What is | x/(x-1)|<= 1/4 ?

The solution to | x/(x-1)|<= 1/4 is -1/3 <= x<= 1/5

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| x/(x-1)|<= 1/4

\frac{x}{x - 1} \leq \frac{1}{4}

- \frac{1}{3} \leq x \leq \frac{1}{5}

Interval Notation

[- \frac{1}{3}, \frac{1}{5}]

Decimal Notation

- 0.33333 \dots \leq x \leq 0.2

Solution steps

\frac{x}{x - 1} \leq \frac{1}{4}

Apply absolute rule: If

∣ u ∣ \leq a, a > 0

then

- a \leq u \leq a

- \frac{1}{4} \leq \frac{x}{x - 1} \leq \frac{1}{4}

\frac{x}{x - 1} \geq - \frac{1}{4} and \frac{x}{x - 1} \leq \frac{1}{4}

(x \leq \frac{1}{5} or x > 1) and - \frac{1}{3} \leq x < 1

Merge Overlapping Intervals

- \frac{1}{3} \leq x \leq \frac{1}{5}

\frac{d ^{2}}{3} - 4 = 0

x^{2} - 12 x + 32 = 0

(- \frac{1}{( 1 + 6 )}) - (- \frac{1}{( 1 + 1 )})

- 8 = \frac{2}{5} u

x^{2} - 9 x - 90