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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 12 and it's opposite -12




Now lets focus on the first equation


Add 7 to both sides


Combine like terms on the right side


Reduce

---

Now lets focus on the second equation



Add 7 to both sides


Combine like terms on the right side


Divide both sides by 2 to isolate x



Reduce





So the solutions to are:

and

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Start with the given inequality.


Distribute.


Combine like terms on the right side.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate . note: Remember, the inequality sign flips when we divide both sides by a negative number.


Reduce.


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

Answer:

So the answer is

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Notice how when x increases by 1, f(x) increases by 4 (eg as x goes from 0 to 1, f(x) goes from 6 to 10)


Since this increase is the same for any x value, this means that f(x) is a linear function.


So to find the equation, all we need are two points.


So let's find the equation through the points (0,6) and (1,10)


First let's find the slope of the line through the points and


Note: is the first point and is the second point .


Start with the slope formula.


Plug in , , , and


Subtract from to get


Subtract from to get


Reduce


So the slope of the line that goes through the points and is


Now let's use the point slope formula:


Start with the point slope formula


Plug in , , and


Distribute


Multiply


Add 6 to both sides.


Combine like terms.


So the equation that goes through the points and is


This means that the function is


Now if we pick a random x value that we haven't used yet, say x=2, and plug it in, we get:




So when , which supports our answer (try other x values to confirm).

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To start things off, let's draw three blank overlapping circles and label them "car", "refrigerator", and "washer". Now draw a large rectangle that surrounds the three circles and label this rectangle U (for the universal set).




Since "Three consumers indicated that they were going to buy all three items", this means that the value "3" goes in the direct center of the overlapping circles.


Now because "9 were going to buy a washer and a refrigerator", this means that some of those 9 people might have bought all three. So to find the ones who ONLY bought a washer and a refrigerator (not all three), we need to subtract 3 from 9 to get: . So this tells us that 6 people bought a washer and a refrigerator (and not all three). So place this value in the overlapping region between the circles labeled "refrigerator" and "washer".

Also, since "15 were going to buy a car and a washer" and we just want to find those who ONLY bought a a car and a washer (not all three), we just subtract 3 from 15 to get . So 12 people bought a car and a washer (not all three). Now place this value in the region between the "car" and "washer" circles.

Since "7 were going t o buy both a car and a refrigerator", and we're only interested in those who ONLY bought the two (not all three), we subtract 3 from 7 to get . So there are 4 people who bought a car and a refrigerator (not all three). Stick this value in the region between the "car" and "refrigerator" circles.


So we should have the following so far:




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


Now because "34 said they were going to buy a new washer" and we're only interested in the number who ONLY bought a washer (not a car or refrigerator), we need to subtract the number who bought all three (3 people), the number who bought a car and washer (12 people), and the number of people who bought a refrigerator and washer (6 people) from 34 to get . So there are 13 people who ONLY bought a washer. Place this value in the "washer" circle (and not in any overlapping regions)


If the method above is a bit cumbersome, then let's try another method in finding the number of people who ONLY bought a car:

Since 12 people bought a car and a washer, 4 people bought a car and a refrigerator, and 3 people bought all three, this means that people bought AT LEAST 2 items (where one of the items is a car). Since we ONLY want the car, we need to subtract this from 33 (the number people who bought a car and maybe something else) to get: . So 15 people ONLY bought a car. Place this value in the "car" circle (and not in any overlapping regions)


Because 4 people bought a refrigerator and a car, 6 people bought a refrigerator and a washer, and 3 bought all three, this means that people bought AT LEAST two items in which one item is a refrigerator. Subtract this value from 18 (the number who bought a refrigerator and maybe something more) to get . So 5 people bought ONLY a refrigerator. Place this value in the "refrigerator" circle (and not in any overlapping regions)


Finally, add up EVERY element that is in every region to get: . So 58 people bought AT LEAST one item (either a car, refrigerator, or washer). Subtract this value from the number who participated (100 people) to get . So 42 people did NOT buy any of the items listed. Place this value outside the three circles but inside the set U (the rectangle).


So we should now have the completed Venn diagram:






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

Now let's answer the questions:

a)
How many were going to buy only a car?

Looking at the diagram (and going over our previous work), we get 15 people only bought a car.

--------

b)
How many were going to buy only a washer?

Similarly, we see from the Venn diagram that 13 people only bought a washer.

-------
c)
How many were going to buy only a refrigerator?

From the diagram, we see that 5 people only bought a refrigerator.

------
d) How many were going to buy a car and a washer but not a refrigerator?

From the diagram, we see that the number that lies in the region between the "car" and "washer" is 12. So 12 people bought a car and a washer but not a refrigerator.

-----
e) How many were going to buy none of these items?

Since the value 42 is outside all of the circles, this means that 42 people did not buy any of these items.

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Start with the given inequality.


Add to both sides.


Combine like terms on the right side.


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

Answer:

So the answer is



So the answer in interval notation is [)


Also, the answer in set-builder notation is


Here's the graph of the solution set

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P(heads AND rolling a 4) = P(heads) * P(rolling a 4)



P(heads AND rolling a 4) = (1/2) * (1/6)


P(heads AND rolling a 4) = (1*1)/(2*6)


P(heads AND rolling a 4) = 1/12

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Recall, the average of a set of numbers is the sum of those numbers divided by the number of values being added.


So the average of a,b,c, and d is

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Start with the given system of equations:



Add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms.


Simplify.


Divide both sides by to isolate .


Reduce.


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


Now go back to the first equation.


Plug in .


Multiply.


Subtract from both sides.


Combine like terms on the right side.


So the solutions are and .


Which form the ordered pair . So the answer is the second choice.


This means that the system is consistent and independent.


Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer.


Graph of (red) and (green)

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Start with the given expression.


Factor to get .


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words, where

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Start with the given system of equations:



Multiply the both sides of the second equation by 3.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms.


Simplify.


Divide both sides by to isolate .


Reduce.


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


Now go back to the first equation.


Plug in .


Multiply.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So the solutions are and .


Which form the ordered pair .


This means that the system is consistent and independent.


Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer.


Graph of (red) and (green)

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If "A" and "B" are two binomials which add to a third binomial "C", then this means that . If we know what "A" and "C" are, then we can subtract "A" from both sides to solve for "B" like so:





So in this case, , , and "B" is our unknown binomial

Start with the given equation.


Plug in and


Distribute


Combine like terms.


So the unknown binomial is

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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,

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Start with the given equation.


Multiply both sides by "c".


Divide both sides by "a".


So the solution is

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Start with the given expression

To expand this, we're going to use binomial expansion. So let's look at Pascal's triangle:
1   
1   1   
1   2   1   
1   3   3   1   
1   4   6   4   1   



Looking at the row that starts with 1,4, etc, we can see that this row has the numbers:

1, 4, 6, 4, and 1

These numbers will be the coefficients of our expansion. So to expand , simply follow this procedure:
Write the first coefficient. Multiply that coefficient with the first binomial term and then the second binomial term . Repeat this until all of the coefficients have been written.

Once that has been done, add up the terms like this:


Notice how the coefficients are in front of each term.



However, we're not done yet.


Looking at the first term , raise to the 4th power and raise to the 0th power.

Looking at the second term raise to the 3rd power and raise to the 1st power.

Continue this until you reach the final term.


Notice how the exponents of are stepping down and the exponents of are stepping up.


So the fully expanded expression should now look like this:





Distribute the exponents


Multiply


Multiply the terms with their coefficients


So expands and simplifies to .


In other words,

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Start with the given equation.


Subtract from both sides.


Combine like terms on the right side.


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

Answer:

So the solution is

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Note: 45 minutes = 0.75 hours


Start with the distance-rate-time formula.


Plug in and


Multiply


So the train travels 52.5 miles.

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"A first number plus twice a second number is 5. Twice the first number plus the second totals 4" translates to the system:



Start with the given system of equations:



Multiply the both sides of the first equation by -2.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms.


Simplify.


Divide both sides by to isolate .


Reduce.


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


Now go back to the first equation.


Plug in .


Multiply.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So the solutions are and .


which means that the first number is 1 and the second number is 2

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Simply graph every inequality on the xy plane. These graphs will form a closed polygon. It turns out that the values that will optimize "c" will be the x and y values that are at the vertices. So simply plug in the x and y values at the vertices to see which value yields the largest/smallest value of "c". Let me know if this makes sense.

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Start with the given expression.


Expand. Remember something like .


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: , , ,


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


Now combine like terms.


So FOILs to .


In other words, .

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Start with the given equation.


Plug in


Multiply -0.5 and 8 to get -4


Evaluate (by use of a calculator) to approximately get


Subtract


Multiply


So the concentration is approximately 98.168.


I'm assuming that we're talking about a percentage. If so, then there's about 98.168% of the drug left over after 8 hours.

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First sort the numbers into ascending order (from least to greatest):
0.2,0.2,0.3,0.4,0.6,0.6,0.9,1.1,1.2,1.3,1.4


Mean:

To find the mean, add up all of the numbers and divide the sum by the number of numbers (which in this case is 11).


So the mean is roughly 0.745454545454545 which rounds to 0.75 (to the nearest hundredth)

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


Median:

To find the median, count off about 5 spaces(it's close to half of the size of the list) until you hit the middle number. Now count count off about 5 spaces from the end of the list until you hit the middle.
0.2, 0.2, 0.3, 0.4, 0.6, , 0.9, 1.1, 1.2, 1.3, 1.4
Median=0.6

So the median is 0.6


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

Mode:

To find the mode, simply look for any repeating numbers. If there is one number that occurs more frequently than any other number, then that number is the mode.

In this case, the numbers 0.6 and 0.2 repeat themselves 2 times.

So the mode is 0.6 and 0.2

Note: you can have more than one mode

---

Range:

To find the range, subtract the smallest number (in this case 0.2) from the largest number (in this case 1.4) like this:




So the range is 1.2

---

Interquartile Range:


First, find the median for the first half of the data set (ignore the median):


Median of 0.2, 0.2, 0.3, 0.4, 0.6 is 0.3 (since this is the middle most value)


Now find the median of the last half of the data set (ignore the median):


Median of 0.9, 1.1, 1.2, 1.3, 1.4 is 1.2 (since this is the middle most value)


So the 1st and 3rd quartiles are 0.3 and 1.2 respectively.


This means that the interquartile range is

1.2 - 0.3 = 0.9


Note: the interquartile range is the difference between the 1st and 3rd quartiles

---

Note: I'm assuming that you want the sample standard deviation.


Standard Deviation:


Step 1) Find the mean:






So the mean is roughly 0.75 (rounded to the nearest hundredth).


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


Step 2) Find the differences between the mean and every value:


Now subtract EVERY value from the mean (0.75) to get:
0.2-0.75 = -0.55, 0.2-0.75 = -0.55, 0.3-0.75 = -0.45, 0.4-0.75 = -0.35, 0.6-0.75 = -0.15, 0.6-0.75 = -0.15, 0.9-0.75 = 0.15, 1.1-0.75 = 0.35, 1.2-0.75 = 0.45, 1.3-0.75 = 0.55, 1.4-0.75 = 0.65,


So you should have a new column of values:
-0.55, -0.55, -0.45, -0.35, -0.15, -0.15, 0.15, 0.35, 0.45, 0.55, 0.65,
----------------------------------------------------------


Step 3) Squaring the differences:


Now square EVERY value in the last column to get:
(-0.55)^2 = 0.3, (-0.55)^2 = 0.3, (-0.45)^2 = 0.2, (-0.35)^2 = 0.12, (-0.15)^2 = 0.02, (-0.15)^2 = 0.02, (0.15)^2 = 0.02, (0.35)^2 = 0.12, (0.45)^2 = 0.2, (0.55)^2 = 0.3, (0.65)^2 = 0.42,


So you should have a new column of values:
0.3, 0.3, 0.2, 0.12, 0.02, 0.02, 0.02, 0.12, 0.2, 0.3, 0.42,
----------------------------------------------------------


Step 4) Adding up the squares:


Now add up the values of the last column:
0.3 + 0.3 + 0.2 + 0.12 + 0.02 + 0.02 + 0.02 + 0.12 + 0.2 + 0.3 + 0.42 = 2.02

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

Step 5) Divide the sum by n-1:


Divide that sum by one less than the number of values in the list (in this case, it is n-1=11-1=10).





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


Step 6) Take the square root of the quotient:


Take the square root of to get


Now round to the nearest hundredth to get .


So the standard deviation is roughly .

---

If you don't like that method, then here's another way:


First sort the numbers into ascending order (from least to greatest):
0.2,0.2,0.3,0.4,0.6,0.6,0.9,1.1,1.2,1.3,1.4


Standard Deviation: where is the average, is the ith number, and is the number of numbers

So we can replace N with 11



Subtract to get 10



Replace with 0.75



Expand the summation (replace each with the respective number)




Subtract the terms in the parenthesis



Square each term



Add up all of the terms



Multiply



Take the square root




Round to the nearest hundredth




So the standard deviation is roughly

---

25th and 75th percentiles:

The 25th and 75th percentiles are simply the first and third quartiles. We found these earlier to be 0.3 and 1.2.



So the 25th and 75th percentiles are 0.3 and 1.2

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Start with the given equation.


Take the square root of both sides.


or Break up the "plus/minus" to form two equations.


or Simplify the square root.


or Add to both sides.


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


Answer:


So the solutions are or .

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Start with the given equation.


Notice we have a quadratic 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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Start with the given equation.


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 .


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

Answer:

So the solution is which approximates to .

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... Start with the given expression.


... Convert to radical notation.


... Break up the root using the identity


... Rewrite -27 as


... Rewrite as


... Take the cube root of to get -3


... Take the cube root of to get



So

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Ignore the other solution, I don't know what that is...




Start with the given system of equations:



Multiply the both sides of the first equation by 3.


Distribute and multiply.


Multiply the both sides of the second equation by -2.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms.


Simplify.


Divide both sides by to isolate .


Reduce.


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


Now go back to the first equation.


Plug in .


Multiply.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So the solutions are and .


Which form the ordered pair .


This means that the system is consistent and independent.


Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer.


Graph of (red) and (green)

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Start with the given inequality.


Subtract from both sides.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

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# 1



Start with the given equation.


Get all terms to the left side.


Notice we have a quadratic in the form of where , , and


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


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 the solutions are or

---

# 2

Start with the given equation.


Multiply EVERY term by 10 to make every number a whole number.


Notice we have a quadratic in the form of where , , and


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


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.


So the solutions are or

---

# 3

Are you looking for the value of "c" that will make a perfect square??? There are no other instructions?

If so, then take half of 10 to get 5. Square 5 to get 25. This means that


So the expression is which is a perfect square.

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The multiplicative inverse of is which simplifies to (simply flip the fraction)

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Let w = speed of the wind (in miles per hour)


Note: if a plane's speed in still air is 180 mph, then the speed with the wind "w" is 180+w (since going with the wind speeds up the plane) and the speed against the wind is 180-w (since going against the wind slows the plane down)


Start with the distance-rate-time formula


Plug in and (see above)


Divide both sides by 180+w.


Rearrange the equation


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




Go back to the distance-rate-time formula


Plug in and (see note above)


Since the times are equal, this means that we can plug in


Multiply both sides by 180+w.


Rearrange the terms.


Distribute.


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 solution is which means that the speed of the wind is 30 mph.