Algebra Examples

$| x - 2 | < \frac{8}{x}$

Step 1

Subtract $\frac{8}{x}$ from both sides of the inequality.

$| x - 2 | - \frac{8}{x} < 0$

Step 2

Simplify $| x - 2 | - \frac{8}{x}$ .

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Step 2.1

To write $| x - 2 |$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{| x - 2 | x}{x} - \frac{8}{x} < 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{| x - 2 | x - 8}{x} < 0$

Step 2.3

Reorder factors in $\frac{| x - 2 | x - 8}{x}$ .

$\frac{x | x - 2 | - 8}{x} < 0$

Step 3

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x = 0$

$x | x - 2 | - 8 = 0$

Step 4

Add $8$ to both sides of the equation.

$x | x - 2 | = 8$

Step 5

Divide each term in $x | x - 2 | = 8$ by $x$ and simplify.

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Step 5.1

Divide each term in $x | x - 2 | = 8$ by $x$ .

$\frac{x | x - 2 |}{x} = \frac{8}{x}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $x$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{x | x - 2 |}{x} = \frac{8}{x}$

Step 5.2.1.2

Divide $| x - 2 |$ by $1$ .

$| x - 2 | = \frac{8}{x}$

Step 6

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x - 2 = \pm \frac{8}{x}$

Step 7

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

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Step 7.1

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

$x - 2 = \frac{8}{x}$

Step 7.2

Add $2$ to both sides of the equation.

$x = \frac{8}{x} + 2$

Step 7.3

Find the LCD of the terms in the equation.

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Step 7.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1$

Step 7.3.2

The LCM of one and any expression is the expression.

$x$

Step 7.4

Multiply each term in $x = \frac{8}{x} + 2$ by $x$ to eliminate the fractions.

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Step 7.4.1

Multiply each term in $x = \frac{8}{x} + 2$ by $x$ .

$x \cdot x = \frac{8}{x} x + 2 x$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Multiply $x$ by $x$ .

$x^{2} = \frac{8}{x} x + 2 x$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Cancel the common factor of $x$ .

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Step 7.4.3.1.1

Cancel the common factor.

$x^{2} = \frac{8}{x} x + 2 x$

Step 7.4.3.1.2

Rewrite the expression.

$x^{2} = 8 + 2 x$

Step 7.5

Solve the equation.

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Step 7.5.1

Subtract $2 x$ from both sides of the equation.

$x^{2} - 2 x = 8$

Step 7.5.2

Subtract $8$ from both sides of the equation.

$x^{2} - 2 x - 8 = 0$

Step 7.5.3

Factor $x^{2} - 2 x - 8$ using the AC method.

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Step 7.5.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 7.5.3.2

Write the factored form using these integers.

$(x - 4) (x + 2) = 0$

Step 7.5.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 4 = 0$

$x + 2 = 0$

Step 7.5.5

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 7.5.5.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 7.5.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 7.5.6

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 7.5.6.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 7.5.6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 7.5.7

The final solution is all the values that make $(x - 4) (x + 2) = 0$ true.

$x = 4, - 2$

Step 7.6

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

$x - 2 = - \frac{8}{x}$

Step 7.7

Add $2$ to both sides of the equation.

$x = - \frac{8}{x} + 2$

Step 7.8

Find the LCD of the terms in the equation.

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Step 7.8.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1$

Step 7.8.2

The LCM of one and any expression is the expression.

$x$

Step 7.9

Multiply each term in $x = - \frac{8}{x} + 2$ by $x$ to eliminate the fractions.

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Step 7.9.1

Multiply each term in $x = - \frac{8}{x} + 2$ by $x$ .

$x \cdot x = - \frac{8}{x} x + 2 x$

Step 7.9.2

Simplify the left side.

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Step 7.9.2.1

Multiply $x$ by $x$ .

$x^{2} = - \frac{8}{x} x + 2 x$

Step 7.9.3

Simplify the right side.

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Step 7.9.3.1

Cancel the common factor of $x$ .

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Step 7.9.3.1.1

Move the leading negative in $- \frac{8}{x}$ into the numerator.

$x^{2} = \frac{- 8}{x} x + 2 x$

Step 7.9.3.1.2

Cancel the common factor.

$x^{2} = \frac{- 8}{x} x + 2 x$

Step 7.9.3.1.3

Rewrite the expression.

$x^{2} = - 8 + 2 x$

Step 7.10

Solve the equation.

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Step 7.10.1

Subtract $2 x$ from both sides of the equation.

$x^{2} - 2 x = - 8$

Step 7.10.2

Add $8$ to both sides of the equation.

$x^{2} - 2 x + 8 = 0$

Step 7.10.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.10.4

Substitute the values $a = 1$ , $b = - 2$ , and $c = 8$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 8)}}{2 \cdot 1}$

Step 7.10.5

Simplify.

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Step 7.10.5.1

Simplify the numerator.

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Step 7.10.5.1.1

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Step 7.10.5.1.2

Multiply $- 4 \cdot 1 \cdot 8$ .

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Step 7.10.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 8}}{2 \cdot 1}$

Step 7.10.5.1.2.2

Multiply $- 4$ by $8$ .

$x = \frac{2 \pm \sqrt{4 - 32}}{2 \cdot 1}$

Step 7.10.5.1.3

Subtract $32$ from $4$ .

$x = \frac{2 \pm \sqrt{- 28}}{2 \cdot 1}$

Step 7.10.5.1.4

Rewrite $- 28$ as $- 1 (28)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 28}}{2 \cdot 1}$

Step 7.10.5.1.5

Rewrite $\sqrt{- 1 (28)}$ as $\sqrt{- 1} \cdot \sqrt{28}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{28}}{2 \cdot 1}$

Step 7.10.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{28}}{2 \cdot 1}$

Step 7.10.5.1.7

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 7.10.5.1.7.1

Factor $4$ out of $28$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (7)}}{2 \cdot 1}$

Step 7.10.5.1.7.2

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Step 7.10.5.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{7})}{2 \cdot 1}$

Step 7.10.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{7}}{2 \cdot 1}$

Step 7.10.5.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i \sqrt{7}}{2}$

Step 7.10.5.3

Simplify $\frac{2 \pm 2 i \sqrt{7}}{2}$ .

$x = 1 \pm i \sqrt{7}$

Step 7.10.6

The final answer is the combination of both solutions.

$x = 1 + i \sqrt{7}, 1 - i \sqrt{7}$

Step 7.11

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

$x = 4, - 2, 1 + i \sqrt{7}, 1 - i \sqrt{7}$

Step 8

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 0$

$x = 4, - 2$

Step 9

Consolidate the solutions.

$x = 0, 4, - 2$

Step 10

Find the domain of $\frac{x | x - 2 | - 8}{x}$ .

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Step 10.1

Set the denominator in $\frac{x | x - 2 | - 8}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 10.2

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

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

Step 11

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 0$

$0 < x < 4$

$x > 4$

Step 12

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 12.1

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

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Step 12.1.1

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

$x = - 4$

Step 12.1.2

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

$| (- 4) - 2 | < \frac{8}{- 4}$

Step 12.1.3

The left side $6$ is not less than the right side $- 2$ , which means that the given statement is false.

False

Step 12.2

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

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Step 12.2.1

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

$x = - 1$

Step 12.2.2

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

$| (- 1) - 2 | < \frac{8}{- 1}$

Step 12.2.3

The left side $3$ is not less than the right side $- 8$ , which means that the given statement is false.

False

Step 12.3

Test a value on the interval $0 < x < 4$ to see if it makes the inequality true.

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Step 12.3.1

Choose a value on the interval $0 < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Step 12.3.2

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

$| (2) - 2 | < \frac{8}{2}$

Step 12.3.3

The left side $0$ is less than the right side $4$ , which means that the given statement is always true.

True

Step 12.4

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 12.4.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 12.4.2

Replace $x$ with $6$ in the original inequality.

$| (6) - 2 | < \frac{8}{6}$

Step 12.4.3

The left side $4$ is not less than the right side $1.\overline{3}$ , which means that the given statement is false.

False

Step 12.5

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

$x < - 2$ False

$- 2 < x < 0$ False

$0 < x < 4$ True

$x > 4$ False

$x < - 2$ False

$- 2 < x < 0$ False

$0 < x < 4$ True

$x > 4$ False

Step 13

The solution consists of all of the true intervals.

$0 < x < 4$

Step 14

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 4$

Interval Notation:

$(0, 4)$

Step 15