Question

Identify the inequality as having either infinitely many solutions or no solution.You may need to show some work to determine the solution.

Can someone please help me with b?I'm not sure if it's supposed to be infinitely many solutions or no solution.​

Answer 1

The writing in red is correct. The inequality sign flips when we divide both sides by a negative value.

To see why the sign flips, think of it like this:

`-2|x-3| > 0\\\\-2y > 0\\\\-2y+2y > 0+2y\\\\0y > 2y\\\\0 > 2y\\\\2y < 0\\\\2|x-3| < 0\\\\`

Where
`y = |x-3|` is used to condense things a bit.

In short, we go from
`-2|x-3| > 0\\\\` to
`2|x-3| < 0\\\\` and we have a sign flip.

The '2' is never negative, and same goes for
`|x-3|`. So overall, the entire left hand side is never negative. So there are no solutions to
`2|x-3| < 0`which further means there are no solutions to
`-2|x-3| > 0`

Put another way, the original inequality has its left side always as some negative number (assuming x is not equal to 3). There's no way to have a negative number larger than 0. So we have a contradiction leading to no solutions.

Answer: There are no solutions