Algebra Examples

$x \geq \frac{4}{x}$

Step 1

Multiply both sides by $x$ .

$x \cdot x = \frac{4}{x} x$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Multiply $x$ by $x$ .

$x^{2} = \frac{4}{x} x$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $x$ .

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Step 2.2.1.1

Cancel the common factor.

$x^{2} = \frac{4}{x} x$

Step 2.2.1.2

Rewrite the expression.

$x^{2} = 4$

Step 3

Solve for $x$ .

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Step 3.1

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

$x = \pm \sqrt{4}$

Step 3.2

Simplify $\pm \sqrt{4}$ .

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Step 3.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 3.2.2

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

$x = \pm 2$

Step 3.3

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

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Step 3.3.1

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

$x = 2$

Step 3.3.2

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

$x = - 2$

Step 3.3.3

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

$x = 2, - 2$

Step 4

Find the domain of $x - \frac{4}{x}$ .

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Step 4.1

Set the denominator in $\frac{4}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 4.2

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.

$x < - 2$

$- 2 < x < 0$

$0 < x < 2$

$x > 2$

Step 6

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

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Step 6.1

Test a value on the interval $x < - 2$ to see if it makes the inequality true.

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Step 6.1.1

Choose a value on the interval $x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Step 6.1.2

Replace $x$ with $- 4$ in the original inequality.

$- 4 \geq \frac{4}{- 4}$

Step 6.1.3

The left side $- 4$ is less than the right side $- 1$ , which means that the given statement is false.

False

Step 6.2

Test a value on the interval $- 2 < x < 0$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $- 2 < x < 0$ and see if this value makes the original inequality true.

$x = - 1$

Step 6.2.2

Replace $x$ with $- 1$ in the original inequality.

$- 1 \geq \frac{4}{- 1}$

Step 6.2.3

The left side $- 1$ is greater than the right side $- 4$ , which means that the given statement is always true.

True

Step 6.3

Test a value on the interval $0 < x < 2$ to see if it makes the inequality true.

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Step 6.3.1

Choose a value on the interval $0 < x < 2$ and see if this value makes the original inequality true.

$x = 1$

Step 6.3.2

Replace $x$ with $1$ in the original inequality.

$1 \geq \frac{4}{1}$

Step 6.3.3

The left side $1$ is less than the right side $4$ , which means that the given statement is false.

False

Step 6.4

Test a value on the interval $x > 2$ to see if it makes the inequality true.

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Step 6.4.1

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Step 6.4.2

Replace $x$ with $4$ in the original inequality.

$4 \geq \frac{4}{4}$

Step 6.4.3

The left side $4$ is greater than the right side $1$ , which means that the given statement is always true.

True

Step 6.5

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

$x < - 2$ False

$- 2 < x < 0$ True

$0 < x < 2$ False

$x > 2$ True

$x < - 2$ False

$- 2 < x < 0$ True

$0 < x < 2$ False

$x > 2$ True

Step 7

The solution consists of all of the true intervals.

$- 2 \leq x < 0$ or $x \geq 2$

Step 8

Convert the inequality to interval notation.

$[- 2, 0) \cup [2, \infty)$

Step 9