Pre-Algebra Examples

$| \frac{2 x - 1}{x} | < 2$

Step 1

Write $| \frac{2 x - 1}{x} | < 2$ 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.

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

Step 1.2

Solve the inequality.

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Step 1.2.1

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

$x = 0$

$2 x - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$2 x = 1$

Step 1.2.3

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 1.2.3.1

Divide each term in $2 x = 1$ by $2$ .

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

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

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

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.4

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

$x = 0$

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

Step 1.2.5

Consolidate the solutions.

$x = 0, \frac{1}{2}$

Step 1.2.6

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

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Step 1.2.6.1

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

$x = 0$

Step 1.2.6.2

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

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

Step 1.2.7

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{1}{2}$

$x > \frac{1}{2}$

Step 1.2.8

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 1.2.8.1

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

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Step 1.2.8.1.1

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

$x = - 2$

Step 1.2.8.1.2

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

$\frac{2 (- 2) - 1}{- 2} \geq 0$

Step 1.2.8.1.3

The left side $2.5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.2.8.2

Test a value on the interval $0 < x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 1.2.8.2.1

Choose a value on the interval $0 < x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0.25$

Step 1.2.8.2.2

Replace $x$ with $0.25$ in the original inequality.

$\frac{2 (0.25) - 1}{0.25} \geq 0$

Step 1.2.8.2.3

The left side $- 2$ is less than the right side $0$ , which means that the given statement is false.

False

Step 1.2.8.3

Test a value on the interval $x > \frac{1}{2}$ to see if it makes the inequality true.

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Step 1.2.8.3.1

Choose a value on the interval $x > \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 3$

Step 1.2.8.3.2

Replace $x$ with $3$ in the original inequality.

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

Step 1.2.8.3.3

The left side $1.\overline{6}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.2.8.4

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

$x < 0$ True

$0 < x < \frac{1}{2}$ False

$x > \frac{1}{2}$ True

$x < 0$ True

$0 < x < \frac{1}{2}$ False

$x > \frac{1}{2}$ True

Step 1.2.9

The solution consists of all of the true intervals.

$x < 0$ or $x \geq \frac{1}{2}$

Step 1.3

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

$\frac{2 x - 1}{x} < 2$

Step 1.4

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

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Step 1.4.1

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

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Step 1.4.1.1

Set the denominator in $\frac{2 x - 1}{x}$ 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 $x < 0 or x \geq \frac{1}{2}$ and $(- \infty, 0) \cup (0, \infty)$ .

$x < 0 or x \geq \frac{1}{2}$

Step 1.5

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

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

Step 1.6

Solve the inequality.

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Step 1.6.1

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

$x = 0$

$2 x - 1 = 0$

Step 1.6.2

Add $1$ to both sides of the equation.

$2 x = 1$

Step 1.6.3

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 1.6.3.1

Divide each term in $2 x = 1$ by $2$ .

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

Step 1.6.3.2

Simplify the left side.

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Step 1.6.3.2.1

Cancel the common factor of $2$ .

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Step 1.6.3.2.1.1

Cancel the common factor.

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

Step 1.6.3.2.1.2

Divide $x$ by $1$ .

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

Step 1.6.4

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

$x = 0$

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

Step 1.6.5

Consolidate the solutions.

$x = 0, \frac{1}{2}$

Step 1.6.6

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

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Step 1.6.6.1

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

$x = 0$

Step 1.6.6.2

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

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

Step 1.6.7

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{1}{2}$

$x > \frac{1}{2}$

Step 1.6.8

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 1.6.8.1

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

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Step 1.6.8.1.1

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

$x = - 2$

Step 1.6.8.1.2

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

$\frac{2 (- 2) - 1}{- 2} < 0$

Step 1.6.8.1.3

The left side $2.5$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 1.6.8.2

Test a value on the interval $0 < x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 1.6.8.2.1

Choose a value on the interval $0 < x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0.25$

Step 1.6.8.2.2

Replace $x$ with $0.25$ in the original inequality.

$\frac{2 (0.25) - 1}{0.25} < 0$

Step 1.6.8.2.3

The left side $- 2$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 1.6.8.3

Test a value on the interval $x > \frac{1}{2}$ to see if it makes the inequality true.

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Step 1.6.8.3.1

Choose a value on the interval $x > \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 3$

Step 1.6.8.3.2

Replace $x$ with $3$ in the original inequality.

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

Step 1.6.8.3.3

The left side $1.\overline{6}$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 1.6.8.4

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

$x < 0$ False

$0 < x < \frac{1}{2}$ True

$x > \frac{1}{2}$ False

$x < 0$ False

$0 < x < \frac{1}{2}$ True

$x > \frac{1}{2}$ False

Step 1.6.9

The solution consists of all of the true intervals.

$0 < x < \frac{1}{2}$

Step 1.7

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

$- \frac{2 x - 1}{x} < 2$

Step 1.8

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

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Step 1.8.1

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

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Step 1.8.1.1

Set the denominator in $\frac{2 x - 1}{x}$ 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 $0 < x < \frac{1}{2}$ and $(- \infty, 0) \cup (0, \infty)$ .

$0 < x < \frac{1}{2}$

Step 1.9

Write as a piecewise.

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

Step 2

Solve $\frac{2 x - 1}{x} < 2$ when $x < 0 or x \geq \frac{1}{2}$ .

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Step 2.1

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

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Step 2.1.1

Subtract $2$ from both sides of the inequality.

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

Step 2.1.2

Simplify $\frac{2 x - 1}{x} - 2$ .

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Step 2.1.2.1

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

$\frac{2 x - 1}{x} - 2 \cdot \frac{x}{x} < 0$

Step 2.1.2.2

Combine $- 2$ and $\frac{x}{x}$ .

$\frac{2 x - 1}{x} + \frac{- 2 x}{x} < 0$

Step 2.1.2.3

Combine the numerators over the common denominator.

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

Step 2.1.2.4

Simplify the numerator.

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Step 2.1.2.4.1

Subtract $2 x$ from $2 x$ .

$\frac{0 - 1}{x} < 0$

Step 2.1.2.4.2

Subtract $1$ from $0$ .

$\frac{- 1}{x} < 0$

Step 2.1.2.5

Move the negative in front of the fraction.

$- \frac{1}{x} < 0$

Step 2.1.3

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

$x = 0$

Step 2.1.4

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

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Step 2.1.4.1

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

$x = 0$

Step 2.1.4.2

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

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

Step 2.1.5

The solution consists of all of the true intervals.

$x > 0$

Step 2.2

Find the intersection of $x > 0$ and $x < 0 or x \geq \frac{1}{2}$ .

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

Step 3

Solve $- \frac{2 x - 1}{x} < 2$ when $0 < x < \frac{1}{2}$ .

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Step 3.1

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

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Step 3.1.1

Subtract $2$ from both sides of the inequality.

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

Step 3.1.2

Simplify $- \frac{2 x - 1}{x} - 2$ .

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Step 3.1.2.1

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

$- \frac{2 x - 1}{x} - 2 \cdot \frac{x}{x} < 0$

Step 3.1.2.2

Combine $- 2$ and $\frac{x}{x}$ .

$- \frac{2 x - 1}{x} + \frac{- 2 x}{x} < 0$

Step 3.1.2.3

Combine the numerators over the common denominator.

$\frac{- (2 x - 1) - 2 x}{x} < 0$

Step 3.1.2.4

Simplify the numerator.

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Step 3.1.2.4.1

Apply the distributive property.

$\frac{- (2 x) - - 1 - 2 x}{x} < 0$

Step 3.1.2.4.2

Multiply $2$ by $- 1$ .

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

Step 3.1.2.4.3

Multiply $- 1$ by $- 1$ .

$\frac{- 2 x + 1 - 2 x}{x} < 0$

Step 3.1.2.4.4

Subtract $2 x$ from $- 2 x$ .

$\frac{- 4 x + 1}{x} < 0$

Step 3.1.2.5

Factor $- 1$ out of $- 4 x$ .

$\frac{- (4 x) + 1}{x} < 0$

Step 3.1.2.6

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

$\frac{- (4 x) - 1 (- 1)}{x} < 0$

Step 3.1.2.7

Factor $- 1$ out of $- (4 x) - 1 (- 1)$ .

$\frac{- (4 x - 1)}{x} < 0$

Step 3.1.2.8

Rewrite $- (4 x - 1)$ as $- 1 (4 x - 1)$ .

$\frac{- 1 (4 x - 1)}{x} < 0$

Step 3.1.2.9

Move the negative in front of the fraction.

$- \frac{4 x - 1}{x} < 0$

Step 3.1.3

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

$x = 0$

$4 x - 1 = 0$

Step 3.1.4

Add $1$ to both sides of the equation.

$4 x = 1$

Step 3.1.5

Divide each term in $4 x = 1$ by $4$ and simplify.

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Step 3.1.5.1

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Step 3.1.5.2

Simplify the left side.

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Step 3.1.5.2.1

Cancel the common factor of $4$ .

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Step 3.1.5.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Step 3.1.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 3.1.6

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

$x = 0$

$x = \frac{1}{4}$

Step 3.1.7

Consolidate the solutions.

$x = 0, \frac{1}{4}$

Step 3.1.8

Find the domain of $- \frac{4 x - 1}{x}$ .

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Step 3.1.8.1

Set the denominator in $\frac{4 x - 1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 3.1.8.2

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

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

Step 3.1.9

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{1}{4}$

$x > \frac{1}{4}$

Step 3.1.10

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 3.1.10.1

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

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Step 3.1.10.1.1

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

$x = - 2$

Step 3.1.10.1.2

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

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

Step 3.1.10.1.3

The left side $- 2.5$ is less than the right side $2$ , which means that the given statement is always true.

True

Step 3.1.10.2

Test a value on the interval $0 < x < \frac{1}{4}$ to see if it makes the inequality true.

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Step 3.1.10.2.1

Choose a value on the interval $0 < x < \frac{1}{4}$ and see if this value makes the original inequality true.

$x = 0.12$

Step 3.1.10.2.2

Replace $x$ with $0.12$ in the original inequality.

$- \frac{2 (0.12) - 1}{0.12} < 2$

Step 3.1.10.2.3

The left side $6.\overline{3}$ is not less than the right side $2$ , which means that the given statement is false.

False

Step 3.1.10.3

Test a value on the interval $x > \frac{1}{4}$ to see if it makes the inequality true.

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Step 3.1.10.3.1

Choose a value on the interval $x > \frac{1}{4}$ and see if this value makes the original inequality true.

$x = 3$

Step 3.1.10.3.2

Replace $x$ with $3$ in the original inequality.

$- \frac{2 (3) - 1}{3} < 2$

Step 3.1.10.3.3

The left side $- 1.\overline{6}$ is less than the right side $2$ , which means that the given statement is always true.

True

Step 3.1.10.4

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

$x < 0$ True

$0 < x < \frac{1}{4}$ False

$x > \frac{1}{4}$ True

$x < 0$ True

$0 < x < \frac{1}{4}$ False

$x > \frac{1}{4}$ True

Step 3.1.11

The solution consists of all of the true intervals.

$x < 0$ or $x > \frac{1}{4}$

Step 3.2

Find the intersection of $x < 0 or x > \frac{1}{4}$ and $0 < x < \frac{1}{2}$ .

$\frac{1}{4} < x < \frac{1}{2}$

Step 4

Find the union of the solutions.

$x > \frac{1}{4}$

Step 5

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

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Step 5.1

Set the denominator in $\frac{2 x - 1}{x}$ 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

The solution consists of all of the true intervals.

$x > \frac{1}{4}$

Step 7