Finite Math Examples

$\sqrt{1 + \sqrt{x}} \sqrt{1 - \sqrt{x}} \sqrt{1 - \sqrt{x}}$

Step 1

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 2

Set the radicand in $\sqrt{1 + \sqrt{x}}$ less than $0$ to find where the expression is undefined.

$1 + \sqrt{x} < 0$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $1$ from both sides of the inequality.

$\sqrt{x} < - 1$

Step 3.2

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

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

Step 3.3

Simplify each side of the inequality.

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

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

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Step 3.3.2.1.1.1

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

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

Step 3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

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

Step 3.3.2.1.1.2.2

Rewrite the expression.

$x^{1} < {(- 1)}^{2}$

Step 3.3.2.1.2

Simplify.

$x < {(- 1)}^{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

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

$x < 1$

Step 3.4

Find the domain of $1 + \sqrt{x}$ .

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Step 3.4.1

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

$x \geq 0$

Step 3.4.2

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

$[0, \infty)$

Step 3.5

Use each root to create test intervals.

$x < 0$

$0 < x < 1$

$x > 1$

Step 3.6

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 3.6.1

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

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Step 3.6.1.1

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

$x = - 2$

Step 3.6.1.2

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

$1 + \sqrt{- 2} < 0$

Step 3.6.1.3

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

False

Step 3.6.2

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

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Step 3.6.2.1

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

$x = 0.5$

Step 3.6.2.2

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

$1 + \sqrt{0.5} < 0$

Step 3.6.2.3

The left side $1.70710678$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 3.6.3

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

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Step 3.6.3.1

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

$x = 4$

Step 3.6.3.2

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

$1 + \sqrt{4} < 0$

Step 3.6.3.3

The left side $3$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 3.6.4

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

$x < 0$ False

$0 < x < 1$ False

$x > 1$ False

$x < 0$ False

$0 < x < 1$ False

$x > 1$ False

Step 3.7

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Step 4

Set the radicand in $\sqrt{1 - \sqrt{x}}$ less than $0$ to find where the expression is undefined.

$1 - \sqrt{x} < 0$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $1$ from both sides of the inequality.

$- \sqrt{x} < - 1$

Step 5.2

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

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

Step 5.3

Simplify each side of the inequality.

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

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

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Step 5.3.2.1.1

Apply the product rule to $- x^{\frac{1}{2}}$ .

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

Step 5.3.2.1.2

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

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

Step 5.3.2.1.3

Multiply ${(x^{\frac{1}{2}})}^{2}$ by $1$ .

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

Step 5.3.2.1.4

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

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Step 5.3.2.1.4.1

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

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

Step 5.3.2.1.4.2

Cancel the common factor of $2$ .

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Step 5.3.2.1.4.2.1

Cancel the common factor.

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

Step 5.3.2.1.4.2.2

Rewrite the expression.

$x^{1} < {(- 1)}^{2}$

Step 5.3.2.1.5

Simplify.

$x < {(- 1)}^{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

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

$x < 1$

Step 5.4

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

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Step 5.4.1

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

$x \geq 0$

Step 5.4.2

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

$[0, \infty)$

Step 5.5

The solution consists of all of the true intervals.

$x > 1$

Step 6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x < 0, x > 1$

$(- \infty, 0) \cup (1, \infty)$

Step 7