Algebra Examples

$| 2 - \frac{3}{x} | \geq 1$

Step 1

Write $| 2 - \frac{3}{x} | \geq 1$ as a piecewise.

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Step 1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$2 - \frac{3}{x} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract $2$ from both sides of the inequality.

$- \frac{3}{x} \geq - 2$

Step 1.2.2

Multiply both sides by $x$ .

$- \frac{3}{x} x = - 2 x$

Step 1.2.3

Simplify the left side.

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Step 1.2.3.1

Cancel the common factor of $x$ .

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Step 1.2.3.1.1

Move the leading negative in $- \frac{3}{x}$ into the numerator.

$\frac{- 3}{x} x = - 2 x$

Step 1.2.3.1.2

Cancel the common factor.

$\frac{- 3}{x} x = - 2 x$

Step 1.2.3.1.3

Rewrite the expression.

$- 3 = - 2 x$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Rewrite the equation as $- 2 x = - 3$ .

$- 2 x = - 3$

Step 1.2.4.2

Divide each term in $- 2 x = - 3$ by $- 2$ and simplify.

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Step 1.2.4.2.1

Divide each term in $- 2 x = - 3$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 3}{- 2}$

Step 1.2.4.2.2

Simplify the left side.

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Step 1.2.4.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.4.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 3}{- 2}$

Step 1.2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 2}$

Step 1.2.4.2.3

Simplify the right side.

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Step 1.2.4.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{3}{2}$

Step 1.2.5

Find the domain of $| 2 - \frac{3}{x} | - 1$ .

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Step 1.2.5.1

Set the denominator in $\frac{3}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 1.2.5.2

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

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

Step 1.2.6

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{3}{2}$

$x > \frac{3}{2}$

Step 1.2.7

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 1.2.7.1

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

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Step 1.2.7.1.1

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

$x = - 2$

Step 1.2.7.1.2

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

$2 - \frac{3}{- 2} \geq 0$

Step 1.2.7.1.3

The left side $3.5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.2.7.2

Test a value on the interval $0 < x < \frac{3}{2}$ to see if it makes the inequality true.

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Step 1.2.7.2.1

Choose a value on the interval $0 < x < \frac{3}{2}$ and see if this value makes the original inequality true.

$x = 1$

Step 1.2.7.2.2

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

$2 - \frac{3}{1} \geq 0$

Step 1.2.7.2.3

The left side $- 1$ is less than the right side $0$ , which means that the given statement is false.

False

Step 1.2.7.3

Test a value on the interval $x > \frac{3}{2}$ to see if it makes the inequality true.

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Step 1.2.7.3.1

Choose a value on the interval $x > \frac{3}{2}$ and see if this value makes the original inequality true.

$x = 4$

Step 1.2.7.3.2

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

$2 - \frac{3}{4} \geq 0$

Step 1.2.7.3.3

The left side $1.25$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.2.7.4

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

$x < 0$ True

$0 < x < \frac{3}{2}$ False

$x > \frac{3}{2}$ True

$x < 0$ True

$0 < x < \frac{3}{2}$ False

$x > \frac{3}{2}$ True

Step 1.2.8

The solution consists of all of the true intervals.

$x < 0$ or $x \geq \frac{3}{2}$

Step 1.3

In the piece where $2 - \frac{3}{x}$ is non-negative, remove the absolute value.

$2 - \frac{3}{x} \geq 1$

Step 1.4

Find the domain of $2 - \frac{3}{x} \geq 1$ and find the intersection with $x < 0 or x \geq \frac{3}{2}$ .

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Step 1.4.1

Find the domain of $2 - \frac{3}{x} \geq 1$ .

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Step 1.4.1.1

Set the denominator in $\frac{3}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 1.4.1.2

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

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

Step 1.4.2

Find the intersection of $x < 0 or x \geq \frac{3}{2}$ and $(- \infty, 0) \cup (0, \infty)$ .

$x < 0 or x \geq \frac{3}{2}$

Step 1.5

To find the interval for the second piece, find where the inside of the absolute value is negative.

$2 - \frac{3}{x} < 0$

Step 1.6

Solve the inequality.

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Step 1.6.1

Subtract $2$ from both sides of the inequality.

$- \frac{3}{x} < - 2$

Step 1.6.2

Multiply both sides by $x$ .

$- \frac{3}{x} x = - 2 x$

Step 1.6.3

Simplify the left side.

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Step 1.6.3.1

Cancel the common factor of $x$ .

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Step 1.6.3.1.1

Move the leading negative in $- \frac{3}{x}$ into the numerator.

$\frac{- 3}{x} x = - 2 x$

Step 1.6.3.1.2

Cancel the common factor.

$\frac{- 3}{x} x = - 2 x$

Step 1.6.3.1.3

Rewrite the expression.

$- 3 = - 2 x$

Step 1.6.4

Solve for $x$ .

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Step 1.6.4.1

Rewrite the equation as $- 2 x = - 3$ .

$- 2 x = - 3$

Step 1.6.4.2

Divide each term in $- 2 x = - 3$ by $- 2$ and simplify.

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Step 1.6.4.2.1

Divide each term in $- 2 x = - 3$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 3}{- 2}$

Step 1.6.4.2.2

Simplify the left side.

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Step 1.6.4.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.6.4.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 3}{- 2}$

Step 1.6.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 2}$

Step 1.6.4.2.3

Simplify the right side.

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Step 1.6.4.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{3}{2}$

Step 1.6.5

Find the domain of $| 2 - \frac{3}{x} | - 1$ .

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Step 1.6.5.1

Set the denominator in $\frac{3}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 1.6.5.2

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

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

Step 1.6.6

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{3}{2}$

$x > \frac{3}{2}$

Step 1.6.7

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 1.6.7.1

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

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Step 1.6.7.1.1

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

$x = - 2$

Step 1.6.7.1.2

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

$2 - \frac{3}{- 2} < 0$

Step 1.6.7.1.3

The left side $3.5$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 1.6.7.2

Test a value on the interval $0 < x < \frac{3}{2}$ to see if it makes the inequality true.

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Step 1.6.7.2.1

Choose a value on the interval $0 < x < \frac{3}{2}$ and see if this value makes the original inequality true.

$x = 1$

Step 1.6.7.2.2

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

$2 - \frac{3}{1} < 0$

Step 1.6.7.2.3

The left side $- 1$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 1.6.7.3

Test a value on the interval $x > \frac{3}{2}$ to see if it makes the inequality true.

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Step 1.6.7.3.1

Choose a value on the interval $x > \frac{3}{2}$ and see if this value makes the original inequality true.

$x = 4$

Step 1.6.7.3.2

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

$2 - \frac{3}{4} < 0$

Step 1.6.7.3.3

The left side $1.25$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 1.6.7.4

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

$x < 0$ False

$0 < x < \frac{3}{2}$ True

$x > \frac{3}{2}$ False

$x < 0$ False

$0 < x < \frac{3}{2}$ True

$x > \frac{3}{2}$ False

Step 1.6.8

The solution consists of all of the true intervals.

$0 < x < \frac{3}{2}$

Step 1.7

In the piece where $2 - \frac{3}{x}$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 - \frac{3}{x}) \geq 1$

Step 1.8

Find the domain of $- (2 - \frac{3}{x}) \geq 1$ and find the intersection with $0 < x < \frac{3}{2}$ .

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Step 1.8.1

Find the domain of $- (2 - \frac{3}{x}) \geq 1$ .

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Step 1.8.1.1

Set the denominator in $\frac{3}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 1.8.1.2

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

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

Step 1.8.2

Find the intersection of $0 < x < \frac{3}{2}$ and $(- \infty, 0) \cup (0, \infty)$ .

$0 < x < \frac{3}{2}$

Step 1.9

Write as a piecewise.

${\begin{cases} 2 - \frac{3}{x} \geq 1 & x < 0 or x \geq \frac{3}{2} \\ - (2 - \frac{3}{x}) \geq 1 & 0 < x < \frac{3}{2} \end{cases}$

Step 1.10

Simplify $- (2 - \frac{3}{x}) \geq 1$ .

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Step 1.10.1

Apply the distributive property.

${\begin{cases} 2 - \frac{3}{x} \geq 1 & x < 0 or x \geq \frac{3}{2} \\ - 1 \cdot 2 - - \frac{3}{x} \geq 1 & 0 < x < \frac{3}{2} \end{cases}$

Step 1.10.2

Multiply $- 1$ by $2$ .

${\begin{cases} 2 - \frac{3}{x} \geq 1 & x < 0 or x \geq \frac{3}{2} \\ - 2 - - \frac{3}{x} \geq 1 & 0 < x < \frac{3}{2} \end{cases}$

Step 1.10.3

Multiply $- - \frac{3}{x}$ .

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Step 1.10.3.1

Multiply $- 1$ by $- 1$ .

${\begin{cases} 2 - \frac{3}{x} \geq 1 & x < 0 or x \geq \frac{3}{2} \\ - 2 + 1 \frac{3}{x} \geq 1 & 0 < x < \frac{3}{2} \end{cases}$

Step 1.10.3.2

Multiply $\frac{3}{x}$ by $1$ .

${\begin{cases} 2 - \frac{3}{x} \geq 1 & x < 0 or x \geq \frac{3}{2} \\ - 2 + \frac{3}{x} \geq 1 & 0 < x < \frac{3}{2} \end{cases}$

Step 2

Solve $2 - \frac{3}{x} \geq 1$ for $x$ .

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Step 2.1

Move all terms not containing $x$ to the right side of the inequality.

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Step 2.1.1

Subtract $2$ from both sides of the inequality.

$- \frac{3}{x} \geq 1 - 2$

Step 2.1.2

Subtract $2$ from $1$ .

$- \frac{3}{x} \geq - 1$

Step 2.2

Multiply both sides by $x$ .

$- \frac{3}{x} x = - x$

Step 2.3

Simplify the left side.

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Step 2.3.1

Cancel the common factor of $x$ .

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Step 2.3.1.1

Move the leading negative in $- \frac{3}{x}$ into the numerator.

$\frac{- 3}{x} x = - x$

Step 2.3.1.2

Cancel the common factor.

$\frac{- 3}{x} x = - x$

Step 2.3.1.3

Rewrite the expression.

$- 3 = - x$

Step 2.4

Solve for $x$ .

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Step 2.4.1

Rewrite the equation as $- x = - 3$ .

$- x = - 3$

Step 2.4.2

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 2.4.2.1

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 2.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 2.5

Find the domain of $- \frac{3}{x} + 1$ .

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Step 2.5.1

Set the denominator in $\frac{3}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 2.5.2

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

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

Step 2.6

Use each root to create test intervals.

$x < 0$

$0 < x < 3$

$x > 3$

Step 2.7

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 2.7.1

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

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Step 2.7.1.1

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

$x = - 2$

Step 2.7.1.2

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

$2 - \frac{3}{- 2} \geq 1$

Step 2.7.1.3

The left side $3.5$ is greater than the right side $1$ , which means that the given statement is always true.

True

Step 2.7.2

Test a value on the interval $0 < x < 3$ to see if it makes the inequality true.

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Step 2.7.2.1

Choose a value on the interval $0 < x < 3$ and see if this value makes the original inequality true.

$x = 2$

Step 2.7.2.2

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

$2 - \frac{3}{2} \geq 1$

Step 2.7.2.3

The left side $0.5$ is less than the right side $1$ , which means that the given statement is false.

False

Step 2.7.3

Test a value on the interval $x > 3$ to see if it makes the inequality true.

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Step 2.7.3.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 2.7.3.2

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

$2 - \frac{3}{6} \geq 1$

Step 2.7.3.3

The left side $1.5$ is greater than the right side $1$ , which means that the given statement is always true.

True

Step 2.7.4

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

$x < 0$ True

$0 < x < 3$ False

$x > 3$ True

$x < 0$ True

$0 < x < 3$ False

$x > 3$ True

Step 2.8

The solution consists of all of the true intervals.

$x < 0$ or $x \geq 3$

Step 3

Solve $- 2 + \frac{3}{x} \geq 1$ for $x$ .

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Step 3.1

Move all terms not containing $x$ to the right side of the inequality.

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Step 3.1.1

Add $2$ to both sides of the inequality.

$\frac{3}{x} \geq 1 + 2$

Step 3.1.2

Add $1$ and $2$ .

$\frac{3}{x} \geq 3$

Step 3.2

Multiply both sides by $x$ .

$\frac{3}{x} x = 3 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $x$ .

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Step 3.3.1.1

Cancel the common factor.

$\frac{3}{x} x = 3 x$

Step 3.3.1.2

Rewrite the expression.

$3 = 3 x$

Step 3.4

Solve for $x$ .

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Step 3.4.1

Rewrite the equation as $3 x = 3$ .

$3 x = 3$

Step 3.4.2

Divide each term in $3 x = 3$ by $3$ and simplify.

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Step 3.4.2.1

Divide each term in $3 x = 3$ by $3$ .

$\frac{3 x}{3} = \frac{3}{3}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $3$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{3}{3}$

Step 3.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{3}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $3$ by $3$ .

$x = 1$

Step 3.5

Find the domain of $\frac{3}{x} - 3$ .

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Step 3.5.1

Set the denominator in $\frac{3}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 3.5.2

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

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

Step 3.6

Use each root to create test intervals.

$x < 0$

$0 < x < 1$

$x > 1$

Step 3.7

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 3.7.1

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

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Step 3.7.1.1

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

$x = - 2$

Step 3.7.1.2

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

$- 2 + \frac{3}{- 2} \geq 1$

Step 3.7.1.3

The left side $- 3.5$ is less than the right side $1$ , which means that the given statement is false.

False

Step 3.7.2

Test a value on the interval $0 < x < 1$ to see if it makes the inequality true.

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Step 3.7.2.1

Choose a value on the interval $0 < x < 1$ and see if this value makes the original inequality true.

$x = 0.5$

Step 3.7.2.2

Replace $x$ with $0.5$ in the original inequality.

$- 2 + \frac{3}{0.5} \geq 1$

Step 3.7.2.3

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

True

Step 3.7.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 3.7.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 3.7.3.2

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

$- 2 + \frac{3}{4} \geq 1$

Step 3.7.3.3

The left side $- 1.25$ is less than the right side $1$ , which means that the given statement is false.

False

Step 3.7.4

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

$x < 0$ False

$0 < x < 1$ True

$x > 1$ False

$x < 0$ False

$0 < x < 1$ True

$x > 1$ False

Step 3.8

The solution consists of all of the true intervals.

$0 < x \leq 1$

Step 4

Find the union of the solutions.

$x < 0$ or $0 < x \leq 1$

Step 5

Find the domain of $| 2 - \frac{3}{x} | - 1$ .

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Step 5.1

Set the denominator in $\frac{3}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 5.2

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

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

Step 6

Use each root to create test intervals.

$x < 0$

$0 < x < 1$

$x > 1$

Step 7

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 7.1

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

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Step 7.1.1

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

$x = - 2$

Step 7.1.2

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

$| 2 - \frac{3}{- 2} | \geq 1$

Step 7.1.3

The left side $3.5$ is greater than the right side $1$ , which means that the given statement is always true.

True

Step 7.2

Test a value on the interval $0 < x < 1$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $0 < x < 1$ and see if this value makes the original inequality true.

$x = 0.5$

Step 7.2.2

Replace $x$ with $0.5$ in the original inequality.

$| 2 - \frac{3}{0.5} | \geq 1$

Step 7.2.3

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

True

Step 7.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 7.3.2

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

$| 2 - \frac{3}{4} | \geq 1$

Step 7.3.3

The left side $1.25$ is greater than the right side $1$ , which means that the given statement is always true.

True

Step 7.4

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

$x < 0$ True

$0 < x < 1$ True

$x > 1$ True

$x < 0$ True

$0 < x < 1$ True

$x > 1$ True

Step 8

The solution consists of all of the true intervals.

$x < 0$ or $0 < x \leq 1$ or $x \geq 1$

Step 9

Combine the intervals.

$x < 0 or x > 0$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x < 0 or x > 0$

Interval Notation:

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

Step 11