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The rule that I use is not the way most people teaching math use, but I like it. Whenever you have an equation with a
POSITIVE coefficient of y > or >= , you must shade ABOVE the line.
POSITIVE coefficient of y < or <= , you must shade BELOW the line.

That's as simple as it gets! Examples:
2x+3y < 6 Shade Below
2x+3y > 6 Shade Above
2x+3y >= 6 Shade Above
2x+3y <= 6 Shade Below

So, what if you have a NEGATIVE COEFFICIENT of Y? Well, if you multiply both sides of an inequality by a negative, this reverses the inequality, and therefore it REVERSES the RULE!!
2x - 3y < 6 same as -2x+3y> -6 Shade Above!!

See if that helps. If you need more, I might be able to send you a section from my book in a word file.

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The concept is the absolute value of x.

The distance of any number x from zero on a number line is |x|.

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1+3+5+7+9+....499
Before you begin, notice that you are adding numbers from 1 to 499, omitting the even numbers in between, so there will be a total of 250 numbers that are added together.

Pair these numbers up by adding 1+499, 3+497, 5+495, etc.

Notice that each sum is 500, and the number of such pairs will be half of the 250 numbers that are added together, or 125. Therefore, the sum will be 500 times 125 = 62500.

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This is one of my favorite questions, and I answer it every day! If you have an EQUATION, then the multiplication property of equations allows you to multiply BOTH SIDES of an EQUATION by the same NON-ZERO number. However, if you have a rational EXPRESSION, then you do NOT have an equation, so the properties of EQUATIONS do NOT apply! Moreover, you can't solve an expression. About all you can do with expressions is add, subtract, multiply, divide, simplify or factor them.

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The equation for this one will be somewhat complicated to solve, but the answer is easy to find. Think about this: Three consecutive numbers whose product is 24. It's a pretty small number, so start small and use trial and error. Start with 2 for the first number. This makes the second number 3, and the third number is 4. Does that work? 2*3*4 = 24. Bingo!!

The equation is to let x = first, x+1= second, and x+2= third.
The equation is x(x+1)(x+2)= 24. This gets ugly fast. If you multiply it out you will get , which is a tough equation to solve without higher math. If that is what you need, then resubmit the question, and someone will solve it for you.

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Let x = 0 and solve for y.

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First to change km to m, multiply by 1000, and to change from m to cm, you must multiply by 100. In other words multiply by 100,000. To change from a time per hour to a time per second, you must divide by 3600 (that is, 60 minutes in an hour and 60 seconds in a minute).

Final answer then is to do take = 666.67 cm/sec.

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If there are 26 problems, let x = number right, and therefore the number wrong will be all the rest, which is 26-x. The equation is due to the fact that 8 times the number right minus 5 times the number wrong equals zero.

8x - 5(26-x) = 0
8x - 130 + 5x = 0
13x - 130=0
13x -130 + 130 = 0+130
13x = 130
x= 10 Questions right
26-10 = 16 Questions wrong

Check: $8*10 - $5*16 = 0

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The probability of having a boy is 1/2, and the probability of having a girl is also 1/2.

To find the probability of having successive children of a particular gender, assuming that the birth of each child is independent of every other child is the product of the probabilities. Suppose the first child born determines the gender, boy or girl, of all the other children. Therefore, if there are 8 children, the probability that the succeeding SEVEN children will match the gender of the first child will be

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First remove the parentheses. Don't forget the negative sign on the 9:
6y-(3y+9)=17
6y - 3y - 9 = 17
3y - 9 = 17

Next add + 9 to each side:
3y- 9 + 9 = 17 + 9
3y = 26

Divide both sides by 26.



It doesn't come out even, but then life doesn't always come out even, does it!

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To find the square root of a fraction, you can take the square root of the numerator divided by the square root of the denominator:





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The equation x+y=4 is in what is called "standard form" where you have an x term plus (or minus) a y term = a number term. When this happens, I think the easiest way to graph it (usually) is to graph the two easiest points that you can, the x and y intercepts. The first points is found by letting x= 0 and find y (that would be the y-intercept!). Second point is found by letting y=0 and find x (that would be the x-intercept!).

In this case, if x = 0, y = 4.
If y=0, then x = 4.

Plot these two points: (0,4) and (4,0). Connect the points with a line, and continue the line in both directions. It should look like this:


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I'm not familiar with the THEA guide book, but when you are graphing an equation like 2x + 3y = 6, I can't imagine why you would want to solve for x. Solving for y is a nice idea, but I don't think that is the easiest way to graph this equation.

Nevertheless, solve for y by adding -2x from both sides:
2x + 3y = 6
2x-2x + 3y = -2x + 6
3y = -2x + 6

Next, divide both sides by 3 in order to solve for y.
3y= -2x + 6



What this means is the the y intercept is 2, and the slope (steepness!) of the line is -2/3.

Start with an xy graph, and go up two units on the y-axis and put a point. Next, with your pencil on this point on the y axis, measure off the slope which is a rise of -2 followed by a run of 3. This means from the point you made at y=2 on the y-axis, count DOWN 2 units, and go RIGHT 3 units, and put another point. Then connect the points with a line, and extend the line in both directions. The result should look like this:


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Oh, I get it! Now we are playing Jeopardy! The answer is 2. What was the question??

"What is the solution to the equation 4y + 16 = 24?"

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Seriously, you probably wanted to graph the equation y=2. Anytime you graph y= a number (with no x in the equation). It is a horizontal line. So on an xy graph, go up two units on the y-axis, and draw a horizontal line. That's it!

If you ever have an equation like x = 2 (with no y in the equation), then you have a vertical line.

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What you describe is called a "cube root": . It means you are asking the question, "What number would you raise to the third power in order to get 64?" Before doing cube roots, you should make a list of what we call the perfect cubes:







Be very familiar with these numbers, and you will be able to do cube roots. Now to answer the question, "What number raised to the third power would give you 64?" Answer is of course 4.



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First the LCD for this problem is . You must multiply the first fraction numerator and denominator by (x-3), and you must multiply the second denominator by x. Did you follow all of that?

.

.

.



I think that's about all you can do with it. You can rearrange the numerator, but it doesn't simplify anything, and it doesn't factor.

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Let V = volume of the smaller box. Since , where x is the length of the side of each square side, each side of this box will be , and the area of each face of the box will be .

Now, the larger box has twice the volume or 2V, where , where y is the length of the side of each square side. Each side of this box will be , and the area of each face of the box will be .

Notice that the larger box has area of each face that is similar to the face of the smaller of the box, except that it is multliplied by power. This computes to a surface area that is about 1.587 times as much as the smaller box, so it should take about 1.587 times as long to paint the box. Hey, it's late in Florida right now, and maybe someone will check this out and make sure I'm right!! Any volunteers??

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It looks right to me. Go ahead and evaluate v(3). I got $12,900. What did you get?

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In my very last solution that I posted not 5 minutes ago, this same concept came up. This is a right triangle whose hypotenuse is the 5 foot ladder. It reaches the wall 3 feet from the wall, which is the first leg, and the second leg is the unknown x, which is the distance up the wall where the ladder reaches at the top.



If you are familiar with the special right triangle whose sides are 3, 4, 5, or any multiple thereof, then you recognize that the unknown side is 4 feet.

If you are NOT familiar with these special triangles, then you solve the equation



x=4 feet.

Either way, the unknown side of the right triangle is 4 feet (up the wall!).

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The board forms a right triangle, where the board itself is the hypotenuse, and the legs will be the sides of the right triangle which are 6 and x, the unknown side on the floor between where the board reaches the floor and the wall.



If you are familiar with the special right triangle whose sides are 3, 4, 5, or any multiple thereof, then you are familiar with a right triangle whose sides are double these numbers: 6, 8, 10, so the unknown side is 8 feet.

If you are NOT familiar with these special triangles, then you solve the equation



x=8 feet.

Either way, the unknown side of the right triangle is 8 feet.

Now, the slope is the rise over the run, where rise is 6 feet, and the run is 8 feet. Therefore the slope is


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The first step is to figure out the pattern. First look to see if there is a common number that you multiply the first number times to get the second, then the third, the fourth, etc. It doesn't look like there is a common factor that will do this, so it is not what we call a geometric sequence.

Second, see if there is a common number that can be added to get from the first number to the second, second to third, etc. Since there is no obvious common denominator, this does not appear to be the case.

So, see if there is a number that you can add to numerator and denominator that would make sense. Also, be aware that in the process of doing this, some of the fractions may have already been reduced to lowest terms. Try adding 1 to the numerator and denominator of 5/7. That would give you 6/8 which reduces to 3/4.

Continue by taking 6/8 and add 1 to the numerator and denominator. This gives 7/9 which is the third number.

Next, add 1 to numerator and denominator of 7/9 which would be 8/10 or 4/5. This is too close to what you have given as 5/4. I think maybe did you copy the problem wrong? If so, the PROBLEM should be
5/7, 3/4, 7/9, 4/5, , ,

The PATTERN would be the
5/7, 6/8, 7/9, 8/10, 9/11, and 9/12 or 3/4

And the SOLUTION would be the last two numbers in the sequence:
9/11 and 3/4.

If indeed you DID copy the problem correctly, then I am wrong. Please disregard this solution, and re-post the question, with my sincere apologies!!

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Looks good to me!!

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Congratulations on writing the fraction in algebra.com notation!

First step is to factor the numerator and denominator:




Next, notice that the factors (x-2) divide out in the numerator and denominator, leaving:



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Is this a right triangle? Assuming that it is a RIGHT triangle, where c is the hypotenuse, and a and b are legs,
, where in this case, the hypotenuse is the longest, which is the unknown, and a and b are 8 and 12 respectively.




or

c is the side of a triangle and therefore cannot be negative.
= feet

You might want to calculate this and round to nearest hundredth:
feet

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First of all, I'm not sure what you want to do with this. Secondly, I'm not sure exactly what you meant for the fraction. Do you mean just the fraction of 10 divided by n? Do you mean the fraction of 3n+10 divided by the quantity of n+17? It's not clear what you want. So, I'll take a shot at what I think you mean, which is exactly what it says:



Find a common denominator which is n, and multiply the first and last terms by .








Factor, just because it does factor!


Does that do it?

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Let me begin by saying that I can't use brackets in algebra.com equation notation, since it has a different meaning --it's used for subscripts. Therefore, where you called for brackets, I'll have to use parentheses.





Subtract from each side and add +6y to each side:





Divide both sides by -6, and you get
y = -1 Which would be answer D.

If you substitute the value of y=-1 back into the original equation, I think you get 15 = 15, which checks.

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If the toothbrush wears down to half of its length in 90 days linearly, then it will wear out the rest of the way in another 90 days, for a total of 180 days to wear completely out.

Since 2 weeks and 4 days is a total of 18 days, that would be 1/10 of the total life of the toothbrush. This means that 10% has been used up, and 90% remains. I would say the answer is E) 9/10.

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In the formula , let F = C, and solve:



Multiply both sides by 5:




So, -40 degrees F = -40 degrees C

Check:




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Divide out the 5 and you get = .


Divide out the 4 and it's = .

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If 1 banana = 2 apples
1 apple = 2 rupees

Then 1 banana = 4 rupees
3 bananas = 12 rupees.

So for 13 rupees you can get 3 bananas and have 1 rupee left over. Can you buy of a banana? If so, then you can buy 3 bananas!

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I have a page on my website that may be what you are looking for. This is the URL if it comes through as such:
http://www2.seminolestate.edu/rrapalje/Tech-Prep-Math/Metric%20System.htm
In case that doesn't show up as a URL that you can click on, go to my website by clicking on my user name anywhere in algebra.com, then look for a topic called "Tech Prep" in the second column near the bottom of the page. Click on that, and look for "Metric System", which is a rather long handout. Scroll down near the end of that "Metric System" page, and you will find a section devoted to "Medical Applications--Dosages". It might help.

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