Solve the Inequality for x 1/(x^2)<0.000001

$\frac{1}{x^{2}} < 0.000001$

Step 1

Multiply both sides by $x^{2}$ .

$\frac{1}{x^{2}} x^{2} = 0.000001 x^{2}$

Step 2

Simplify the left side.

Cancel the common factor of $x^{2}$ .

Cancel the common factor.

$\frac{1}{x^{2}} x^{2} = 0.000001 x^{2}$

Rewrite the expression.

$1 = 0.000001 x^{2}$

Step 3

Solve for $x$ .

Rewrite the equation as $0.000001 x^{2} = 1$ .

$0.000001 x^{2} = 1$

Divide each term in $0.000001 x^{2} = 1$ by $0.000001$ and simplify.

Divide each term in $0.000001 x^{2} = 1$ by $0.000001$ .

$\frac{0.000001 x^{2}}{0.000001} = \frac{1}{0.000001}$

Simplify the left side.

Cancel the common factor of $0.000001$ .

Cancel the common factor.

$\frac{0.000001 x^{2}}{0.000001} = \frac{1}{0.000001}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{1}{0.000001}$

Simplify the right side.

Divide $1$ by $0.000001$ .

$x^{2} = 1000000$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1000000}$

Simplify $\pm \sqrt{1000000}$ .

Rewrite $1000000$ as $1000^{2}$ .

$x = \pm \sqrt{1000^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 1000$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 1000$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1000$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1000, - 1000$

Step 4

Find the domain of $\frac{1}{x^{2}} - 0.000001$ .

Set the denominator in $\frac{1}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Solve for $x$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

The domain is all values of $x$ that make the expression defined.

$(- \infty, 0) \cup (0, \infty)$

Step 5

Use each root to create test intervals.

$x < - 1000$

$- 1000 < x < 0$

$0 < x < 1000$

$x > 1000$

Step 6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < - 1000$ to see if it makes the inequality true.

Choose a value on the interval $x < - 1000$ and see if this value makes the original inequality true.

$x = - 1002$

Replace $x$ with $- 1002$ in the original inequality.

$\frac{1}{{(- 1002)}^{2}} < 0.000001$

The left side $9.96011968 E - 7$ is less than the right side $1 E - 6$ , which means that the given statement is always true.

True

Test a value on the interval $- 1000 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 1000 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Replace $x$ with $- 2$ in the original inequality.

$\frac{1}{{(- 2)}^{2}} < 0.000001$

The left side $0.25$ is not less than the right side $1 E - 6$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 1000$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 1000$ and see if this value makes the original inequality true.

$x = 2$

Replace $x$ with $2$ in the original inequality.

$\frac{1}{{(2)}^{2}} < 0.000001$

The left side $0.25$ is not less than the right side $1 E - 6$ , which means that the given statement is false.

False

Test a value on the interval $x > 1000$ to see if it makes the inequality true.

Choose a value on the interval $x > 1000$ and see if this value makes the original inequality true.

$x = 1002$

Replace $x$ with $1002$ in the original inequality.

$\frac{1}{{(1002)}^{2}} < 0.000001$

The left side $9.96011968 E - 7$ is less than the right side $1 E - 6$ , which means that the given statement is always true.

True

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 1000$ True

$- 1000 < x < 0$ False

$0 < x < 1000$ False

$x > 1000$ True

$x < - 1000$ True

$- 1000 < x < 0$ False

$0 < x < 1000$ False

$x > 1000$ True

Step 7

The solution consists of all of the true intervals.

$x < - 1000$ or $x > 1000$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < - 1000 or x > 1000$

Interval Notation:

$(- \infty, - 1000) \cup (1000, \infty)$

Step 9

Solve the Inequality for x 1/(x^2)<0.000001

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