triangle is 26 units. Using pythagoreans theorem, we get:




Now looking at the left most smaller triangle, we see that the legs are 8 and "h" while the hypotenuse is "x". So once again, with pythagoreans theorem, we can say:




Now solving for , we get


Finally, the right most smaller triangle has a base of 18 and "h" and a hypotenuse of y. So this gives us:



---------


Start with the first equation.


Plug in . In other words, replace with


Square 18 to get 324. Square 26 to get 676.


Square 18 to get 324. Square 26 to get 676.


Plug in


Square 8 to get 64


Combine like terms.


Subtract 260 from both sides.


Divide both sides by 2.


Take the square root of both sides. Note: only the positive square root is considered.


So the length of x is which is about 14.422 units (this isn't important).


Go back to the second equation


Plug in


Square the square root to eliminate it.


Square 8 to get 64


Combine like terms.


Take the square root of both sides. Once again, only the positive square root is considered.

So the height is 12 units which means that altitude is 12 units.

You can put this solution on YOUR website!
First draw the rectangle with the point P



Now draw in the segments AP, BD, and PD. Take note of the unknown variables I'm assigning. If you look closely, you'll notice that a trapezoid forms due to these extra lines segments. Through the use of pythagoreans theorem, we get the length of AP of 13 and BD of 20





Now, it turns out that the ratio of the parallel sides 5 and 16 are the same as the ratio of the lengths of the cut diagonals. So the following ratios are true:

and


So let's solve for x:

Start with the first ratio


Cross multiply


Distribute and multiply.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Divide both sides by to isolate .


Reduce.


So the approximate length of x is 3.095 units. This means that TP is 3.095 units. This means that the other length is . So TA is 9.905 units.


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

Now let's solve for y:

Start with the second ratio


Cross multiply


Distribute and multiply.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Divide both sides by to isolate .


Reduce.


So the approximate length of y is 4.762 which means that the other length is . So TB is 4.762 units and TD is 15.238 units.

You can put this solution on YOUR website!
If we draw the picture, we get




From the picture, we can see that there is one large triangle and there is one smaller triangle towards the top.

It turns out that these triangles are similar. So this means that these ratios hold:

and


So let's find the length of x:

Start with the first ratio.


Cross multiply


Distribute and multiply.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Divide.


So the length of x is 4.5 units.

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

Now let's find the length of y:

Start with the second ratio


Cross multiply


Distribute and multiply.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So the length of y is 8 units.


So the lengths of the non-parallel sides are 4.5 and 8 units.

You can put this solution on YOUR website!
First plot the two points (0,0) and (3,4). Now draw a right triangle with the hypotenuse that goes through the two points. With the use of pythagoreans theorem, we find that the hypotenuse is 5 units (since ). So our drawing looks like this:



Now if draw the vectors and draw the parallelogram, we can see that the height of the parallelogram is the y-coordinate of the point (3,4). So the first height is 4.



So from the drawing, we see that the base is 8 units and the height is 4 units.


Now if we compute the area, we get:




Now let the vector from (0,0) to (3,4) be the new base. So the base is now 5. The area is still 32. So this means that



Solving for h, we get

So the second height is 6.4

You can put this solution on YOUR website!

Start with the given equation.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Simplify.


Since this equation is never true for any x value, this means that there are no solutions.

You can put this solution on YOUR website!

Start with the given equation.


Distribute.


Subtract from both sides.


Add to both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Simplify.


Since this equation is always true for any x value, this means x can equal any number. So there are an infinite number of solutions.

You can put this solution on YOUR website!
Start with the given ratio


Multiply both sides by 24


Multiply


Simplify



So our answer is

You can put this solution on YOUR website!
Start with the given expression.


Factor to get


So the LCD is


Multiply the first fraction by . Multiply the third fraction by . Doing this will make each denominator the LCD



Combine the fractions.


Distribute


Since the denominators are equal, we can combine the numerators of the common denominator.


Combine like terms.


Factor out the GCF 4 from the numerator.


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to


In other words, where or

You can put this solution on YOUR website!
# 1

Start with the given expression.


Multiply the first term by . Multiply the second term by . Multiply the third term by .


Combine the fractions.


FOIL the terms in the numerator.


Distribute.



Combine the fractions.


Distribute


Combine like terms.



So simplifies to


In other words, where , , or

---

---

# 2



Start with the given expression.


Factor to get .


Factor to get .


Factor to get .


Combine the fractions.


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words, where , , or

---

# 3



Start with the given expression.


Multiply the first fraction by the reciprocal of the second fraction .


Factor to get .


Factor to get .


Factor to get .


Combine the fractions.


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words, where , , or

You can put this solution on YOUR website!



Since and are given zeros this means that:


and


Get all terms to the left side in each case


and


Now use the zero product property in reverse to join the factors.





Expand and multiply




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

So the polynomial with roots of and is





Notice how if we graph , we can visually verify our answer

Graph of with roots of and

You can put this solution on YOUR website!

Start with the circumference of a circle formula


Plug in


Multiply 2, and (you can use 3.14 for pi) to get



So the circumference of a circle with a radius of 4.1 meters is 25.8 meters (rounded to the nearest tenth)

You can put this solution on YOUR website!
Let L=length and W=width


Since the "length of a top of a board is 6m greater than width", this means that the first equation is . Also, since the "area is 72m^2", this tells us that or


Start with the given equation.


Plug in and .


Distribute


Subtract 72 from both sides.


Let's use the quadratic formula to solve for W


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


Since a negative width is not possible, the only possible width is


So the width is 6 m


Go back to the first equation.


Plug in


Add


So the length is 12 m

You can put this solution on YOUR website!
Start with the given equation.



Use the Change of Base formula to rewrite each log. Remember, the Change of Base formula is:


Rewrite as . Rewrite as .


Rewrite each log using the identity


Now to make things simple, let . So this means that the equation is now





Notice now that the LCD is 6z


Multiply both sides by the LCD to clear the fractions.


Distribute and multiply.


Factor out the GCF


Add


Rearrange the terms.


Divide both sides by 11.


Now replace "z" with .


Rearrange the terms.


Divide both sides by .


Use the change of base formula to rewrite the left side.


Rewrite the equation using the property: ====>


So the answer is which approximates to

You can put this solution on YOUR website!
Technique #1 Factoring:

First let's factor


Start with the given expression


Factor out the GCF


Now let's focus on the inner expression

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

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 :
1,2,3,6,7,9,14,18,21,42,63,126
-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-63,-126


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


These factors pair up and multiply to .
1*(-126)
2*(-63)
3*(-42)
6*(-21)
7*(-18)
9*(-14)
(-1)*(126)
(-2)*(63)
(-3)*(42)
(-6)*(21)
(-7)*(18)
(-9)*(14)

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	-126	1+(-126)=-125
2	-63	2+(-63)=-61
3	-42	3+(-42)=-39
6	-21	6+(-21)=-15
7	-18	7+(-18)=-11
9	-14	9+(-14)=-5
-1	126	-1+126=125
-2	63	-2+63=61
-3	42	-3+42=39
-6	21	-6+21=15
-7	18	-7+18=11
-9	14	-9+14=5

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


So factors to


Set the factored expression equal to zero


Now set each factor equal to zero:

or

or Now solve for x in each case


So our answers are

or

---

Technique #2 Quadratic Formula:



Start with the given equation.


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 .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So our answers are or

---

Technique # 3 Completing the square


Start with the given expression


Factor out the leading coefficient


Take half of the x coefficient to get (ie ).

Now square to get (ie )




Now add and subtract this value inside the parenthesis. Notice how . Since we're adding 0, we're not changing the equation.



Now factor to get


Combine like terms


Distribute


Multiply



So after completing the square, becomes .


So is equivalent to


Start with completed square equation.



Add to both sides.


Divide both sides by 12.


Take the square root of both sides.


or Break up the expression


or Take the square root of to get



or Subtract from both sides.


or Combine like terms and simplify.


So the answers are or

---

Technique # 4 Graphing

Simply graph to get


Graph of


Now use the calculator's zero function to find the zeros at and

You can put this solution on YOUR website!
Start with the given expression.


Distribute.


Group like terms.


Combine like terms.


Simplify


So

You can put this solution on YOUR website!
# 1

Start with the given zeros

and


Get all terms to the left side in each case (ie subtract 1 from both sides in the first equation and add 8 to both sides in the second equation)


and


Now use the zero product property in reverse to join the factors.





FOIL and multiply





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

So the polynomial with zeros of and is





Notice how if we graph , we can see that the polynomial has roots of and

Graph of with roots of and

---

# 2

Remember the quadratic formula is


Where the discriminant is . So the quadratic formula could also look like


If D<0, then we'll be taking the square root of a negative number (which we cannot do). So this results in 2 complex solutions (ie no real solutions).



For example, let's find the discriminant for


From we can see that , , and


Start with the discriminant formula


Plug in , , and


Square to get


Multiply to get


Subtract from to get


Since the discriminant is less than zero, this means that there are two complex solutions (or no real solutions)


Now let's use the quadratic formula to find the solutions of


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 Break up the fraction for each case.


or Reduce.


So our answers are or


Since our answers are complex (non real), this verifies our original claim.

You can put this solution on YOUR website!
Start with the given expression.


Factor out the GCF 2 from to get


Highlight the common terms.



Cancel out the common terms.


Simplify.



So simplifies to .


In other words,

You can put this solution on YOUR website!
Start with the given expression.


Plug in x= 49 and y= -7


Divide


So when x= 49 and y= -7

You can put this solution on YOUR website!
Start with the given expression.


Factor out the GCF -6 from to get


Highlight the common terms.


Cancel out the common terms.





So simplifies to

In other words

You can put this solution on YOUR website!
Start with the given equation.


Subtract from both sides.


Add to both sides.


Combine like terms.


Divide both sides by .


Divide.


So the answer is

You can put this solution on YOUR website!
Start with the given equation.


Multiply both sides by 4.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


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

Answer:

So the answer is


Which approximates to

You can put this solution on YOUR website!

Start with the given equation.


Start with the given expression.


Distribute.


Distribute.


Combine like terms on the left side.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

You can put this solution on YOUR website!
Start with the given equation.


Distribute.


Combine like terms on the left side.


Combine like terms on the right side.


Subtract from both sides.


Add to both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

You can put this solution on YOUR website!
First, we need to find the zeros.


Set the expression equal to zero.



Now set each factor equal to zero:

or

Now solve for x for each factor:

or

So the zeros of are or



Now average the zeros to get


So the axis of symmetry (ie the x-coordinate of the vertex) is


Now plug in to find the y coordinate of the vertex


Start with the given function.



Plug in


Rewrite as


Multiply


Add


Multiply


So the y coordinate of the vertex is


This means that the vertex is at the point (-2,125). Since the vertex is either the highest or lowest point (in this case the highest), this means that that the maximum value of the function is

You can put this solution on YOUR website!
First, we need to find the zeros.


Set the expression equal to zero.



Now set each factor equal to zero:

or

Now solve for x for each factor:

or

So the zeros of are or



Now average the zeros to get


So the axis of symmetry (ie the x-coordinate of the vertex) is


Now plug in to find the y coordinate of the vertex


Start with the given function.



Plug in


Rewrite as


Multiply


Add


Multiply


So the y coordinate of the vertex is


This means that the vertex is at the point (-2,125). Since the vertex is either the highest or lowest point (in this case the highest), this means that that the maximum value of the function is

You can put this solution on YOUR website!

Start with the given equation.



Multiply both sides by the LCD . Doing so will eliminate the fractions.



Distribute. Notice how the denominators cancel out.


Simplify.


Subtract from both sides.


Simplify.


Combine like terms.


So the answer is