Solve |(2x 1)/(x-1)|≥3

Evaluate

\frac{2 x \times 1}{x - 1} \geq 3

Find the domain

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\frac{2 x \times 1}{x - 1} \geq 3, x \neq = 1

Simplify

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

Calculate

\frac{2 ∣ x ∣ - 3 ∣ x - 1 ∣}{∣ x - 1 ∣} \geq 0

Separate the inequality into 2 possible cases

{2 ∣ x ∣ - 3 ∣ x - 1 ∣ \geq 0 ∣ x - 1 ∣ > 0 {2 ∣ x ∣ - 3 ∣ x - 1 ∣ \leq 0 ∣ x - 1 ∣ < 0

Solve the inequality

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Evaluate

2 ∣ x ∣ - 3 ∣ x - 1 ∣ \geq 0

Separate the inequality into 4 possible cases

2 x - 3 (x - 1) \geq 0, x \geq 0, x - 1 \geq 0 2 x - 3 (- (x - 1)) \geq 0, x \geq 0, x - 1 < 0 2 (- x) - 3 (x - 1) \geq 0, x < 0, x - 1 \geq 0 2 (- x) - 3 (- (x - 1)) \geq 0, x < 0, x - 1 < 0

Evaluate

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x \leq 3, x \geq 0, x - 1 \geq 0 2 x - 3 (- (x - 1)) \geq 0, x \geq 0, x - 1 < 0 2 (- x) - 3 (x - 1) \geq 0, x < 0, x - 1 \geq 0 2 (- x) - 3 (- (x - 1)) \geq 0, x < 0, x - 1 < 0

Evaluate

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x \leq 3, x \geq 0, x \geq 1 2 x - 3 (- (x - 1)) \geq 0, x \geq 0, x - 1 < 0 2 (- x) - 3 (x - 1) \geq 0, x < 0, x - 1 \geq 0 2 (- x) - 3 (- (x - 1)) \geq 0, x < 0, x - 1 < 0

Evaluate

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x \leq 3, x \geq 0, x \geq 1 x \geq \frac{3}{5}, x \geq 0, x - 1 < 0 2 (- x) - 3 (x - 1) \geq 0, x < 0, x - 1 \geq 0 2 (- x) - 3 (- (x - 1)) \geq 0, x < 0, x - 1 < 0

Evaluate

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x \leq 3, x \geq 0, x \geq 1 x \geq \frac{3}{5}, x \geq 0, x < 1 2 (- x) - 3 (x - 1) \geq 0, x < 0, x - 1 \geq 0 2 (- x) - 3 (- (x - 1)) \geq 0, x < 0, x - 1 < 0

Evaluate

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x \leq 3, x \geq 0, x \geq 1 x \geq \frac{3}{5}, x \geq 0, x < 1 x \leq \frac{3}{5}, x < 0, x - 1 \geq 0 2 (- x) - 3 (- (x - 1)) \geq 0, x < 0, x - 1 < 0

Evaluate

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x \leq 3, x \geq 0, x \geq 1 x \geq \frac{3}{5}, x \geq 0, x < 1 x \leq \frac{3}{5}, x < 0, x \geq 1 2 (- x) - 3 (- (x - 1)) \geq 0, x < 0, x - 1 < 0

Evaluate

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x \leq 3, x \geq 0, x \geq 1 x \geq \frac{3}{5}, x \geq 0, x < 1 x \leq \frac{3}{5}, x < 0, x \geq 1 x \geq 3, x < 0, x - 1 < 0

Evaluate

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x \leq 3, x \geq 0, x \geq 1 x \geq \frac{3}{5}, x \geq 0, x < 1 x \leq \frac{3}{5}, x < 0, x \geq 1 x \geq 3, x < 0, x < 1

Find the intersection

1 \leq x \leq 3 x \geq \frac{3}{5}, x \geq 0, x < 1 x \leq \frac{3}{5}, x < 0, x \geq 1 x \geq 3, x < 0, x < 1

Find the intersection

1 \leq x \leq 3 \frac{3}{5} \leq x < 1 x \leq \frac{3}{5}, x < 0, x \geq 1 x \geq 3, x < 0, x < 1

Find the intersection

1 \leq x \leq 3 \frac{3}{5} \leq x < 1 x \in \emptyset x \geq 3, x < 0, x < 1

Find the intersection

1 \leq x \leq 3 \frac{3}{5} \leq x < 1 x \in \emptyset x \in \emptyset

Find the union

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

{\frac{3}{5} \leq x \leq 3 ∣ x - 1 ∣ > 0 {2 ∣ x ∣ - 3 ∣ x - 1 ∣ \leq 0 ∣ x - 1 ∣ < 0

Solve the inequality

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{\frac{3}{5} \leq x \leq 3 x \neq = 1 {2 ∣ x ∣ - 3 ∣ x - 1 ∣ \leq 0 ∣ x - 1 ∣ < 0

Solve the inequality

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Evaluate

2 ∣ x ∣ - 3 ∣ x - 1 ∣ \leq 0

Separate the inequality into 4 possible cases

2 x - 3 (x - 1) \leq 0, x \geq 0, x - 1 \geq 0 2 x - 3 (- (x - 1)) \leq 0, x \geq 0, x - 1 < 0 2 (- x) - 3 (x - 1) \leq 0, x < 0, x - 1 \geq 0 2 (- x) - 3 (- (x - 1)) \leq 0, x < 0, x - 1 < 0

Evaluate

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x \geq 3, x \geq 0, x - 1 \geq 0 2 x - 3 (- (x - 1)) \leq 0, x \geq 0, x - 1 < 0 2 (- x) - 3 (x - 1) \leq 0, x < 0, x - 1 \geq 0 2 (- x) - 3 (- (x - 1)) \leq 0, x < 0, x - 1 < 0

Evaluate

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x \geq 3, x \geq 0, x \geq 1 2 x - 3 (- (x - 1)) \leq 0, x \geq 0, x - 1 < 0 2 (- x) - 3 (x - 1) \leq 0, x < 0, x - 1 \geq 0 2 (- x) - 3 (- (x - 1)) \leq 0, x < 0, x - 1 < 0

Evaluate

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x \geq 3, x \geq 0, x \geq 1 x \leq \frac{3}{5}, x \geq 0, x - 1 < 0 2 (- x) - 3 (x - 1) \leq 0, x < 0, x - 1 \geq 0 2 (- x) - 3 (- (x - 1)) \leq 0, x < 0, x - 1 < 0

Evaluate

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x \geq 3, x \geq 0, x \geq 1 x \leq \frac{3}{5}, x \geq 0, x < 1 2 (- x) - 3 (x - 1) \leq 0, x < 0, x - 1 \geq 0 2 (- x) - 3 (- (x - 1)) \leq 0, x < 0, x - 1 < 0

Evaluate

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x \geq 3, x \geq 0, x \geq 1 x \leq \frac{3}{5}, x \geq 0, x < 1 x \geq \frac{3}{5}, x < 0, x - 1 \geq 0 2 (- x) - 3 (- (x - 1)) \leq 0, x < 0, x - 1 < 0

Evaluate

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x \geq 3, x \geq 0, x \geq 1 x \leq \frac{3}{5}, x \geq 0, x < 1 x \geq \frac{3}{5}, x < 0, x \geq 1 2 (- x) - 3 (- (x - 1)) \leq 0, x < 0, x - 1 < 0

Evaluate

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x \geq 3, x \geq 0, x \geq 1 x \leq \frac{3}{5}, x \geq 0, x < 1 x \geq \frac{3}{5}, x < 0, x \geq 1 x \leq 3, x < 0, x - 1 < 0

Evaluate

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x \geq 3, x \geq 0, x \geq 1 x \leq \frac{3}{5}, x \geq 0, x < 1 x \geq \frac{3}{5}, x < 0, x \geq 1 x \leq 3, x < 0, x < 1

Find the intersection

x \geq 3 x \leq \frac{3}{5}, x \geq 0, x < 1 x \geq \frac{3}{5}, x < 0, x \geq 1 x \leq 3, x < 0, x < 1

Find the intersection

x \geq 3 0 \leq x \leq \frac{3}{5} x \geq \frac{3}{5}, x < 0, x \geq 1 x \leq 3, x < 0, x < 1

Find the intersection

x \geq 3 0 \leq x \leq \frac{3}{5} x \in \emptyset x \leq 3, x < 0, x < 1

Find the intersection

x \geq 3 0 \leq x \leq \frac{3}{5} x \in \emptyset x < 0

Find the union

x \in (- \infty, \frac{3}{5}] \cup [3, + \infty)

{\frac{3}{5} \leq x \leq 3 x \neq = 1 {x \in (- \infty, \frac{3}{5}] \cup [3, + \infty) ∣ x - 1 ∣ < 0

Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is false for any value of x

{\frac{3}{5} \leq x \leq 3 x \neq = 1 {x \in (- \infty, \frac{3}{5}] \cup [3, + \infty) x \in / R

Find the intersection

x \in [\frac{3}{5}, 1) \cup (1, 3] {x \in (- \infty, \frac{3}{5}] \cup [3, + \infty) x \in / R

Find the intersection

x \in [\frac{3}{5}, 1) \cup (1, 3] x \in / R

Find the union

x \in [\frac{3}{5}, 1) \cup (1, 3]

Check if the solution is in the defined range

x \in [\frac{3}{5}, 1) \cup (1, 3], x \neq = 1

Solution

x \in [\frac{3}{5}, 1) \cup (1, 3]