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Thanks for trying, but I already solved it. It was 1/8.

Ummm I have another few questions:

1. What is the sum of all integer solutions to \(|n| < |n-3| < 9 \) ?

2. Find x, given that \(\dfrac{\sqrt{x}}{x\sqrt{3}+\sqrt{2}} = \dfrac{1}{2x\sqrt{6}+4}\)

Here's a logical (graphical) solution for the first one

Looking at the left side of the equation :

(x -5x + 12)^2 + 1  =

(12 - 4x)^2 + 1  =

144 - 96x + 16x^2 + 1 =

16x^2 - 96x + 145      (1)

This is a parabola that turns upward    with the x coordinate of the vertex at   - (-96)/ [ 2 * 16]  =

96 / 32  =  3

Plugging this value  back into (1) resulrs in yhe y coordinate of the  vertex = 

16 (3)^2  - 96(3)   + 145  =  1

So....the LOWEST point on this graph =  ( 3,1)

Meanwhile....the graph of  -abs (x)  reaches its HIGHEST point at (0,0)

Thus the graph of the left side of this equation never intersects the graph of the right side of the equation....therefore.....there are no real  solutions

CORRECTED ANSWER   !!!!

Here's the second one

log₃ 27sqrt (3)  = 

log₃ 27 + log₃ sqrt (3) = 

log ₃ 3^3  + log ₃ 3^(1/2) =

(3) log ₃ 3  + (1/2) log₃ 3  =

(3) (1) + (1/2) (1)  =

3 + 1/2  =

7/ 2

To find out how much money Kenny saved, we need to analyze the given information step by step.

Let's assume Kenny's total amount of money is represented by "x."

According to the problem, Kenny spent 1/4 of his money on a bag. This means he spent (1/4)x on the bag.

Next, we are told that Kenny spent $120 on a belt. So, the total amount of money he spent on the bag and belt is (1/4)x + $120.

We also know that Kenny saved the rest of his money after buying the bag and belt. Therefore, the amount he saved can be calculated as x - [(1/4)x + $120].

The problem states that Kenny spent twice as much money as what he saved. So, we can set up an equation:

(1/4)x + $120 = 2 * [x - ((1/4)x + $120)]

Now, let's solve this equation to find the value of x, which represents Kenny's total amount of money:

(1/4)x + $120 = 2x - (1/2)x - $240
Multiplying through by 4 to eliminate fractions:
x + $480 = 8x - 2x - $960
Combining like terms:
x + $480 = 6x - $960
Subtracting x from both sides:
$480 = 5x - $960
Adding $960 to both sides:
$1440 = 5x
Dividing both sides by 5:
$288 = x

Therefore, Kenny had a total of $288.

To find out how much money Kenny saved, we substitute this value back into our earlier expression:

Amount saved = x - [(1/4)x + $120]
Amount saved = $288 - [(1/4) * $288 + $120]
Amount saved = $288 - [$72 + $120]
Amount saved = $288 - $192
Amount saved = $96

Hence, Kenny saved $96.
 

Here's what I think :

Total possible outcomes =  6^4  = 1296

Sets that produce a product of  6

(1,1,1,6).....permute this = 4 P1  = 4 possible sets

(1, 1, 2, 3) .....permute this = 4P2  = 12 possible sets

Probability =   [ 4 + 12 ] / 1296 =  16 / 1296 =    1 / 81

Answering  this in  a more "mathematical" way

Let T be the total games to  be  played

Number  of games  played =  .75T

Number of games won so  far =  .42 ( .75) T = .315 T

Games left to be played =  .25T

Number of games needed to  win  to  get to .50T =   [ .50T - .315T]  = .185T

Pct of games left we need to  win  to  get to .50T =

.185T / .25T = .74 =   74%

Convert meters to  feet

[1 foot / .3048 m ] *  [1m ]  =    3.28 ft

So

12 m (3.28) = 39.36 ft

And 120m  =  393.60 ft

No, of gallons required  =  Area of wall / Area covered by one can of paint =

[ 39.36 * 393.60 ] / 150 = 103.28 cans......since we must buy "whole" cans....round up  to 104 cans

Here's what I think

Total possible teams = C(14, 6)  =  3003 teams

Total teams that include all three triplets =   choose the three triplets and then, from the remaining 11 people, choose any  3 of them =   C(11,3)  = 165  teams

Total teams  that don't include the three triplets =  3003 - 165  =  2838 teams

Let's say in total there are 100 games.

So far they played 75 games. 

42% of 75 is 31.5 games so the team has won 31.5 games so far and lost 43.5 (lets pretend 0.5 games is ok).

They have 25 games left, so 31.5 + x = 43.5 + (25-x)

Solving the equation, we get 2x-25=12 and x = 37/2 = 18.5.

The percentage would then be 18.5/25 = 185/250 = 37/50 = 74/100 = 74%. 

Hope this helped!

To make this easier, let us suppose that the total games to be played = 200

So far,  we have played  .75 (200) =  150 games

And we  have won  .42 (150) =  63 games

And we  must  win 100 to have the same number of wins as  losses

So....we need 100 - 63 =  37 more wins to  get to 100

And we have 50 games  left to play

So....we have to  win 37 out of 50 games  remaining to get to 100 wins

So

37/50  = 74/ 100  =  .74

So....we must  win 74% of the remaining games  to  have the same number of wins as losses

For the circumradius we have

( product of the sides)  / (4 * Area)

You should be able to  finish this one......

Inradius =   Area /  semi-perimeter

semiperimeter  (s)  =   ( 7 + 5 + 4) / 2  =  16 / 2  =  8  

Using Heron's Formula to  find the area =   sqrt [ s ( s -a) (s - b) (s -c) ]  where a,b,c  are the sides

So

Area = sqrt  [ 8 (8 -7) (8 -5) ( 8-4) ]  = sqrt [ 8 * 1 * 3 * 4 ] =  sqrt [ 96] =  sqrt [ 16 * 6] = 4sqrt (6)

So

inradius=   4sqrt (6) / 8  =   sqrt (6) / 2  

Juan =   c cheesecakes

Clarence = 3c - 6     cheesecakes

Darlene =   (3c - 6)/2 + 4 =   (3c - 6 + 8) / 2  =   (3c + 2) / 2 =   (3/2)c + 1    cheesecakes

Total number of  cheesecakes = 

c + 3c -6 + (3/2)c + 1  =     

4c + (3/2)c - 5  =

(8/2)c + (3/2)c - 5 =

(11/2)c - 5

Thank you !!!