Question

How do you solve `2(3x-6)=3(2x-4)`?

Answer 1

There is no solution for `x` as it cancels out.

`color(red)("Is the question correct?")`

Explanation:

Multiply out the brackets

`6x-12=6x-12`

Add 12 to both sides

`6x=6x` `color(purple)(larr" Which is true but of no use!")`

Answer 2

`x in (-oo, oo)`

Explanation:

Given:

`2(3x-6) = 3(2x-4)`

Expand both sides to get:

`6x-12 = 6x-12`

This is true for any value of `x`, so the solution space is the whole of the Real numbers, in interval notation `(-oo, oo)`

Or did you simply want to prove the equality?

We can do that like this:

`2(3x-6) = (2*3x)-(2*6) = 6x-12 = (3*2x)-(3*4) = 3(2x-4)`

Answer 3

`x` can have any value. `x in RR`

Explanation:

`2(3x-6)= 3(2x-4)`

`6x-12 = 6x-12" "larr` both sides are the same?

`6x -6x = -12+12" "larr` follow the normal process

`0 = 0" "larr` This is TRUE, but what is `x`??

We have ended up with a true statement but there is no `x` left to solve for?

This is a special kind of equation called an identity which will be true for every value of `x`. `x` can be any Real number.

This is indicated by the TRUE statement we get.
This can be in the form `x=x`, `12=12` or similar.

The answer is therefore "x can have any value"

There are 4 different types of solutions which we can get to an equation.

`x = "a specific number"` (or numbers) as the solution(s). These are the normal equations - linear, quadratic, cubic etc.

`x = 0`
(Indicated by equations such as 23x = 12x, which at first glance seem impossible, because different quantities of x are equal to each other.) The only possible solution is if `x=0`

`x " can have any value " x in RR`
This is indicated by answers such as `0=0, " or " 8=8"` which are TRUE statements but there is no `x` term.

`x " has no value/ is undefined"`
This result is indicated by an answer such as `0=8` which is a FALSE statement but there is no `x` term.