Precalculus Examples

$\arcsin (\sqrt{\frac{2 x}{1 - 2 x}})$

Step 1

Set the radicand in $\sqrt{\frac{2 x}{1 - 2 x}}$ greater than or equal to $0$ to find where the expression is defined.

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

Step 2

Solve for $x$ .

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Step 2.1

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

$2 x = 0$

$1 - 2 x = 0$

Step 2.2

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x = 0$ by $2$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $2$ .

$x = 0$

Step 2.3

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $- 2$ .

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Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Divide $x$ by $1$ .

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Dividing two negative values results in a positive value.

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

Step 2.5

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 2.6

Consolidate the solutions.

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

Step 2.7

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

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Step 2.7.1

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

$1 - 2 x = 0$

Step 2.7.2

Solve for $x$ .

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Step 2.7.2.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 2.7.2.2

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

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Step 2.7.2.2.1

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

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

Step 2.7.2.2.2

Simplify the left side.

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Step 2.7.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.7.2.2.2.1.1

Cancel the common factor.

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

Step 2.7.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.7.2.2.3

Simplify the right side.

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Step 2.7.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 2.7.3

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

$(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$

Step 2.8

Use each root to create test intervals.

$x < 0$

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

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

Step 2.9

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 2.9.1

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

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Step 2.9.1.1

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

$x = - 2$

Step 2.9.1.2

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

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

Step 2.9.1.3

The left side $- 0.8$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.9.2

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

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Step 2.9.2.1

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 2.9.2.2

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

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

Step 2.9.2.3

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

True

Step 2.9.3

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

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Step 2.9.3.1

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

$x = 3$

Step 2.9.3.2

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

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

Step 2.9.3.3

The left side $- 1.2$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.9.4

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 2.10

The solution consists of all of the true intervals.

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

Step 3

Set the argument in $\arcsin (\sqrt{\frac{2 x}{1 - 2 x}})$ greater than or equal to $- 1$ to find where the expression is defined.

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

Step 4

Solve for $x$ .

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Step 4.1

To remove the radical on the left side of the inequality, square both sides of the inequality.

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

Step 4.2

Simplify each side of the inequality.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{2 x}{1 - 2 x}}$ as ${(\frac{2 x}{1 - 2 x})}^{\frac{1}{2}}$ .

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

Multiply the exponents in ${({(\frac{2 x}{1 - 2 x})}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

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

Step 4.2.2.1.1.2.2

Rewrite the expression.

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

Step 4.2.2.1.2

Simplify.

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

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raise $- 1$ to the power of $2$ .

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

Step 4.3

Solve for $x$ .

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Step 4.3.1

Subtract $1$ from both sides of the inequality.

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

Step 4.3.2

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

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Step 4.3.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{1 - 2 x}{1 - 2 x}$ .

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

Step 4.3.2.2

Combine $- 1$ and $\frac{1 - 2 x}{1 - 2 x}$ .

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

Step 4.3.2.3

Combine the numerators over the common denominator.

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

Step 4.3.2.4

Simplify the numerator.

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Step 4.3.2.4.1

Apply the distributive property.

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

Step 4.3.2.4.2

Multiply $- 1$ by $1$ .

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

Step 4.3.2.4.3

Multiply $- 2$ by $- 1$ .

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

Step 4.3.2.4.4

Add $2 x$ and $2 x$ .

$\frac{4 x - 1}{1 - 2 x} \geq 0$

Step 4.3.3

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

$4 x - 1 = 0$

$1 - 2 x = 0$

Step 4.3.4

Add $1$ to both sides of the equation.

$4 x = 1$

Step 4.3.5

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

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Step 4.3.5.1

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

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

Step 4.3.5.2

Simplify the left side.

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Step 4.3.5.2.1

Cancel the common factor of $4$ .

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Step 4.3.5.2.1.1

Cancel the common factor.

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

Step 4.3.5.2.1.2

Divide $x$ by $1$ .

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

Step 4.3.6

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 4.3.7

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

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Step 4.3.7.1

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

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

Step 4.3.7.2

Simplify the left side.

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Step 4.3.7.2.1

Cancel the common factor of $- 2$ .

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Step 4.3.7.2.1.1

Cancel the common factor.

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

Step 4.3.7.2.1.2

Divide $x$ by $1$ .

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

Step 4.3.7.3

Simplify the right side.

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Step 4.3.7.3.1

Dividing two negative values results in a positive value.

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

Step 4.3.8

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

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

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

Step 4.3.9

Consolidate the solutions.

$x = \frac{1}{4}, \frac{1}{2}$

Step 4.4

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

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Step 4.4.1

Set the radicand in $\sqrt{2 x (1 - 2 x)}$ greater than or equal to $0$ to find where the expression is defined.

$2 x (1 - 2 x) \geq 0$

Step 4.4.2

Solve for $x$ .

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Step 4.4.2.1

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

$x = 0$

$1 - 2 x = 0$

Step 4.4.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 4.4.2.3

Set $1 - 2 x$ equal to $0$ and solve for $x$ .

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Step 4.4.2.3.1

Set $1 - 2 x$ equal to $0$ .

$1 - 2 x = 0$

Step 4.4.2.3.2

Solve $1 - 2 x = 0$ for $x$ .

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Step 4.4.2.3.2.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 4.4.2.3.2.2

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

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Step 4.4.2.3.2.2.1

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

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

Step 4.4.2.3.2.2.2

Simplify the left side.

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Step 4.4.2.3.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 4.4.2.3.2.2.2.1.1

Cancel the common factor.

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

Step 4.4.2.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.4.2.3.2.2.3

Simplify the right side.

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Step 4.4.2.3.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 4.4.2.4

The final solution is all the values that make $2 x (1 - 2 x) \geq 0$ true.

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

Step 4.4.2.5

Use each root to create test intervals.

$x < 0$

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

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

Step 4.4.2.6

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 4.4.2.6.1

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

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Step 4.4.2.6.1.1

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

$x = - 2$

Step 4.4.2.6.1.2

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

$2 (- 2) (1 - 2 \cdot - 2) \geq 0$

Step 4.4.2.6.1.3

The left side $- 20$ is less than the right side $0$ , which means that the given statement is false.

False

Step 4.4.2.6.2

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

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Step 4.4.2.6.2.1

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 4.4.2.6.2.2

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

$2 (0.25) (1 - 2 \cdot 0.25) \geq 0$

Step 4.4.2.6.2.3

The left side $0.25$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 4.4.2.6.3

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

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Step 4.4.2.6.3.1

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

$x = 3$

Step 4.4.2.6.3.2

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

$2 (3) (1 - 2 \cdot 3) \geq 0$

Step 4.4.2.6.3.3

The left side $- 30$ is less than the right side $0$ , which means that the given statement is false.

False

Step 4.4.2.6.4

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 4.4.2.7

The solution consists of all of the true intervals.

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

Step 4.4.3

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

$1 - 2 x = 0$

Step 4.4.4

Solve for $x$ .

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Step 4.4.4.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 4.4.4.2

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

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Step 4.4.4.2.1

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

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

Step 4.4.4.2.2

Simplify the left side.

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Step 4.4.4.2.2.1

Cancel the common factor of $- 2$ .

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Step 4.4.4.2.2.1.1

Cancel the common factor.

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

Step 4.4.4.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.4.4.2.3

Simplify the right side.

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Step 4.4.4.2.3.1

Dividing two negative values results in a positive value.

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

Step 4.4.5

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

$[0, \frac{1}{2})$

Step 4.5

Use each root to create test intervals.

$x < 0$

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

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

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

Step 4.6

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 4.6.1

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

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Step 4.6.1.1

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

$x = - 2$

Step 4.6.1.2

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

$\sqrt{\frac{2 (- 2)}{1 - 2 \cdot - 2}} \geq - 1$

Step 4.6.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 4.6.2

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

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Step 4.6.2.1

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 4.6.2.2

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

$\sqrt{\frac{2 (0.12)}{1 - 2 \cdot 0.12}} \geq - 1$

Step 4.6.2.3

The left side $0.56195148$ is greater than the right side $- 1$ , which means that the given statement is always true.

True

Step 4.6.3

Test a value on the interval $\frac{1}{4} < x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 4.6.3.1

Choose a value on the interval $\frac{1}{4} < x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0.38$

Step 4.6.3.2

Replace $x$ with $0.38$ in the original inequality.

$\sqrt{\frac{2 (0.38)}{1 - 2 \cdot 0.38}} \geq - 1$

Step 4.6.3.3

The left side $1.77951304$ is greater than the right side $- 1$ , which means that the given statement is always true.

True

Step 4.6.4

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

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Step 4.6.4.1

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

$x = 3$

Step 4.6.4.2

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

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

Step 4.6.4.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 4.6.5

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

$x < 0$ False

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

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

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

$x < 0$ False

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

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

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

Step 4.7

The solution consists of all of the true intervals.

$0 \leq x \leq \frac{1}{4}$ or $\frac{1}{4} \leq x < \frac{1}{2}$

Step 4.8

Combine the intervals.

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

Step 5

Set the argument in $\arcsin (\sqrt{\frac{2 x}{1 - 2 x}})$ less than or equal to $1$ to find where the expression is defined.

$\sqrt{\frac{2 x}{1 - 2 x}} \leq 1$

Step 6

Solve for $x$ .

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Step 6.1

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{\frac{2 x}{1 - 2 x}}}^{2} \leq 1^{2}$

Step 6.2

Simplify each side of the inequality.

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Step 6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{2 x}{1 - 2 x}}$ as ${(\frac{2 x}{1 - 2 x})}^{\frac{1}{2}}$ .

${({(\frac{2 x}{1 - 2 x})}^{\frac{1}{2}})}^{2} \leq 1^{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

Multiply the exponents in ${({(\frac{2 x}{1 - 2 x})}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(\frac{2 x}{1 - 2 x})}^{\frac{1}{2} \cdot 2} \leq 1^{2}$

Step 6.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.2.1

Cancel the common factor.

${(\frac{2 x}{1 - 2 x})}^{\frac{1}{2} \cdot 2} \leq 1^{2}$

Step 6.2.2.1.1.2.2

Rewrite the expression.

${(\frac{2 x}{1 - 2 x})}^{1} \leq 1^{2}$

Step 6.2.2.1.2

Simplify.

$\frac{2 x}{1 - 2 x} \leq 1^{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

One to any power is one.

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

Step 6.3

Solve for $x$ .

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Step 6.3.1

Subtract $1$ from both sides of the inequality.

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

Step 6.3.2

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

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Step 6.3.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{1 - 2 x}{1 - 2 x}$ .

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

Step 6.3.2.2

Combine $- 1$ and $\frac{1 - 2 x}{1 - 2 x}$ .

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

Step 6.3.2.3

Combine the numerators over the common denominator.

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

Step 6.3.2.4

Simplify the numerator.

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Step 6.3.2.4.1

Apply the distributive property.

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

Step 6.3.2.4.2

Multiply $- 1$ by $1$ .

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

Step 6.3.2.4.3

Multiply $- 2$ by $- 1$ .

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

Step 6.3.2.4.4

Add $2 x$ and $2 x$ .

$\frac{4 x - 1}{1 - 2 x} \leq 0$

Step 6.3.3

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

$4 x - 1 = 0$

$1 - 2 x = 0$

Step 6.3.4

Add $1$ to both sides of the equation.

$4 x = 1$

Step 6.3.5

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

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Step 6.3.5.1

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

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

Step 6.3.5.2

Simplify the left side.

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Step 6.3.5.2.1

Cancel the common factor of $4$ .

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Step 6.3.5.2.1.1

Cancel the common factor.

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

Step 6.3.5.2.1.2

Divide $x$ by $1$ .

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

Step 6.3.6

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 6.3.7

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

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Step 6.3.7.1

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

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

Step 6.3.7.2

Simplify the left side.

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Step 6.3.7.2.1

Cancel the common factor of $- 2$ .

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Step 6.3.7.2.1.1

Cancel the common factor.

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

Step 6.3.7.2.1.2

Divide $x$ by $1$ .

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

Step 6.3.7.3

Simplify the right side.

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Step 6.3.7.3.1

Dividing two negative values results in a positive value.

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

Step 6.3.8

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

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

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

Step 6.3.9

Consolidate the solutions.

$x = \frac{1}{4}, \frac{1}{2}$

Step 6.4

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

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Step 6.4.1

Set the radicand in $\sqrt{2 x (1 - 2 x)}$ greater than or equal to $0$ to find where the expression is defined.

$2 x (1 - 2 x) \geq 0$

Step 6.4.2

Solve for $x$ .

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Step 6.4.2.1

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

$x = 0$

$1 - 2 x = 0$

Step 6.4.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 6.4.2.3

Set $1 - 2 x$ equal to $0$ and solve for $x$ .

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Step 6.4.2.3.1

Set $1 - 2 x$ equal to $0$ .

$1 - 2 x = 0$

Step 6.4.2.3.2

Solve $1 - 2 x = 0$ for $x$ .

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Step 6.4.2.3.2.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 6.4.2.3.2.2

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

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Step 6.4.2.3.2.2.1

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

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

Step 6.4.2.3.2.2.2

Simplify the left side.

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Step 6.4.2.3.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 6.4.2.3.2.2.2.1.1

Cancel the common factor.

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

Step 6.4.2.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 6.4.2.3.2.2.3

Simplify the right side.

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Step 6.4.2.3.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 6.4.2.4

The final solution is all the values that make $2 x (1 - 2 x) \geq 0$ true.

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

Step 6.4.2.5

Use each root to create test intervals.

$x < 0$

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

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

Step 6.4.2.6

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 6.4.2.6.1

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

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Step 6.4.2.6.1.1

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

$x = - 2$

Step 6.4.2.6.1.2

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

$2 (- 2) (1 - 2 \cdot - 2) \geq 0$

Step 6.4.2.6.1.3

The left side $- 20$ is less than the right side $0$ , which means that the given statement is false.

False

Step 6.4.2.6.2

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

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Step 6.4.2.6.2.1

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 6.4.2.6.2.2

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

$2 (0.25) (1 - 2 \cdot 0.25) \geq 0$

Step 6.4.2.6.2.3

The left side $0.25$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.4.2.6.3

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

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Step 6.4.2.6.3.1

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

$x = 3$

Step 6.4.2.6.3.2

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

$2 (3) (1 - 2 \cdot 3) \geq 0$

Step 6.4.2.6.3.3

The left side $- 30$ is less than the right side $0$ , which means that the given statement is false.

False

Step 6.4.2.6.4

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 6.4.2.7

The solution consists of all of the true intervals.

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

Step 6.4.3

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

$1 - 2 x = 0$

Step 6.4.4

Solve for $x$ .

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Step 6.4.4.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 6.4.4.2

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

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Step 6.4.4.2.1

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

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

Step 6.4.4.2.2

Simplify the left side.

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Step 6.4.4.2.2.1

Cancel the common factor of $- 2$ .

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Step 6.4.4.2.2.1.1

Cancel the common factor.

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

Step 6.4.4.2.2.1.2

Divide $x$ by $1$ .

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

Step 6.4.4.2.3

Simplify the right side.

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Step 6.4.4.2.3.1

Dividing two negative values results in a positive value.

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

Step 6.4.5

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

$[0, \frac{1}{2})$

Step 6.5

Use each root to create test intervals.

$x < 0$

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

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

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

Step 6.6

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 6.6.1

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

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Step 6.6.1.1

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

$x = - 2$

Step 6.6.1.2

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

$\sqrt{\frac{2 (- 2)}{1 - 2 \cdot - 2}} \leq 1$

Step 6.6.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 6.6.2

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

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Step 6.6.2.1

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 6.6.2.2

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

$\sqrt{\frac{2 (0.12)}{1 - 2 \cdot 0.12}} \leq 1$

Step 6.6.2.3

The left side $0.56195148$ is less than the right side $1$ , which means that the given statement is always true.

True

Step 6.6.3

Test a value on the interval $\frac{1}{4} < x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 6.6.3.1

Choose a value on the interval $\frac{1}{4} < x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0.38$

Step 6.6.3.2

Replace $x$ with $0.38$ in the original inequality.

$\sqrt{\frac{2 (0.38)}{1 - 2 \cdot 0.38}} \leq 1$

Step 6.6.3.3

The left side $1.77951304$ is greater than the right side $1$ , which means that the given statement is false.

False

Step 6.6.4

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

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Step 6.6.4.1

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

$x = 3$

Step 6.6.4.2

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

$\sqrt{\frac{2 (3)}{1 - 2 \cdot 3}} \leq 1$

Step 6.6.4.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 6.6.5

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

$x < 0$ False

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

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

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

$x < 0$ False

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

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

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

Step 6.7

The solution consists of all of the true intervals.

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

Step 7

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

$1 - 2 x = 0$

Step 8

Solve for $x$ .

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Step 8.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $- 2$ .

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Step 8.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.1.2

Divide $x$ by $1$ .

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

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Dividing two negative values results in a positive value.

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

Step 9

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

Interval Notation:

$[0, \frac{1}{4}]$

Set-Builder Notation:

${x | 0 \leq x \leq \frac{1}{4}}$

Step 10