Solve x^2-x-6x-7<0

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for x

\frac{- 77 + 7}{2} < x < \frac{77 + 7}{2}

Alternative Form

x \in (\frac{- 77 + 7}{2}, \frac{77 + 7}{2})

Evaluate

x^{2} - x - 6 x - 7 < 0

Subtract the terms

More Steps

x^{2} - 7 x - 7 < 0

Rewrite the expression

x^{2} - 7 x - 7 = 0

Add or subtract both sides

x^{2} - 7 x = 7

Add the same value to both sides

x^{2} - 7 x + \frac{49}{4} = 7 + \frac{49}{4}

Simplify the expression

(x - \frac{7}{2})^{2} = \frac{77}{4}

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

x - \frac{7}{2} = \pm \frac{77}{4}

Simplify the expression

x - \frac{7}{2} = \pm \frac{77}{2}

Separate the equation into 2 possible cases

x - \frac{7}{2} = \frac{77}{2} x - \frac{7}{2} = - \frac{77}{2}

Solve the equation

More Steps

x = \frac{77 + 7}{2} x - \frac{7}{2} = - \frac{77}{2}

Solve the equation

More Steps

x = \frac{77 + 7}{2} x = \frac{- 77 + 7}{2}

Determine the test intervals using the critical values

x < \frac{- 77 + 7}{2} \frac{- 77 + 7}{2} < x < \frac{77 + 7}{2} x > \frac{77 + 7}{2}

Choose a value form each interval

x_{1} = - 2 x_{2} = 4 x_{3} = 9

To determine if x < \frac{- 77 + 7}{2} is the solution to the inequality,test if the chosen value x = - 2 satisfies the initial inequality

More Steps

x < \frac{- 77 + 7}{2} is not a solution x_{2} = 4 x_{3} = 9

To determine if \frac{- 77 + 7}{2} < x < \frac{77 + 7}{2} is the solution to the inequality,test if the chosen value x = 4 satisfies the initial inequality

More Steps

x < \frac{- 77 + 7}{2} is not a solution \frac{- 77 + 7}{2} < x < \frac{77 + 7}{2} is the solution x_{3} = 9

To determine if x > \frac{77 + 7}{2} is the solution to the inequality,test if the chosen value x = 9 satisfies the initial inequality

More Steps

x < \frac{- 77 + 7}{2} is not a solution \frac{- 77 + 7}{2} < x < \frac{77 + 7}{2} is the solution x > \frac{77 + 7}{2} is not a solution

Solution

\frac{- 77 + 7}{2} < x < \frac{77 + 7}{2}

Alternative Form

x \in (\frac{- 77 + 7}{2}, \frac{77 + 7}{2})

Show Solution