Solve 3x^2-4x-6<0

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for x

\frac{- 22 + 2}{3} < x < \frac{22 + 2}{3}

Alternative Form

x \in (\frac{- 22 + 2}{3}, \frac{22 + 2}{3})

Evaluate

3 x^{2} - 4 x - 6 < 0

Rewrite the expression

3 x^{2} - 4 x - 6 = 0

Add or subtract both sides

3 x^{2} - 4 x = 6

Divide both sides

\frac{3 x ^{2} - 4 x}{3} = \frac{6}{3}

Evaluate

x^{2} - \frac{4}{3} x = 2

Add the same value to both sides

x^{2} - \frac{4}{3} x + \frac{4}{9} = 2 + \frac{4}{9}

Simplify the expression

(x - \frac{2}{3})^{2} = \frac{22}{9}

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

x - \frac{2}{3} = \pm \frac{22}{9}

Simplify the expression

x - \frac{2}{3} = \pm \frac{22}{3}

Separate the equation into 2 possible cases

x - \frac{2}{3} = \frac{22}{3} x - \frac{2}{3} = - \frac{22}{3}

Solve the equation

More Steps

x = \frac{22 + 2}{3} x - \frac{2}{3} = - \frac{22}{3}

Solve the equation

More Steps

x = \frac{22 + 2}{3} x = \frac{- 22 + 2}{3}

Determine the test intervals using the critical values

x < \frac{- 22 + 2}{3} \frac{- 22 + 2}{3} < x < \frac{22 + 2}{3} x > \frac{22 + 2}{3}

Choose a value form each interval

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

To determine if x < \frac{- 22 + 2}{3} is the solution to the inequality,test if the chosen value x = - 2 satisfies the initial inequality

More Steps

x < \frac{- 22 + 2}{3} is not a solution x_{2} = 1 x_{3} = 3

To determine if \frac{- 22 + 2}{3} < x < \frac{22 + 2}{3} is the solution to the inequality,test if the chosen value x = 1 satisfies the initial inequality

More Steps

x < \frac{- 22 + 2}{3} is not a solution \frac{- 22 + 2}{3} < x < \frac{22 + 2}{3} is the solution x_{3} = 3

To determine if x > \frac{22 + 2}{3} is the solution to the inequality,test if the chosen value x = 3 satisfies the initial inequality

More Steps

x < \frac{- 22 + 2}{3} is not a solution \frac{- 22 + 2}{3} < x < \frac{22 + 2}{3} is the solution x > \frac{22 + 2}{3} is not a solution

Solution

\frac{- 22 + 2}{3} < x < \frac{22 + 2}{3}

Alternative Form

x \in (\frac{- 22 + 2}{3}, \frac{22 + 2}{3})

Show Solution