Solve root(2)(2x)- root(2)(x-1)>1

Question

Solve the inequality

x \in [1, 2) \cup (2, + \infty)

Evaluate

2 x - x - 1 > 1

Find the domain

More Steps

2 x - x - 1 > 1, x \geq 1

Move the expression to the left side

2 x - x - 1 - 1 > 0

Move the expression to the right side

2 x > x - 1 + 1

Raise both sides of the inequality to the power of 2

2 x > (x - 1 + 1)^{2}

Expand the expression

2 x > x + 2 x - 1

Move the expression to the left side

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

Subtract the terms

More Steps

x - 2 x - 1 > 0

Move the expression to the right side

- 2 x - 1 > - x

Change the signs on both sides of the inequality and flip the inequality sign

2 x - 1 < x

Separate the inequality into 2 possible cases

2 x - 1 < x, x \geq 0 2 x - 1 < x, x < 0

Solve the inequality

More Steps

x \neq = 2, x \geq 0 2 x - 1 < x, x < 0

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

x \neq = 2, x \geq 0 x \in \emptyset, x < 0

Find the intersection

x \in [0, 2) \cup (2, + \infty) x \in \emptyset, x < 0

Find the intersection

x \in [0, 2) \cup (2, + \infty) x \in \emptyset

Find the union

x \in [0, 2) \cup (2, + \infty)

Check if the solution is in the defined range

x \in [0, 2) \cup (2, + \infty), x \geq 1

Solution

x \in [1, 2) \cup (2, + \infty)

Show Solution