Algebra Examples

$5 \sqrt{x - 1} > 10$

Step 1

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

${(5 \sqrt{x - 1})}^{2} > 10^{2}$

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

${(5 {(x - 1)}^{\frac{1}{2}})}^{2} > 10^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Apply the product rule to $5 {(x - 1)}^{\frac{1}{2}}$ .

$5^{2} {({(x - 1)}^{\frac{1}{2}})}^{2} > 10^{2}$

Step 2.2.1.2

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

$25 {({(x - 1)}^{\frac{1}{2}})}^{2} > 10^{2}$

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

$25 {(x - 1)}^{\frac{1}{2} \cdot 2} > 10^{2}$

Step 2.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.2.1.3.2.1

Cancel the common factor.

$25 {(x - 1)}^{\frac{1}{2} \cdot 2} > 10^{2}$

Step 2.2.1.3.2.2

Rewrite the expression.

$25 {(x - 1)}^{1} > 10^{2}$

Step 2.2.1.4

Simplify.

$25 (x - 1) > 10^{2}$

Step 2.2.1.5

Apply the distributive property.

$25 x + 25 \cdot - 1 > 10^{2}$

Step 2.2.1.6

Multiply $25$ by $- 1$ .

$25 x - 25 > 10^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$25 x - 25 > 100$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Add $25$ to both sides of the inequality.

$25 x > 100 + 25$

Step 3.1.2

Add $100$ and $25$ .

$25 x > 125$

Step 3.2

Divide each term in $25 x > 125$ by $25$ and simplify.

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Step 3.2.1

Divide each term in $25 x > 125$ by $25$ .

$\frac{25 x}{25} > \frac{125}{25}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $25$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{25 x}{25} > \frac{125}{25}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{125}{25}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $125$ by $25$ .

$x > 5$

Step 4

Find the domain of $5 \sqrt{x - 1} - 10$ .

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Step 4.1

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

$x - 1 \geq 0$

Step 4.2

Add $1$ to both sides of the inequality.

$x \geq 1$

Step 4.3

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

$[1, \infty)$

Step 5

The solution consists of all of the true intervals.

$x > 5$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x > 5$

Interval Notation:

$(5, \infty)$

Step 7