Question

Why can't we do crossing over in inequalities?

If we have the inequality `12/(x-4)<=4`, why can't we multiply both sides by `x-4`?

Answer 1

Because you might be mutliplying both sides by a negative quantity.

Explanation:

Given any inequality, some operations that you can perform and preserve the truth or falsity of the inequality are:

• Add or subtract the same expression on both sides.
• Multiply or divide both sides by a positive expression.
• Multiply or divide both sides by a negative expression and reverse the inequality (i.e. `<` becomes `>`, `>=` becomes `<=`, etc.).
• Apply the same strictly monotonically increasing function to both sides.
• Apply the same strictly monotonically decreasing function to both sides and reverse the inequality.

Given:

`12/(x-4) <= 4`

we would like to get `(x-4)` out of the denominator, but multiplying both sides by `(x-4)` would result in the wrong inequality when `x < 4`.

One option is to note that `x = 4` is not in the solution space (since it results in division by `0`), then multiply both sides by `(x-4)^2` for the cases where `x != 4`, since when `x != 4` we have `(x-4)^2 > 0`.

We find:

`12(x-4) <= 4(x-4)^2 = 4(x^2-8x+16)`

Then we can divide both ends by `4` to get:

`3x-12 <= x^2-8x+16`

Then subtract `3x-12` from both sides to get:

`0 <= x^2-11x+28 = (x-4)(x-7)`

Note this inequality would be true for `x in (-oo, 4] uu [7, oo)`

but the point `x=4` is excluded.

So the solution of the original inequality is:

`x in (-oo, 4) uu [7, oo)`