Solve -3n^2\ge-4n-3

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for n

\frac{- 13 + 2}{3} \leq n \leq \frac{13 + 2}{3}

Alternative Form

n \in [\frac{- 13 + 2}{3}, \frac{13 + 2}{3}]

Evaluate

- 3 n^{2} \geq - 4 n - 3

Move the expression to the left side

- 3 n^{2} - (- 4 n - 3) \geq 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

- 3 n^{2} + 4 n + 3 \geq 0

Rewrite the expression

- 3 n^{2} + 4 n + 3 = 0

Add or subtract both sides

- 3 n^{2} + 4 n = - 3

Divide both sides

\frac{- 3 n ^{2} + 4 n}{- 3} = \frac{- 3}{- 3}

Evaluate

n^{2} - \frac{4}{3} n = 1

Add the same value to both sides

n^{2} - \frac{4}{3} n + \frac{4}{9} = 1 + \frac{4}{9}

Simplify the expression

(n - \frac{2}{3})^{2} = \frac{13}{9}

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

n - \frac{2}{3} = \pm \frac{13}{9}

Simplify the expression

n - \frac{2}{3} = \pm \frac{13}{3}

Separate the equation into 2 possible cases

n - \frac{2}{3} = \frac{13}{3} n - \frac{2}{3} = - \frac{13}{3}

Solve the equation

More Steps

n = \frac{13 + 2}{3} n - \frac{2}{3} = - \frac{13}{3}

Solve the equation

More Steps

n = \frac{13 + 2}{3} n = \frac{- 13 + 2}{3}

Determine the test intervals using the critical values

n < \frac{- 13 + 2}{3} \frac{- 13 + 2}{3} < n < \frac{13 + 2}{3} n > \frac{13 + 2}{3}

Choose a value form each interval

n_{1} = - 2 n_{2} = 1 n_{3} = 3

To determine if n < \frac{- 13 + 2}{3} is the solution to the inequality,test if the chosen value n = - 2 satisfies the initial inequality

More Steps

n < \frac{- 13 + 2}{3} is not a solution n_{2} = 1 n_{3} = 3

To determine if \frac{- 13 + 2}{3} < n < \frac{13 + 2}{3} is the solution to the inequality,test if the chosen value n = 1 satisfies the initial inequality

More Steps

n < \frac{- 13 + 2}{3} is not a solution \frac{- 13 + 2}{3} < n < \frac{13 + 2}{3} is the solution n_{3} = 3

To determine if n > \frac{13 + 2}{3} is the solution to the inequality,test if the chosen value n = 3 satisfies the initial inequality

More Steps

n < \frac{- 13 + 2}{3} is not a solution \frac{- 13 + 2}{3} < n < \frac{13 + 2}{3} is the solution n > \frac{13 + 2}{3} is not a solution

The original inequality is a nonstrict inequality,so include the critical value in the solution

\frac{- 13 + 2}{3} \leq n \leq \frac{13 + 2}{3} is the solution

Solution

\frac{- 13 + 2}{3} \leq n \leq \frac{13 + 2}{3}

Alternative Form

n \in [\frac{- 13 + 2}{3}, \frac{13 + 2}{3}]

Show Solution