Melody

What has subtraction got to do with mode? 

This is not an inequality - it is an equal sign so it is just an equation.

2-3x=4-x

2-3x  +x  =  4-x  +x

2-2x  =  4

-2         -2

-2x = 2

x=-1

check

LHS=2-3*-1=2+3=5

RHS=4- -1 = 5

LHS=RHS so the answer is definitely     x=-1

WELL THAT IS EXCELLENT!     

In a family with 7 children, excluding multiple births, what is the probability of having 7 girls? Assume that a girl is as likely as a boy at each birth.

1 / 2^7   = 1/128

There is another complex root but ok :)

find four consecutive integers such that five times the fourth minus the second is equal to the third

I am going to let the forth be x so the numbers will be

x-3, x-2, x-1, x

now

5x - (x-2) = x-1

5x - x + 2 = x-1

4x      + 2 = x-1

3x            =  -3

  x            =  -1

so the numbers are 

-4, -3, -2, and -1

check,

-5- -3 = -2   true  

Hi Hamil..

so how exactly would i "factor" an equation...?

WEll it reallt depends on what level of question you are talking about BUT

if I want to factorize  3x+9  I have to look for highest number or letter that goes into both.

3x=3*x

9  = 3*3

they have both got 3 for a factor so

3x+9

= 3*x +3*3

=3(x+3)

you can expand the bracket to check  that this is correct.  

So

very generally speaking - FACTORIZE  means PUT IN BRACKETS      

(-5SQRT(3)-5i)(2i)

\((-5\sqrt3-5i)(2i)\\ =2i(-5\sqrt3-5i)\\ =-10i\sqrt3-10*i*i\\ =-10i\sqrt3-10*-1\\ =-10i\sqrt3+10\\ =10-10\sqrt3\;i\\ \)

if x is real, how can x^2+1=0 ???    

I do not think that this is a question related to imaginary numbers!

How can you simplify the equation y = sin(atan(x))?

atan of some number is between -90 and +90 degrees   (by usual definition)

so sin of it can be positive or negative

Let's consider the case where the angle  is an acute angle

Consider (draw) a right angled triangle triangle and put theta as an acute angle

opp side is x

adj side is 1

so atan(x)=theta

and hypotenuse is  \(\sqrt{1+x^2}\)

so

    \(sin\theta = \frac{opp}{hyp}\\ sin\theta = \frac{x}{1+x^2}\\ sin(atan(x)) =\pm \; \frac{x}{1+x^2}\\\)