Solve \left|1-x\right|-\left|3x-2\right|<=-1 - ScanMath

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Type your math problem... \left|1-x\right|-\left|3x-2\right|<=-1

Question

Solve the inequality

x \in (- \infty, 0] \cup [1, + \infty)

Evaluate

∣ 1 - x ∣ - ∣ 3 x - 2 ∣ \leq - 1

Separate the inequality into 4 possible cases

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

Evaluate

More Steps

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

Evaluate

More Steps

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

Evaluate

More Steps

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

Evaluate

More Steps

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

Evaluate

More Steps

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

Evaluate

More Steps

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

Evaluate

More Steps

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

Evaluate

More Steps

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

Evaluate

More Steps

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

Evaluate

More Steps

x \geq 1, x \leq 1, x \geq \frac{2}{3} x \leq 0, x \leq 1, x < \frac{2}{3} x \geq 1, x > 1, x \geq \frac{2}{3} x \leq \frac{1}{2}, 1 - x < 0, 3 x - 2 < 0

Evaluate

More Steps

x \geq 1, x \leq 1, x \geq \frac{2}{3} x \leq 0, x \leq 1, x < \frac{2}{3} x \geq 1, x > 1, x \geq \frac{2}{3} x \leq \frac{1}{2}, x > 1, 3 x - 2 < 0

Evaluate

More Steps

x \geq 1, x \leq 1, x \geq \frac{2}{3} x \leq 0, x \leq 1, x < \frac{2}{3} x \geq 1, x > 1, x \geq \frac{2}{3} x \leq \frac{1}{2}, x > 1, x < \frac{2}{3}

Find the intersection

x = 1 x \leq 0, x \leq 1, x < \frac{2}{3} x \geq 1, x > 1, x \geq \frac{2}{3} x \leq \frac{1}{2}, x > 1, x < \frac{2}{3}

Find the intersection

x = 1 x \leq 0 x \geq 1, x > 1, x \geq \frac{2}{3} x \leq \frac{1}{2}, x > 1, x < \frac{2}{3}

Find the intersection

x = 1 x \leq 0 x > 1 x \leq \frac{1}{2}, x > 1, x < \frac{2}{3}

Find the intersection

x = 1 x \leq 0 x > 1 x \in \emptyset

Solution

x \in (- \infty, 0] \cup [1, + \infty)

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