Swaroopdev wrote:

Hi

Bunuel,

Could you please explain this step |x−5|=5−x? --> is x−5≤0? --> is x≤5?. I know for all values less than or equal to 5 holds good for this equation, however, how do we arrive to that final equation using |x−5|=5−x.

What i do understand is that absolute value of anything has two values. When i simplify the equation above i get x=5 (when i take positive value) and x-5=x-5 (when i take negative value).

Thanks.

Let me try to answer.

You know that \(\sqrt{x^2} = |x|\) ---> \(\sqrt{x^2} = \pm x\). Thus \(\sqrt{(x-5)^2} = |x-5|\) --->\(\sqrt{(x-5)^2} = \pm (x-5)\)....(1)

Additionally, you know that \(\sqrt {x} \geq 0\) (for GMAT purposes!)....(2)

Thus based on (1) and (2) above,

Coming back to your question, the questions asks whether \(\sqrt{(x-5)^2} = 5-x\). This can only be possible when \(x\leq 5\) as if x>5 then 5-x will become <0 and will go against (2) above. The most x can go is = 5....(3)

Also, from (1), \(\sqrt{x^2} = - x\) only when x<0 ...(4)

Thus from (3) and (4), you get the rephrase of the question asked as is \(x \leq 5\) ?

Hope this helps.