Pre-Algebra Examples

$\sqrt{x} < - 1$

Step 1

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

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

Step 2

Simplify each side of the inequality.

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Step 2.1

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 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$x < 1$

Step 3

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

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Step 3.1

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

$x \geq 0$

Step 3.2

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

$[0, \infty)$

Step 4

Use each root to create test intervals.

$x < 0$

$0 < x < 1$

$x > 1$

Step 5

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 5.1

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

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Step 5.1.1

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

$x = - 2$

Step 5.1.2

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

$\sqrt{- 2} < - 1$

Step 5.1.3

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

False

Step 5.2

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

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Step 5.2.1

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

$x = 0.5$

Step 5.2.2

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

$\sqrt{0.5} < - 1$

Step 5.2.3

The left side $0.70710678$ is not less than the right side $- 1$ , which means that the given statement is false.

False

Step 5.3

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

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Step 5.3.1

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

$x = 4$

Step 5.3.2

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

$\sqrt{4} < - 1$

Step 5.3.3

The left side $2$ is not less than the right side $- 1$ , which means that the given statement is false.

False

Step 5.4

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 6

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

No solution

Step 7