Algebra Examples

$2 + \sqrt{5 x - 1} > 5$

Step 1

Move all terms not containing $\sqrt{5 x - 1}$ to the right side of the inequality.

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Step 1.1

Subtract $2$ from both sides of the inequality.

$\sqrt{5 x - 1} > 5 - 2$

Step 1.2

Subtract $2$ from $5$ .

$\sqrt{5 x - 1} > 3$

Step 2

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

${\sqrt{5 x - 1}}^{2} > 3^{2}$

Step 3

Simplify each side of the inequality.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x - 1}$ as ${(5 x - 1)}^{\frac{1}{2}}$ .

${({(5 x - 1)}^{\frac{1}{2}})}^{2} > 3^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(5 x - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(5 x - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(5 x - 1)}^{\frac{1}{2} \cdot 2} > 3^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(5 x - 1)}^{\frac{1}{2} \cdot 2} > 3^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(5 x - 1)}^{1} > 3^{2}$

Step 3.2.1.2

Simplify.

$5 x - 1 > 3^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $3$ to the power of $2$ .

$5 x - 1 > 9$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms not containing $x$ to the right side of the inequality.

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Step 4.1.1

Add $1$ to both sides of the inequality.

$5 x > 9 + 1$

Step 4.1.2

Add $9$ and $1$ .

$5 x > 10$

Step 4.2

Divide each term in $5 x > 10$ by $5$ and simplify.

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Step 4.2.1

Divide each term in $5 x > 10$ by $5$ .

$\frac{5 x}{5} > \frac{10}{5}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} > \frac{10}{5}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{10}{5}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $10$ by $5$ .

$x > 2$

Step 5

Find the domain of $- 3 + \sqrt{5 x - 1}$ .

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Step 5.1

Set the radicand in $\sqrt{5 x - 1}$ greater than or equal to $0$ to find where the expression is defined.

$5 x - 1 \geq 0$

Step 5.2

Solve for $x$ .

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Step 5.2.1

Add $1$ to both sides of the inequality.

$5 x \geq 1$

Step 5.2.2

Divide each term in $5 x \geq 1$ by $5$ and simplify.

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Step 5.2.2.1

Divide each term in $5 x \geq 1$ by $5$ .

$\frac{5 x}{5} \geq \frac{1}{5}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{1}{5}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{1}{5}$

Step 5.3

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

$[\frac{1}{5}, \infty)$

Step 6

The solution consists of all of the true intervals.

$x > 2$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x > 2$

Interval Notation:

$(2, \infty)$

Step 8