Question

How many solutions does the equation `-4(x+5) = -4x - 20` have?

Answer 1

If you count `0=0` as the solution then one, However. If you count the number of `x` values that are required to achieve this final value then there are an infinite number of solutions for `x`

Explanation:

Expanding the bracket gives:

`-4x-20=-4x-20`

`-4x+4x=20-20`

`0=0` Which is true

The after consulting with friend I have been advised that the following applies:

The outcome is always zero on both sides and thus always true. From this it means that despite this one outcome you can allocated any value to `x`. So there is a infinite number of solutions.

Answer 2

This equation has an (uncountable) infinity of solutions, since it is satisfied by any value of `x`.

Explanation:

One way to look at this is to start with an equation that you can see is satisfied by any value of `x`, then derive the given equation from it.

Start with:

`x = x`

Add `5` to both sides to get:

`x+5 = x+5`

Multiply both sides by `-4` to get:

`-4(x+5) = -4(x+5)`

Expand the right hand side using distributivity to get:

`-4(x+5) = -4x-20`

So this holds for any value of `x`

So there are infinitely many solutions (in fact uncountably infinitely many solutions).