Solve |2/x-12|>6/5

Evaluate

\frac{2}{x} - 12 > \frac{6}{5}

Find the domain

\frac{2}{x} - 12 > \frac{6}{5}, x \neq = 0

Simplify

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2 \frac{1}{x} - 6 > \frac{6}{5}

Calculate

\frac{10 ∣ 1 - 6 x ∣ - 6 ∣ x ∣}{5 ∣ x ∣} > 0

Separate the inequality into 2 possible cases

{10 ∣ 1 - 6 x ∣ - 6 ∣ x ∣ > 0 5 ∣ x ∣ > 0 {10 ∣ 1 - 6 x ∣ - 6 ∣ x ∣ < 0 5 ∣ x ∣ < 0

Solve the inequality

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Evaluate

10 ∣ 1 - 6 x ∣ - 6 ∣ x ∣ > 0

Separate the inequality into 4 possible cases

10 (1 - 6 x) - 6 x > 0, 1 - 6 x \geq 0, x \geq 0 10 (1 - 6 x) - 6 (- x) > 0, 1 - 6 x \geq 0, x < 0 10 (- (1 - 6 x)) - 6 x > 0, 1 - 6 x < 0, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) > 0, 1 - 6 x < 0, x < 0

Evaluate

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x < \frac{5}{33}, 1 - 6 x \geq 0, x \geq 0 10 (1 - 6 x) - 6 (- x) > 0, 1 - 6 x \geq 0, x < 0 10 (- (1 - 6 x)) - 6 x > 0, 1 - 6 x < 0, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) > 0, 1 - 6 x < 0, x < 0

Evaluate

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x < \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 10 (1 - 6 x) - 6 (- x) > 0, 1 - 6 x \geq 0, x < 0 10 (- (1 - 6 x)) - 6 x > 0, 1 - 6 x < 0, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) > 0, 1 - 6 x < 0, x < 0

Evaluate

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x < \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 x < \frac{5}{27}, 1 - 6 x \geq 0, x < 0 10 (- (1 - 6 x)) - 6 x > 0, 1 - 6 x < 0, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) > 0, 1 - 6 x < 0, x < 0

Evaluate

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x < \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 x < \frac{5}{27}, x \leq \frac{1}{6}, x < 0 10 (- (1 - 6 x)) - 6 x > 0, 1 - 6 x < 0, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) > 0, 1 - 6 x < 0, x < 0

Evaluate

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x < \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 x < \frac{5}{27}, x \leq \frac{1}{6}, x < 0 x > \frac{5}{27}, 1 - 6 x < 0, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) > 0, 1 - 6 x < 0, x < 0

Evaluate

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x < \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 x < \frac{5}{27}, x \leq \frac{1}{6}, x < 0 x > \frac{5}{27}, x > \frac{1}{6}, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) > 0, 1 - 6 x < 0, x < 0

Evaluate

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x < \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 x < \frac{5}{27}, x \leq \frac{1}{6}, x < 0 x > \frac{5}{27}, x > \frac{1}{6}, x \geq 0 x > \frac{5}{33}, 1 - 6 x < 0, x < 0

Evaluate

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x < \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 x < \frac{5}{27}, x \leq \frac{1}{6}, x < 0 x > \frac{5}{27}, x > \frac{1}{6}, x \geq 0 x > \frac{5}{33}, x > \frac{1}{6}, x < 0

Find the intersection

0 \leq x < \frac{5}{33} x < \frac{5}{27}, x \leq \frac{1}{6}, x < 0 x > \frac{5}{27}, x > \frac{1}{6}, x \geq 0 x > \frac{5}{33}, x > \frac{1}{6}, x < 0

Find the intersection

0 \leq x < \frac{5}{33} x < 0 x > \frac{5}{27}, x > \frac{1}{6}, x \geq 0 x > \frac{5}{33}, x > \frac{1}{6}, x < 0

Find the intersection

0 \leq x < \frac{5}{33} x < 0 x > \frac{5}{27} x > \frac{5}{33}, x > \frac{1}{6}, x < 0

Find the intersection

0 \leq x < \frac{5}{33} x < 0 x > \frac{5}{27} x \in \emptyset

Find the union

x \in (- \infty, \frac{5}{33}) \cup (\frac{5}{27}, + \infty)

{x \in (- \infty, \frac{5}{33}) \cup (\frac{5}{27}, + \infty) 5 ∣ x ∣ > 0 {10 ∣ 1 - 6 x ∣ - 6 ∣ x ∣ < 0 5 ∣ x ∣ < 0

Solve the inequality

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{x \in (- \infty, \frac{5}{33}) \cup (\frac{5}{27}, + \infty) x \neq = 0 {10 ∣ 1 - 6 x ∣ - 6 ∣ x ∣ < 0 5 ∣ x ∣ < 0

Solve the inequality

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Evaluate

10 ∣ 1 - 6 x ∣ - 6 ∣ x ∣ < 0

Separate the inequality into 4 possible cases

10 (1 - 6 x) - 6 x < 0, 1 - 6 x \geq 0, x \geq 0 10 (1 - 6 x) - 6 (- x) < 0, 1 - 6 x \geq 0, x < 0 10 (- (1 - 6 x)) - 6 x < 0, 1 - 6 x < 0, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) < 0, 1 - 6 x < 0, x < 0

Evaluate

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x > \frac{5}{33}, 1 - 6 x \geq 0, x \geq 0 10 (1 - 6 x) - 6 (- x) < 0, 1 - 6 x \geq 0, x < 0 10 (- (1 - 6 x)) - 6 x < 0, 1 - 6 x < 0, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) < 0, 1 - 6 x < 0, x < 0

Evaluate

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x > \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 10 (1 - 6 x) - 6 (- x) < 0, 1 - 6 x \geq 0, x < 0 10 (- (1 - 6 x)) - 6 x < 0, 1 - 6 x < 0, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) < 0, 1 - 6 x < 0, x < 0

Evaluate

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x > \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 x > \frac{5}{27}, 1 - 6 x \geq 0, x < 0 10 (- (1 - 6 x)) - 6 x < 0, 1 - 6 x < 0, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) < 0, 1 - 6 x < 0, x < 0

Evaluate

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x > \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 x > \frac{5}{27}, x \leq \frac{1}{6}, x < 0 10 (- (1 - 6 x)) - 6 x < 0, 1 - 6 x < 0, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) < 0, 1 - 6 x < 0, x < 0

Evaluate

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x > \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 x > \frac{5}{27}, x \leq \frac{1}{6}, x < 0 x < \frac{5}{27}, 1 - 6 x < 0, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) < 0, 1 - 6 x < 0, x < 0

Evaluate

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x > \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 x > \frac{5}{27}, x \leq \frac{1}{6}, x < 0 x < \frac{5}{27}, x > \frac{1}{6}, x \geq 0 10 (- (1 - 6 x)) - 6 (- x) < 0, 1 - 6 x < 0, x < 0

Evaluate

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x > \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 x > \frac{5}{27}, x \leq \frac{1}{6}, x < 0 x < \frac{5}{27}, x > \frac{1}{6}, x \geq 0 x < \frac{5}{33}, 1 - 6 x < 0, x < 0

Evaluate

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x > \frac{5}{33}, x \leq \frac{1}{6}, x \geq 0 x > \frac{5}{27}, x \leq \frac{1}{6}, x < 0 x < \frac{5}{27}, x > \frac{1}{6}, x \geq 0 x < \frac{5}{33}, x > \frac{1}{6}, x < 0

Find the intersection

\frac{5}{33} < x \leq \frac{1}{6} x > \frac{5}{27}, x \leq \frac{1}{6}, x < 0 x < \frac{5}{27}, x > \frac{1}{6}, x \geq 0 x < \frac{5}{33}, x > \frac{1}{6}, x < 0

Find the intersection

\frac{5}{33} < x \leq \frac{1}{6} x \in \emptyset x < \frac{5}{27}, x > \frac{1}{6}, x \geq 0 x < \frac{5}{33}, x > \frac{1}{6}, x < 0

Find the intersection

\frac{5}{33} < x \leq \frac{1}{6} x \in \emptyset \frac{1}{6} < x < \frac{5}{27} x < \frac{5}{33}, x > \frac{1}{6}, x < 0

Find the intersection

\frac{5}{33} < x \leq \frac{1}{6} x \in \emptyset \frac{1}{6} < x < \frac{5}{27} x \in \emptyset

Find the union

\frac{5}{33} < x < \frac{5}{27}

{x \in (- \infty, \frac{5}{33}) \cup (\frac{5}{27}, + \infty) x \neq = 0 {\frac{5}{33} < x < \frac{5}{27} 5 ∣ x ∣ < 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

{x \in (- \infty, \frac{5}{33}) \cup (\frac{5}{27}, + \infty) x \neq = 0 {\frac{5}{33} < x < \frac{5}{27} x \in / R

Find the intersection

x \in (- \infty, 0) \cup (0, \frac{5}{33}) \cup (\frac{5}{27}, + \infty) {\frac{5}{33} < x < \frac{5}{27} x \in / R

Find the intersection

x \in (- \infty, 0) \cup (0, \frac{5}{33}) \cup (\frac{5}{27}, + \infty) x \in / R

Find the union

x \in (- \infty, 0) \cup (0, \frac{5}{33}) \cup (\frac{5}{27}, + \infty)

Check if the solution is in the defined range

x \in (- \infty, 0) \cup (0, \frac{5}{33}) \cup (\frac{5}{27}, + \infty), x \neq = 0

Solution

x \in (- \infty, 0) \cup (0, \frac{5}{33}) \cup (\frac{5}{27}, + \infty)