Solve |x-|x-2||=1

Question

Solve the equation

x_{1} = \frac{1}{2}, x_{2} = \frac{3}{2}

Alternative Form

x_{1} = 0.5, x_{2} = 1.5

Evaluate

∣ x - ∣ x - 2 ∣ ∣ = 1

Separate the equation into 2 possible cases

x - ∣ x - 2 ∣ = 1 x - ∣ x - 2 ∣ = - 1

Solve the equation for x

More Steps

Evaluate

x - ∣ x - 2 ∣ = 1

Move the expression to the left side

x - ∣ x - 2 ∣ - 1 = 0

Separate the equation into 2 possible cases

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

The statement is false for any value of x

More Steps

Evaluate

x - (x - 2) - 1 = 0

Calculate

x - x + 2 - 1 = 0

Calculate the sum or difference

1 = 0

The statement is false for any value of x

x \in \emptyset

x \in \emptyset, x - 2 \geq 0 x - (- (x - 2)) - 1 = 0, x - 2 < 0

Solve the inequality

More Steps

Evaluate

x - 2 \geq 0

Move the constant to the right side

x \geq 0 + 2

Removing 0 doesn't change the value,so remove it from the expression

x \geq 2

x \in \emptyset, x \geq 2 x - (- (x - 2)) - 1 = 0, x - 2 < 0

Solve the equation

More Steps

Evaluate

x - (- (x - 2)) - 1 = 0

Calculate

x + x - 2 - 1 = 0

Calculate the sum or difference

2 x - 3 = 0

Move the constant to the right-hand side and change its sign

2 x = 0 + 3

Removing 0 doesn't change the value,so remove it from the expression

2 x = 3

Divide both sides

\frac{2 x}{2} = \frac{3}{2}

Divide the numbers

x = \frac{3}{2}

x \in \emptyset, x \geq 2 x = \frac{3}{2}, x - 2 < 0

Solve the inequality

More Steps

Evaluate

x - 2 < 0

Move the constant to the right side

x < 0 + 2

Removing 0 doesn't change the value,so remove it from the expression

x < 2

x \in \emptyset, x \geq 2 x = \frac{3}{2}, x < 2

Find the intersection

x \in \emptyset x = \frac{3}{2}, x < 2

Find the intersection

x \in \emptyset x = \frac{3}{2}

Find the union

x = \frac{3}{2}

x = \frac{3}{2} x - ∣ x - 2 ∣ = - 1

Solve the equation for x

More Steps

Evaluate

x - ∣ x - 2 ∣ = - 1

Move the expression to the left side

x - ∣ x - 2 ∣ - (- 1) = 0

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

x - ∣ x - 2 ∣ + 1 = 0

Separate the equation into 2 possible cases

x - (x - 2) + 1 = 0, x - 2 \geq 0 x - (- (x - 2)) + 1 = 0, x - 2 < 0

The statement is false for any value of x

More Steps

Evaluate

x - (x - 2) + 1 = 0

Calculate

x - x + 2 + 1 = 0

Calculate the sum or difference

3 = 0

The statement is false for any value of x

x \in \emptyset

x \in \emptyset, x - 2 \geq 0 x - (- (x - 2)) + 1 = 0, x - 2 < 0

Solve the inequality

More Steps

Evaluate

x - 2 \geq 0

Move the constant to the right side

x \geq 0 + 2

Removing 0 doesn't change the value,so remove it from the expression

x \geq 2

x \in \emptyset, x \geq 2 x - (- (x - 2)) + 1 = 0, x - 2 < 0

Solve the equation

More Steps

Evaluate

x - (- (x - 2)) + 1 = 0

Calculate

x + x - 2 + 1 = 0

Calculate the sum or difference

2 x - 1 = 0

Move the constant to the right-hand side and change its sign

2 x = 0 + 1

Removing 0 doesn't change the value,so remove it from the expression

2 x = 1

Divide both sides

\frac{2 x}{2} = \frac{1}{2}

Divide the numbers

x = \frac{1}{2}

x \in \emptyset, x \geq 2 x = \frac{1}{2}, x - 2 < 0

Solve the inequality

More Steps

Evaluate

x - 2 < 0

Move the constant to the right side

x < 0 + 2

Removing 0 doesn't change the value,so remove it from the expression

x < 2

x \in \emptyset, x \geq 2 x = \frac{1}{2}, x < 2

Find the intersection

x \in \emptyset x = \frac{1}{2}, x < 2

Find the intersection

x \in \emptyset x = \frac{1}{2}

Find the union

x = \frac{1}{2}

x = \frac{3}{2} x = \frac{1}{2}

Solution

x_{1} = \frac{1}{2}, x_{2} = \frac{3}{2}

Alternative Form

x_{1} = 0.5, x_{2} = 1.5

Show Solution

Solve the equation

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