Pre-Algebra Examples

$\sqrt{x} < 10$

Step 1

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

${\sqrt{x}}^{2} < 10^{2}$

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 $x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} < 10^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} < 10^{2}$

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.

$x^{\frac{1}{2} \cdot 2} < 10^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

$x^{1} < 10^{2}$

Step 2.2.1.2

Simplify.

$x < 10^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $10$ to the power of $2$ .

$x < 100$

Step 3

Find the domain of $\sqrt{x} - 10$ .

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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.

$x < 0$

$0 < x < 100$

$x > 100$

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.

$\sqrt{- 2} < 10$

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 $0 < x < 100$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $0 < x < 100$ and see if this value makes the original inequality true.

$x = 2$

Step 5.2.2

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

$\sqrt{2} < 10$

Step 5.2.3

The left side $1.41421356$ is less than the right side $10$ , which means that the given statement is always true.

True

Step 5.3

Test a value on the interval $x > 100$ to see if it makes the inequality true.

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Step 5.3.1

Choose a value on the interval $x > 100$ and see if this value makes the original inequality true.

$x = 102$

Step 5.3.2

Replace $x$ with $102$ in the original inequality.

$\sqrt{102} < 10$

Step 5.3.3

The left side $10.09950493$ is not less than the right side $10$ , 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

$0 < x < 100$ True

$x > 100$ False

$x < 0$ False

$0 < x < 100$ True

$x > 100$ False

Step 6

The solution consists of all of the true intervals.

$0 < x < 100$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 100$

Interval Notation:

$(0, 100)$

Step 8