Algebra Examples

$\sqrt{x} \leq 5$

Step 1

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

${\sqrt{x}}^{2} \leq 5^{2}$

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

${(x^{\frac{1}{2}})}^{2} \leq 5^{2}$

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

$x^{\frac{1}{2} \cdot 2} \leq 5^{2}$

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.

$x^{\frac{1}{2} \cdot 2} \leq 5^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

$x^{1} \leq 5^{2}$

Step 2.2.1.2

Simplify.

$x \leq 5^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $5$ to the power of $2$ .

$x \leq 25$

Step 3

Find the domain of $\sqrt{x} - 5$ .

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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.

$x < 0$

$0 < x < 25$

$x > 25$

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.

$\sqrt{- 2} \leq 5$

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 $0 < x < 25$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $0 < x < 25$ and see if this value makes the original inequality true.

$x = 2$

Step 5.2.2

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

$\sqrt{2} \leq 5$

Step 5.2.3

The left side $1.41421356$ is less than the right side $5$ , which means that the given statement is always true.

True

Step 5.3

Test a value on the interval $x > 25$ to see if it makes the inequality true.

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Step 5.3.1

Choose a value on the interval $x > 25$ and see if this value makes the original inequality true.

$x = 28$

Step 5.3.2

Replace $x$ with $28$ in the original inequality.

$\sqrt{28} \leq 5$

Step 5.3.3

The left side $5.29150262$ is greater than the right side $5$ , 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

$0 < x < 25$ True

$x > 25$ False

$x < 0$ False

$0 < x < 25$ True

$x > 25$ False

Step 6

The solution consists of all of the true intervals.

$0 \leq x \leq 25$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$0 \leq x \leq 25$

Interval Notation:

$[0, 25]$

Step 8