TMUA Practice Question — Worked Solution

P2·Algebra & Surds·5.0Mock 0 · P2 · Q4

proof errorrational inequalitymultiplying by an expression of unknown signif and only ifinequalities

Consider the following claim about a real number

x

with

x > 0

and

x \neq = 1

\frac{1}{x - 1} < \frac{2}{x} if and only if x > 2.

A student offers the following proof of the claim.

(I)

Since

x > 0

, multiplying both sides of the inequality by

x

gives

\frac{x}{x - 1} < 2

(II)

Since

\frac{x}{x - 1} = \frac{( x - 1 ) + 1}{x - 1} = 1 + \frac{1}{x - 1}

, this is

1 + \frac{1}{x - 1} < 2

, that is,

\frac{1}{x - 1} < 1

(III)

Multiplying both sides by

x - 1

, which is not zero, gives

1 < x - 1

(IV)

That is,

x > 2

(V)

Each step above can be reversed, so for

x > 0

with

x \neq = 1

, the inequality

\frac{1}{x - 1} < \frac{2}{x}

holds exactly when

x > 2

Which one of the following is true?

Discussion

No discussions yet. Stuck, or want to share an approach? Start the first one.