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After looking at all that effort I and our guest have gone to I expect it would be a lot easier just to solve all the possibilities and then substitute them back in and discard the answers that don't work.

|3x-2|-1=|5-x|

So

1) solve   3x-2-1=5-x   then check the answer works

2) solve    3x-2-1=-5+x   then check the answer works

3) solve    -3x+2-1=5-x   then check the answer works

4) solve    -3x+2-1=-5+x   then check the answer works

These double absoutes are always long and tricky.

|3x-2|-1=|5-x|

There are 4 possibilities that might lead to different answers.

\(1)\quad If\quad3x-2\ge0\quad and \quad 5-x\ge 0\\ \qquad x\ge 2/3\qquad and \qquad x\le 5\\ \qquad \qquad 2/3\le x\le5\\ then\\ 3x-2-1=5-x\\ \qquad\;\; \quad4x=8\\ \qquad\; \qquad x=2\\ So \;\;x=2\;\;is\;\;a\;\;solution\\~\\ 2)\quad If\quad3x-2\le0\quad and \quad 5-x\le 0\\ \qquad x\le 2/3\qquad and \qquad x\ge 5\\ \qquad No\;\;solution\\ \)

\(3)\quad If\quad3x-2\ge0\quad and \quad 5-x\le 0\\ \qquad x\ge 2/3\qquad and \qquad x\ge 5\\ \qquad\;\text{The union is }\;\; x\ge 5\\ \qquad3x-2-1=-(5-x)\\ \qquad3x-3=-5+x\\ \qquad2x=-2\\ \qquad x=-1\\ \qquad \text {This is not greater than 5 so there is no solution here} \)

\(4)\quad If\quad3x-2\le0\quad and \quad 5-x\ge 0\\ \qquad x\le 2/3\qquad and \qquad x\le 5\\ \qquad\;\text{The union is }\;\; x\le 2/3\\ \qquad -3x+2-1=5-x\\ \qquad \qquad -2x=4\\ \qquad \qquad x=-2\\ \qquad \text {This is less than 2/3 so x=-2 is a second solution}\\~\\ \text{So the 2 solutions are } \;\;x=2 \;\;and \;\;x=-2 \)

Well, we have 8+(-8)+a.

8+(-8) simply becomes 8-8. The negative cancels the positive.

So we have 8-8+a, which is 0+a. And that is 'a.'

So the answer is 'a.'

Correct me if I'm wrong.

Thanks. I think I get it now.

\(a\,=\,b\,*\,\frac{c^2}{d^2} \\~\\ a\,=\,\frac{bc^2}{d^2}\)

                                    Multiply both sides of the equation by  d²  .

\(ad^2\,=\,bc^2\)

                                    Divide both sides of the equation by  a  .

\(d^2\,=\,\frac{bc^2}{a}\)

                                    Take the  ± square root  of both sides.

\(d\,=\,\pm\sqrt{\frac{bc^2}{a}}\)

Where is the problem? I don't see anything !!!.

Solve for x over the real numbers:
abs(2 - 3 x) = abs(5 - x) + 1

Split the equation into two possible cases:
2 - 3 x = abs(5 - x) + 1 or 2 - 3 x = -abs(5 - x) - 1

Subtract abs(5 - x) + 1 from both sides:
1 - 3 x - abs(5 - x) = 0 or 2 - 3 x = -abs(5 - x) - 1

Subtract 1 - 3 x from both sides:
-abs(5 - x) = 3 x - 1 or 2 - 3 x = -abs(5 - x) - 1

Multiply both sides by -1:
abs(5 - x) = 1 - 3 x or 2 - 3 x = -abs(5 - x) - 1

Split the equation into two possible cases:
5 - x = 1 - 3 x or 5 - x = 3 x - 1 or 2 - 3 x = -abs(5 - x) - 1

Subtract 5 - 3 x from both sides:
2 x = -4 or 5 - x = 3 x - 1 or 2 - 3 x = -abs(5 - x) - 1

Divide both sides by 2:
x = -2 or 5 - x = 3 x - 1 or 2 - 3 x = -abs(5 - x) - 1

Subtract 3 x + 5 from both sides:
x = -2 or -4 x = -6 or 2 - 3 x = -abs(5 - x) - 1

Divide both sides by -4:
x = -2 or x = 3/2 or 2 - 3 x = -abs(5 - x) - 1

Add abs(5 - x) + 1 to both sides:
x = -2 or x = 3/2 or 3 - 3 x + abs(5 - x) = 0

Subtract 3 - 3 x from both sides:
x = -2 or x = 3/2 or abs(5 - x) = 3 x - 3

Split the equation into two possible cases:
x = -2 or x = 3/2 or 5 - x = 3 x - 3 or 5 - x = 3 - 3 x

Subtract 3 x + 5 from both sides:
x = -2 or x = 3/2 or -4 x = -8 or 5 - x = 3 - 3 x

Divide both sides by -4:
x = -2 or x = 3/2 or x = 2 or 5 - x = 3 - 3 x

Subtract 5 - 3 x from both sides:
x = -2 or x = 3/2 or x = 2 or 2 x = -2

Divide both sides by 2:
x = -2 or x = 3/2 or x = 2 or x = -1

x = -2    or       x = 2

\(\begin{array} &&1\frac12\,\times\,\frac32 \\~\\ =&(1+\frac12)\,\times\,\frac32 \\~\\ =&(\frac22+\frac12)\,\times\,\frac32 \\~\\ =&\frac32\,\times\,\frac32 \\~\\ =&\frac{3\,\times\,3}{2\,\times\,2} \\~\\ =&\frac94 \end{array}\)

y  = -10 / x

Domain  ...   all   x  except  x =   0

Range  ....    all   y  except  y =   0

Compare.....this is the graph of 1/x  reflected across the x axis and vertically stretched by a factor of 10

Here's the graph : https://www.desmos.com/calculator/sbfex2cui8

y  = 1  / [ x + 3]   + 3                             

Domain  ....all x  except  x  = -3

Range .....  all y except  y  = 3

Compare ........This  is the  graph  of 1/x  shifted to the left  3 units and up 3 units

Here's the graph : https://www.desmos.com/calculator/oesuut3smm

4/[4+4+7] =4/15 x 70 =$20 one donation

Same for ratio of 4 -The second donation =$20

7/[4+4+7] = 7/15 x 70 =$35 the third donation.

I think you mean 17 less than the sum of a number and 7  = 

( N + 7)   -  17          where N is the number

 (x^5+17x^6+16x+14)/(x^2+14)  =

                      17x^4  + x^3  - 238x^2  - 14x +  3332

x^2 + 14  [   17x^6  + x^5  + 0x^4  + 0 x^3  +   0x^2  +  16x + 14) ] 

                     17x^6  +        +238x^4

                     _________________________________________

                                   x^5  -  238x^4  + 0x^3

                                   x^5                   + 14x^3

                                  ____________________

                                            -238x^4  -  14x^3  + 0x^2

                                            -238x^4                -3332x^2

                                            ______________________

                                                            -14x^3 + 3332x^2  + 16x 

                                                            -14x^3                   -196x

                                                            ____________________

                                                                          3332x^2  + 212x + 14

                                                                          3332x^2              + 46648

                                                                          _____________________

                                                                                           212x - 46634

So...the result  is    17x^4  + x^3  - 238x^2  - 14x +  3332   with R  [ 212x - 46634 ] / [ x^2 + 14 ]

Note that we have

119 + 120 + 121 +  ..... +  218 + 219 + 220

We can rearrange this as

(119 + 220 )   + (120 + 219)  + ( 121 + 218) +  ........

Note that each pair sums to 339

And we have  [ 220 - 119 + 1] / 2  = 51 pairs

So....51 (339)  = 17,289

Note that (2¹) (2²) (2³) (2⁴)......(2⁹⁹)(2¹⁰⁰)   can be rearranged as

(2¹)(2¹⁰⁰) * ( 2²)(2⁹⁹)  * ( 2³)(2⁹⁸)  =

(2¹⁰¹)⁵⁰   =  

(2⁵⁰)¹⁰¹ =

( [(2^10)]^5)^101

Note that  2^10  ends in 24

And 24^(2n + 1)  ends in 24 for n = any whole number......so we can write

[( ......24)^5]^101  =

[ (......24)^(2(2) + 1) ]^101  =

[(.....24) ] ^(2(50) + 1)  =

24

And the product of the tens digit and  ones digit  = 8

It is 8

8^(-5)  / 8^3    = 1/8^5       X   1/8^3     add indices to get       1/8^8

or   8^(-5)  / 8^3   =  8^(-5)    X  8^(-3)       add indices   to get 8^(-8)    = 1/8^8,same thing.

Remember that  X^(-n)    = 1/ X^n
 

maths iz numbers my friend :3

Yeah, he’s related to Einstein – the same way a turkey is related to Einstein. 

Oh....thanks. 

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