Solve n n-1 /2 +1 <= 42148572

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for n

- \frac{168594286}{2} \leq n \leq \frac{168594286}{2}

Alternative Form

n \in [- \frac{168594286}{2}, \frac{168594286}{2}]

Evaluate

n \times n - \frac{1}{2} + 1 \leq 42148572

Simplify

More Steps

n^{2} + \frac{1}{2} \leq 42148572

Move the expression to the left side

n^{2} + \frac{1}{2} - 42148572 \leq 0

Subtract the numbers

More Steps

n^{2} - \frac{84297143}{2} \leq 0

Rewrite the expression

n^{2} - \frac{84297143}{2} = 0

Move the constant to the right-hand side and change its sign

n^{2} = 0 + \frac{84297143}{2}

Add the terms

n^{2} = \frac{84297143}{2}

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

n = \pm \frac{84297143}{2}

Simplify the expression

More Steps

n = \pm \frac{168594286}{2}

Separate the equation into 2 possible cases

n = \frac{168594286}{2} n = - \frac{168594286}{2}

Determine the test intervals using the critical values

n < - \frac{168594286}{2} - \frac{168594286}{2} < n < \frac{168594286}{2} n > \frac{168594286}{2}

Choose a value form each interval

n_{1} = - 6493 n_{2} = 0 n_{3} = 6493

To determine if n < - \frac{168594286}{2} is the solution to the inequality,test if the chosen value n = - 6493 satisfies the initial inequality

More Steps

n < - \frac{168594286}{2} is not a solution n_{2} = 0 n_{3} = 6493

To determine if - \frac{168594286}{2} < n < \frac{168594286}{2} is the solution to the inequality,test if the chosen value n = 0 satisfies the initial inequality

More Steps

n < - \frac{168594286}{2} is not a solution - \frac{168594286}{2} < n < \frac{168594286}{2} is the solution n_{3} = 6493

To determine if n > \frac{168594286}{2} is the solution to the inequality,test if the chosen value n = 6493 satisfies the initial inequality

More Steps

n < - \frac{168594286}{2} is not a solution - \frac{168594286}{2} < n < \frac{168594286}{2} is the solution n > \frac{168594286}{2} is not a solution

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

- \frac{168594286}{2} \leq n \leq \frac{168594286}{2} is the solution

Solution

- \frac{168594286}{2} \leq n \leq \frac{168594286}{2}

Alternative Form

n \in [- \frac{168594286}{2}, \frac{168594286}{2}]

Show Solution