Algebra Examples

$\sqrt{2 x - 14} + 4 > 12$ ?

Step 1

Move all terms not containing $\sqrt{2 x - 14}$ to the right side of the inequality.

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Step 1.1

Subtract $4$ from both sides of the inequality.

$\sqrt{2 x - 14} > 12 - 4$

Step 1.2

Subtract $4$ from $12$ .

$\sqrt{2 x - 14} > 8$

Step 2

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

${\sqrt{2 x - 14}}^{2} > 8^{2}$

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 $\sqrt{2 x - 14}$ as ${(2 x - 14)}^{\frac{1}{2}}$ .

${({(2 x - 14)}^{\frac{1}{2}})}^{2} > 8^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(2 x - 14)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(2 x - 14)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(2 x - 14)}^{\frac{1}{2} \cdot 2} > 8^{2}$

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.

${(2 x - 14)}^{\frac{1}{2} \cdot 2} > 8^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 x - 14)}^{1} > 8^{2}$

Step 3.2.1.2

Simplify.

$2 x - 14 > 8^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $8$ to the power of $2$ .

$2 x - 14 > 64$

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 $14$ to both sides of the inequality.

$2 x > 64 + 14$

Step 4.1.2

Add $64$ and $14$ .

$2 x > 78$

Step 4.2

Divide each term in $2 x > 78$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $2 x > 78$ by $2$ .

$\frac{2 x}{2} > \frac{78}{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} > \frac{78}{2}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{78}{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $78$ by $2$ .

$x > 39$

Step 5

Find the domain of $\sqrt{2 (x - 7)} - 8$ .

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Step 5.1

Set the radicand in $\sqrt{2 (x - 7)}$ greater than or equal to $0$ to find where the expression is defined.

$2 (x - 7) \geq 0$

Step 5.2

Solve for $x$ .

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Step 5.2.1

Divide each term in $2 (x - 7) \geq 0$ by $2$ and simplify.

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Step 5.2.1.1

Divide each term in $2 (x - 7) \geq 0$ by $2$ .

$\frac{2 (x - 7)}{2} \geq \frac{0}{2}$

Step 5.2.1.2

Simplify the left side.

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Step 5.2.1.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.2.1.1

Cancel the common factor.

$\frac{2 (x - 7)}{2} \geq \frac{0}{2}$

Step 5.2.1.2.1.2

Divide $x - 7$ by $1$ .

$x - 7 \geq \frac{0}{2}$

Step 5.2.1.3

Simplify the right side.

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Step 5.2.1.3.1

Divide $0$ by $2$ .

$x - 7 \geq 0$

Step 5.2.2

Add $7$ to both sides of the inequality.

$x \geq 7$

Step 5.3

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

$[7, \infty)$

Step 6

The solution consists of all of the true intervals.

$x > 39$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x > 39$

Interval Notation:

$(39, \infty)$

Step 8