Solve (x 1)⋅(x-1)<2024

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for x

\frac{- 8097 + 1}{2} < x < \frac{8097 + 1}{2}

Alternative Form

x \in (\frac{- 8097 + 1}{2}, \frac{8097 + 1}{2})

Evaluate

(x \times 1) (x - 1) < 2024

Remove the parentheses

x \times 1 \times (x - 1) < 2024

Multiply the terms

x (x - 1) < 2024

Move the expression to the left side

x (x - 1) - 2024 < 0

Expand the expression

More Steps

x^{2} - x - 2024 < 0

Rewrite the expression

x^{2} - x - 2024 = 0

Add or subtract both sides

x^{2} - x = 2024

Add the same value to both sides

x^{2} - x + \frac{1}{4} = 2024 + \frac{1}{4}

Simplify the expression

(x - \frac{1}{2})^{2} = \frac{8097}{4}

Take the root of both sides of the equation and remember to use both positive and negative roots

x - \frac{1}{2} = \pm \frac{8097}{4}

Simplify the expression

x - \frac{1}{2} = \pm \frac{8097}{2}

Separate the equation into 2 possible cases

x - \frac{1}{2} = \frac{8097}{2} x - \frac{1}{2} = - \frac{8097}{2}

Solve the equation

More Steps

x = \frac{8097 + 1}{2} x - \frac{1}{2} = - \frac{8097}{2}

Solve the equation

More Steps

x = \frac{8097 + 1}{2} x = \frac{- 8097 + 1}{2}

Determine the test intervals using the critical values

x < \frac{- 8097 + 1}{2} \frac{- 8097 + 1}{2} < x < \frac{8097 + 1}{2} x > \frac{8097 + 1}{2}

Choose a value form each interval

x_{1} = - 45 x_{2} = 1 x_{3} = 46

To determine if x < \frac{- 8097 + 1}{2} is the solution to the inequality,test if the chosen value x = - 45 satisfies the initial inequality

More Steps

x < \frac{- 8097 + 1}{2} is not a solution x_{2} = 1 x_{3} = 46

To determine if \frac{- 8097 + 1}{2} < x < \frac{8097 + 1}{2} is the solution to the inequality,test if the chosen value x = 1 satisfies the initial inequality

More Steps

x < \frac{- 8097 + 1}{2} is not a solution \frac{- 8097 + 1}{2} < x < \frac{8097 + 1}{2} is the solution x_{3} = 46

To determine if x > \frac{8097 + 1}{2} is the solution to the inequality,test if the chosen value x = 46 satisfies the initial inequality

More Steps

x < \frac{- 8097 + 1}{2} is not a solution \frac{- 8097 + 1}{2} < x < \frac{8097 + 1}{2} is the solution x > \frac{8097 + 1}{2} is not a solution

Solution

\frac{- 8097 + 1}{2} < x < \frac{8097 + 1}{2}

Alternative Form

x \in (\frac{- 8097 + 1}{2}, \frac{8097 + 1}{2})

Show Solution