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@OP: I'm not sure why you posted this as off-topic

@guest that answered: I'm not sure if you can see the LaTeX, but \overline{} means "repeating decimal", not fraction. (the command for fraction is \frac{}{} or \dfrac{}{} depending on what size you want it to be)

The answer begins here:

Because my LaTeX is not working, I will do my best to write the fractions and repeating decimals.

We want to convert this to a fraction:

     ––

0.510, which is 0.51010101010101010101010.....

To begin, let's call this fraction "x". Based on what we know:

1000x = 510.10101010...

10x = 5.101010101010...

Subtract one from the other:

990x = 505

Divide by 990:

x = 505/990 = 101/198

:)

Let us simplify this mess: \(5m-3>8m+21 \implies -3>3m+21 \implies -24 > 3m \implies -8>m\)

That may not have rendered correctly, so screenshot is attached just in case. (I simplified it down to -8>m, so if you want an integer value: -9 is the answer. If you want in decimal form, it is a number with a lot of decimal places which I shall leave up to you to figure out :) )

Hope this helps!

m can be at most 8.

(a,b) = (8/5,9/5).

f^{-1}(-3)+f^{-1}(0)+f^{-1}(3) = -8.

0.5/10= 5/100 so 1/20

Hello dp1806!

\(Let\ f(x) = \begin{cases} 3x + 5 &\text{if }x<-3\ then\ {\color{blue}f(x)\in \{\mathbb{R}<-4\}} \\ 7-4x&\text{if }x\ge -3\ then\ \color{blue}f(x)\in \{\mathbb{R}\le19\} \end{cases} \\ Find\ f(5) \)

           \(f(5)=3x+5=20\\ f(5)=7-4x=-13\)

  !

We can immediately see by inspection that when 2x - 3 = 0, both sides of the equation are zero.

This gives us our first soution:  2x - 3 =0  so  x = 3/2

Now divide through by 2x - 3 (which we can do for the other solutions where 2x - 3 is not zero).

We are left with (2x - 3)² = 4(2x - 3) - 3 

Let u = 2x - 3, substitute in to the above and rearrange to get:  u²  - 4u + 3 = 0

This quadratic factorises as (u - 1)(u - 3) = 0, giving u = 1 and u = 3

These, in turn, give:  2x - 3 = 1  means x = 2, and 2x - 3 = 3 means x = 3.

You will need to know what "slope intercept" form is to solve these probelems. Basically, they are the equation of a line in this form:

y = [slope]x + [y-intercept]

Therefore, for each line, we need to find the slope and the y-intercept.

Here are a few basic things you need to know before we continue:

What is a slope?

The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).

- Khan Academy

What is the y-intercept?

It is the y coordinate at the point where the line touches the y-axis.

How to tell if a slope is positive or negative?

Found these pictures on quora. The second one is a funny way to remeber:

1)

SLOPE:

To find this, let's find two points on the line: (2, 0) and (0, 3)

                             rise        -3

We see that the  -------- = ----      (AKA the difference of y over difference of x)

                             run         2

Y-INTERCEPT:
 

This is easy. It is on the graph. It basically is where the line touches the y-axis. This is at y = 3

THE EQUATION:

y = -3/2 x + 3

2)

SLOPE:

To find this, let's find two points on the line: (2, 2) and (0, 3)

                             rise        -1

We see that the  -------- = -----      

                             run         2

Y-INTERCEPT:
 

This is at y = 3

THE EQUATION:

y = -1/2 x + 3

3)

SLOPE:

To find this, let's find two points on the line: (3, 3) and (0, 1)

                             rise        3

We see that the  -------- = -----      

                             run         2

Y-INTERCEPT:
 

This is at y = 1

THE EQUATION:

y = 3/2 x + 1

:)

Probability = (1111, 1112, 1113, 1114, 1115, 1116, 1121, 1131, 1141, 1151, 1161, 1211, 1311, 1411, 1511, 1611, 2111, 3111, 4111, 5111, 6111)>Total = 21 * 6 = 126 / 6^4 =7 / 72

For lines like such to be parallel, the slope (the m) has to be parallel. We first need to arrange all of them into slope intercept form:

y = [slope]x + [y - intercept]

Line 1: y = -1/4 x + 4

already in the right form! Great 👍 

Line 2: x + 4y = 3

Move the "x" to the right side:

4y = 3 - x

4y = -x + 3

Divide by 4

y = -1/4 x + 3/4
 

Line 3: y = -1/2 x - 6

already in the right form! Great 👍

Line 4: y + 1 = -2(x - 5)

Distribute:

y + 1 = -2x + 10

Subtract 1:

y = -2x + 9

We see that the only two lines that have the same slope is line 1 and line 2.

Double check answer with graph:

Let x be the number of flowers with 3 petals and 30-x the number of flowers with 4 petals. We form an equation:

3x+4(30-x)=102

3x+120-4x=102

-x+120=102

-x=-18

x=18 3 petal flowers,

So there are 30-18=12 4 petal flowers.

OK Kiddo! This is the best I came up with. Maybe others will try it:

1 - 1/2 =1/2+(1/3*1/2)=2/3 - (1/4*2/3)=1/2+(1/5*1/2)=3/5 - (1/6*3/5) =1/2+(1/7*1/2)=4/7 - (1/8*4/7)=1/2+(1/9*1/2) =5/9 - (1/0*5/9) =1/2 - This is what is left in the first jar after 10 pourings!!

Instead of circles and whatnot, we can instead call this a "heptagon"

AKA this shape:

Anyways, we can reword this question: How many diagonals / sides does a 7-sided heptagon have?

Sure, you can go draw all the lines. I'll wait for you. But there is a quicker, mathematical way to do it.

We see that there are 7 dots to choose from. From these 7 points, how many more points can we use to choose the second dot? There are 7 - 1 = 8 points (because you can't your original point)

Now, we multiply:

7 x 8 = 56

But, remember, the order does not matter! This means that we need to divide by 2! to cancel out the repeats.

56/2! = 56/2 = 28 diagonals / chords       <------ answer that guest got!

have you finished drawing the lines yet? if i were you, i would not draw 28 lines!

:)

answered.

I think I have thought this through but I have not completed all the maths.

I let XYZ be circumscribed by a circle of radius 3  (I find it easier to use actual numbers)

area Triangle XGY = 0.5*3*3sin120 =(9sqrt3)/4

area Triangle X'GY' = 2/3 of that = (3sqrt3)/2

Now find the area of one little triangle.

the equation of XZ is  y=sqrt3x+3

point A has a y value of 1 so the x value is (-2sqrt3)/3    

Y' is the intersection of   x^2+y^2=4  and  y=1   so I think it is  (-sqrt3,1)

Distance AY'=  sqrt3/3

So from that you can work out the area of each of the little triangles.

Then you can work out the rest.

(maybe there is a better way, but that will work)

You will need to check all my logic and arithmetic.

Instead of using words to think about it, let's use variables and numbers!

What the problem is asking is: For what numbers 1 - 100 is n^2 - 1 prime?

First, we need to know what n^2 - 1 is. This is (n-1)(n+1). (Difference of squares?)

We see that for it to be a prime number, n - 1 has to be 1. The only number that this works is 2.

So the answer is 1/100

Guest. Do you understand now?

:)

The number of possible chords is 28.

c = 13.

The answer is x = 12.

Haha yes! I was really stuck on what errors I had made....I just realized that I got the diameter of the small circle wrong, thanks to asinus! 

Nice work asinus and guest :)

Oh dear ilorty, you have made a couple of basic errors   

sqrt 50 is   only a half of the diagonal of the square.  So it is in fact the radius of the big circle

Also half of sqrt(50) is definitely not sqrt(25)

Never mind, it happens to us all.    

You can draw the perpendiculars from the center of a circle to the nearest sides of the square.  Then the length of that small square's diagonal would be the radius times the square root of two.  Do the same with the other circle, then add those two diagonals and the two radii between them, and that gives you the diagonal of the large square in terms of r.  Divide that by the square root of two and you have the side of the square in terms of r. 

Then you can express the area of the square in terms of r and the area of the circles in terms of r, and subtract the latter from the former. 

_.