What is 1-|(x-2)/x |<-3 ?

The solution to 1-|(x-2)/x |<-3 is -2/3 <x<0\lor 0<x< 2/5

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1-|(x-2)/x |<-3

1 - \frac{x - 2}{x} < - 3

- \frac{2}{3} < x < 0 or 0 < x < \frac{2}{5}

Interval Notation

(- \frac{2}{3}, 0) \cup (0, \frac{2}{5})

Decimal Notation

- 0.66666 \dots < x < 0 or 0 < x < 0.4

Solution steps

1 - \frac{x - 2}{x} < - 3

Move

1

to the right side

- \frac{x - 2}{x} < - 4

Multiply both sides by

- 1

\frac{x - 2}{x} > 4

Apply absolute rule: If

∣ u ∣ > a, a > 0

then

u < - a

u > a

\frac{x - 2}{x} < - 4 or \frac{x - 2}{x} > 4

\frac{x - 2}{x} < - 4 or \frac{x - 2}{x} > 4

- \frac{2}{3} < x < 0 or 0 < x < \frac{2}{5}

∣ x + 1 ∣ > x - 3

∣ 2 x - 3 ∣ < x + 1

x^{2} - 4 x < 5, ∣ x + 1 ∣ < 3

\frac{( 3 x )}{4 - 2} < 8

∣ 19 + 3 x ∣ \geq 46

What is 1-|(x-2)/x |<-3 ?
The solution to 1-|(x-2)/x |<-3 is -2/3 <x<0\lor 0<x< 2/5