5lb Manhattan Chapter 9 Q20


how to approach such question?

p -|p| > k -|k| which tell us that p is positive and k is negative. So A

could you explain with an example?

Don’t worry, his reasoning is wrong. Could you show your attempt instead?

As a hint, consider the definition of an absolute value function:

|x| = \begin{cases}x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Now consider 4 cases in tandem with the above definition:

  1. p \geq 0 and k \geq 0

  2. p \geq 0 and k < 0

  3. p < 0 and k \geq 0

  4. p < 0 and k < 0

Why it is wrong? can’t we rearrange the inequality?

Why is it wrong?

Let p = -3 and k = -5, then you have that -6 > -10, which is a true statement. Clearly, p isn’t positive as you mentioned above.

Can’t we rearrange inequality

Yes you can, but your implied conclusion after doing that wasn’t right.