Solve -4(2n-5) 3n>12

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for n

\frac{- 17 + 5}{4} < n < \frac{17 + 5}{4}

Alternative Form

n \in (\frac{- 17 + 5}{4}, \frac{17 + 5}{4})

Evaluate

- 4 (2 n - 5) \times 3 n > 12

Multiply

More Steps

- 12 n (2 n - 5) > 12

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

12 n (2 n - 5) < - 12

Move the expression to the left side

12 n (2 n - 5) - (- 12) < 0

Subtract the terms

More Steps

24 n^{2} - 60 n + 12 < 0

Rewrite the expression

24 n^{2} - 60 n + 12 = 0

Add or subtract both sides

24 n^{2} - 60 n = - 12

Divide both sides

\frac{24 n ^{2} - 60 n}{24} = \frac{- 12}{24}

Evaluate

n^{2} - \frac{5}{2} n = - \frac{1}{2}

Add the same value to both sides

n^{2} - \frac{5}{2} n + \frac{25}{16} = - \frac{1}{2} + \frac{25}{16}

Simplify the expression

(n - \frac{5}{4})^{2} = \frac{17}{16}

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

n - \frac{5}{4} = \pm \frac{17}{16}

Simplify the expression

n - \frac{5}{4} = \pm \frac{17}{4}

Separate the equation into 2 possible cases

n - \frac{5}{4} = \frac{17}{4} n - \frac{5}{4} = - \frac{17}{4}

Solve the equation

More Steps

n = \frac{17 + 5}{4} n - \frac{5}{4} = - \frac{17}{4}

Solve the equation

More Steps

n = \frac{17 + 5}{4} n = \frac{- 17 + 5}{4}

Determine the test intervals using the critical values

n < \frac{- 17 + 5}{4} \frac{- 17 + 5}{4} < n < \frac{17 + 5}{4} n > \frac{17 + 5}{4}

Choose a value form each interval

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

To determine if n < \frac{- 17 + 5}{4} is the solution to the inequality,test if the chosen value n = - 1 satisfies the initial inequality

More Steps

n < \frac{- 17 + 5}{4} is not a solution n_{2} = 1 n_{3} = 3

To determine if \frac{- 17 + 5}{4} < n < \frac{17 + 5}{4} is the solution to the inequality,test if the chosen value n = 1 satisfies the initial inequality

More Steps

n < \frac{- 17 + 5}{4} is not a solution \frac{- 17 + 5}{4} < n < \frac{17 + 5}{4} is the solution n_{3} = 3

To determine if n > \frac{17 + 5}{4} is the solution to the inequality,test if the chosen value n = 3 satisfies the initial inequality

More Steps

n < \frac{- 17 + 5}{4} is not a solution \frac{- 17 + 5}{4} < n < \frac{17 + 5}{4} is the solution n > \frac{17 + 5}{4} is not a solution

Solution

\frac{- 17 + 5}{4} < n < \frac{17 + 5}{4}

Alternative Form

n \in (\frac{- 17 + 5}{4}, \frac{17 + 5}{4})

Show Solution