Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

**Wch12267****Guest**

Plz tell me a way to solve this:

Determine the largest real value of o such that the inequality sqrt(1-o)-sqrt(1+o)is greater than or equal to 1 holds

I am a grade 8 student, plz help! Thx!

**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,462

hi Wch12267

Welcome to the forum.

I'm going to use x rather than o here to avoid confusion with zero.

root(1-x) cannot have a real value if x is positive as the square root of a negative doesn't exist in real numbers.

So I did a bit of trial and improvement. I set up Excel with 11 lines of formulas, testing from -1 to 0 in steps -0.9, -0.8 and so on. I could see a solution lay between -0.9 and -0.8 to I repeated the search between these and gradually 'homed in' on -0.866. That looked to me like root(3)/2 so I tried that directly and the expression evaluated to 1.

Now Excel only works with 'so many' digits of accuracy so this isn't a proof of the solution, but it gave me enough to switch to an algebraic method. What I'm about to show you has to be used with caution as it can lead to values that aren't solutions as well as values that are. Let me show you why:

Suppose we have a simple equation like x = 5. That's easy to 'solve' ; the answer is 5.

But if I square it: x^2 = 25 ; this has 2 solutions, x = 5 and x = -5. The second isn't a solution of my original equation! So if you ever use this technique, always check the values you end up with, to make sure they really do solve the original problem.

I'll replace the inequality with an equals.

square both sides

Simplify and rearrange

Square again:

The 'other' value can be disregarded as it is positive and I've already discounted that.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob Bundy

Offline