Algebra Examples

$\sqrt{x + 10} - \sqrt{x + 2} > 2$

Step 1

Add $\sqrt{x + 2}$ to both sides of the inequality.

$\sqrt{x + 10} > 2 + \sqrt{x + 2}$

Step 2

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

${\sqrt{x + 10}}^{2} > {(2 + \sqrt{x + 2})}^{2}$

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 $\sqrt{x + 10}$ as ${(x + 10)}^{\frac{1}{2}}$ .

${({(x + 10)}^{\frac{1}{2}})}^{2} > {(2 + \sqrt{x + 2})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(x + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(x + 10)}^{\frac{1}{2} \cdot 2} > {(2 + \sqrt{x + 2})}^{2}$

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.

${(x + 10)}^{\frac{1}{2} \cdot 2} > {(2 + \sqrt{x + 2})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

$x + 10 > {(2 + \sqrt{x + 2})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(2 + \sqrt{x + 2})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(2 + \sqrt{x + 2})}^{2}$ as $(2 + \sqrt{x + 2}) (2 + \sqrt{x + 2})$ .

$x + 10 > (2 + \sqrt{x + 2}) (2 + \sqrt{x + 2})$

Step 3.3.1.2

Expand $(2 + \sqrt{x + 2}) (2 + \sqrt{x + 2})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$x + 10 > 2 (2 + \sqrt{x + 2}) + \sqrt{x + 2} (2 + \sqrt{x + 2})$

Step 3.3.1.2.2

Apply the distributive property.

$x + 10 > 2 \cdot 2 + 2 \sqrt{x + 2} + \sqrt{x + 2} (2 + \sqrt{x + 2})$

Step 3.3.1.2.3

Apply the distributive property.

$x + 10 > 2 \cdot 2 + 2 \sqrt{x + 2} + \sqrt{x + 2} \cdot 2 + \sqrt{x + 2} \sqrt{x + 2}$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $2$ by $2$ .

$x + 10 > 4 + 2 \sqrt{x + 2} + \sqrt{x + 2} \cdot 2 + \sqrt{x + 2} \sqrt{x + 2}$

Step 3.3.1.3.1.2

Move $2$ to the left of $\sqrt{x + 2}$ .

$x + 10 > 4 + 2 \sqrt{x + 2} + 2 \cdot \sqrt{x + 2} + \sqrt{x + 2} \sqrt{x + 2}$

Step 3.3.1.3.1.3

Multiply $\sqrt{x + 2} \sqrt{x + 2}$ .

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Step 3.3.1.3.1.3.1

Raise $\sqrt{x + 2}$ to the power of $1$ .

$x + 10 > 4 + 2 \sqrt{x + 2} + 2 \sqrt{x + 2} + {\sqrt{x + 2}}^{1} \sqrt{x + 2}$

Step 3.3.1.3.1.3.2

Raise $\sqrt{x + 2}$ to the power of $1$ .

$x + 10 > 4 + 2 \sqrt{x + 2} + 2 \sqrt{x + 2} + {\sqrt{x + 2}}^{1} {\sqrt{x + 2}}^{1}$

Step 3.3.1.3.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x + 10 > 4 + 2 \sqrt{x + 2} + 2 \sqrt{x + 2} + {\sqrt{x + 2}}^{1 + 1}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$x + 10 > 4 + 2 \sqrt{x + 2} + 2 \sqrt{x + 2} + {\sqrt{x + 2}}^{2}$

Step 3.3.1.3.1.4

Rewrite ${\sqrt{x + 2}}^{2}$ as $x + 2$ .

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Step 3.3.1.3.1.4.1

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

$x + 10 > 4 + 2 \sqrt{x + 2} + 2 \sqrt{x + 2} + {({(x + 2)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.4.2

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

$x + 10 > 4 + 2 \sqrt{x + 2} + 2 \sqrt{x + 2} + {(x + 2)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$x + 10 > 4 + 2 \sqrt{x + 2} + 2 \sqrt{x + 2} + {(x + 2)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$x + 10 > 4 + 2 \sqrt{x + 2} + 2 \sqrt{x + 2} + {(x + 2)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$x + 10 > 4 + 2 \sqrt{x + 2} + 2 \sqrt{x + 2} + {(x + 2)}^{1}$

Step 3.3.1.3.1.4.5

Simplify.

$x + 10 > 4 + 2 \sqrt{x + 2} + 2 \sqrt{x + 2} + x + 2$

Step 3.3.1.3.2

Add $4$ and $2$ .

$x + 10 > 6 + 2 \sqrt{x + 2} + 2 \sqrt{x + 2} + x$

Step 3.3.1.3.3

Add $2 \sqrt{x + 2}$ and $2 \sqrt{x + 2}$ .

$x + 10 > 6 + 4 \sqrt{x + 2} + x$

Step 4

Solve for $4 \sqrt{x + 2}$ .

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Step 4.1

Rewrite so $4 \sqrt{x + 2}$ is on the left side of the inequality.

$6 + 4 \sqrt{x + 2} + x < x + 10$

Step 4.2

Move all terms not containing $4 \sqrt{x + 2}$ to the right side of the inequality.

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Step 4.2.1

Subtract $6$ from both sides of the inequality.

$4 \sqrt{x + 2} + x < x + 10 - 6$

Step 4.2.2

Subtract $x$ from both sides of the inequality.

$4 \sqrt{x + 2} < x + 10 - 6 - x$

Step 4.2.3

Combine the opposite terms in $x + 10 - 6 - x$ .

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Step 4.2.3.1

Subtract $x$ from $x$ .

$4 \sqrt{x + 2} < 0 + 10 - 6$

Step 4.2.3.2

Add $0$ and $10$ .

$4 \sqrt{x + 2} < 10 - 6$

Step 4.2.4

Subtract $6$ from $10$ .

$4 \sqrt{x + 2} < 4$

Step 5

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

${(4 \sqrt{x + 2})}^{2} < 4^{2}$

Step 6

Simplify each side of the inequality.

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Step 6.1

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

${(4 {(x + 2)}^{\frac{1}{2}})}^{2} < 4^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(4 {(x + 2)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $4 {(x + 2)}^{\frac{1}{2}}$ .

$4^{2} {({(x + 2)}^{\frac{1}{2}})}^{2} < 4^{2}$

Step 6.2.1.2

Raise $4$ to the power of $2$ .

$16 {({(x + 2)}^{\frac{1}{2}})}^{2} < 4^{2}$

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

$16 {(x + 2)}^{\frac{1}{2} \cdot 2} < 4^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$16 {(x + 2)}^{\frac{1}{2} \cdot 2} < 4^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$16 {(x + 2)}^{1} < 4^{2}$

Step 6.2.1.4

Simplify.

$16 (x + 2) < 4^{2}$

Step 6.2.1.5

Apply the distributive property.

$16 x + 16 \cdot 2 < 4^{2}$

Step 6.2.1.6

Multiply $16$ by $2$ .

$16 x + 32 < 4^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Raise $4$ to the power of $2$ .

$16 x + 32 < 16$

Step 7

Solve for $x$ .

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Step 7.1

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

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Step 7.1.1

Subtract $32$ from both sides of the inequality.

$16 x < 16 - 32$

Step 7.1.2

Subtract $32$ from $16$ .

$16 x < - 16$

Step 7.2

Divide each term in $16 x < - 16$ by $16$ and simplify.

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Step 7.2.1

Divide each term in $16 x < - 16$ by $16$ .

$\frac{16 x}{16} < \frac{- 16}{16}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $16$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{16 x}{16} < \frac{- 16}{16}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 16}{16}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $- 16$ by $16$ .

$x < - 1$

Step 8

Find the domain of $\sqrt{x + 10} - \sqrt{x + 2} - 2$ .

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Step 8.1

Set the radicand in $\sqrt{x + 10}$ greater than or equal to $0$ to find where the expression is defined.

$x + 10 \geq 0$

Step 8.2

Subtract $10$ from both sides of the inequality.

$x \geq - 10$

Step 8.3

Set the radicand in $\sqrt{x + 2}$ greater than or equal to $0$ to find where the expression is defined.

$x + 2 \geq 0$

Step 8.4

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 8.5

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

$[- 2, \infty)$

Step 9

Use each root to create test intervals.

$x < - 2$

$- 2 < x < - 1$

$x > - 1$

Step 10

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 10.1

Test a value on the interval $x < - 2$ to see if it makes the inequality true.

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Step 10.1.1

Choose a value on the interval $x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Step 10.1.2

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

$\sqrt{(- 4) + 10} - \sqrt{(- 4) + 2} > 2$

Step 10.1.3

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

False

Step 10.2

Test a value on the interval $- 2 < x < - 1$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $- 2 < x < - 1$ and see if this value makes the original inequality true.

$x = - 1.5$

Step 10.2.2

Replace $x$ with $- 1.5$ in the original inequality.

$\sqrt{(- 1.5) + 10} - \sqrt{(- 1.5) + 2} > 2$

Step 10.2.3

The left side $2.20836916$ is greater than the right side $2$ , which means that the given statement is always true.

True

Step 10.3

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

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Step 10.3.1

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

$x = 0$

Step 10.3.2

Replace $x$ with $0$ in the original inequality.

$\sqrt{(0) + 10} - \sqrt{(0) + 2} > 2$

Step 10.3.3

The left side $1.74806409$ is not greater than the right side $2$ , which means that the given statement is false.

False

Step 10.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 2$ False

$- 2 < x < - 1$ True

$x > - 1$ False

$x < - 2$ False

$- 2 < x < - 1$ True

$x > - 1$ False

Step 11

The solution consists of all of the true intervals.

$- 2 < x < - 1$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < - 1$

Interval Notation:

$(- 2, - 1)$

Step 13