Linear Algebra Examples

$\sqrt{\sqrt{15 x + 19}} = \sqrt{2 x + 3}$

Step 1

Set the radicand in $\sqrt{15 x + 19}$ greater than or equal to $0$ to find where the expression is defined.

$15 x + 19 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $19$ from both sides of the inequality.

$15 x \geq - 19$

Step 2.2

Divide each term in $15 x \geq - 19$ by $15$ and simplify.

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Step 2.2.1

Divide each term in $15 x \geq - 19$ by $15$ .

$\frac{15 x}{15} \geq \frac{- 19}{15}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $15$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{15 x}{15} \geq \frac{- 19}{15}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 19}{15}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{19}{15}$

Step 3

Set the radicand in $\sqrt{\sqrt{15 x + 19}}$ greater than or equal to $0$ to find where the expression is defined.

$\sqrt{15 x + 19} \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

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

${\sqrt{15 x + 19}}^{2} \geq 0^{2}$

Step 4.2

Simplify each side of the inequality.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15 x + 19}$ as ${(15 x + 19)}^{\frac{1}{2}}$ .

${({(15 x + 19)}^{\frac{1}{2}})}^{2} \geq 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(15 x + 19)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(15 x + 19)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

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

${(15 x + 19)}^{\frac{1}{2} \cdot 2} \geq 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(15 x + 19)}^{\frac{1}{2} \cdot 2} \geq 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(15 x + 19)}^{1} \geq 0^{2}$

Step 4.2.2.1.2

Simplify.

$15 x + 19 \geq 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raising $0$ to any positive power yields $0$ .

$15 x + 19 \geq 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Subtract $19$ from both sides of the inequality.

$15 x \geq - 19$

Step 4.3.2

Divide each term in $15 x \geq - 19$ by $15$ and simplify.

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Step 4.3.2.1

Divide each term in $15 x \geq - 19$ by $15$ .

$\frac{15 x}{15} \geq \frac{- 19}{15}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $15$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{15 x}{15} \geq \frac{- 19}{15}$

Step 4.3.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 19}{15}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{19}{15}$

Step 4.4

Find the domain of $\sqrt{15 x + 19}$ .

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Step 4.4.1

Set the radicand in $\sqrt{15 x + 19}$ greater than or equal to $0$ to find where the expression is defined.

$15 x + 19 \geq 0$

Step 4.4.2

Solve for $x$ .

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Step 4.4.2.1

Subtract $19$ from both sides of the inequality.

$15 x \geq - 19$

Step 4.4.2.2

Divide each term in $15 x \geq - 19$ by $15$ and simplify.

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Step 4.4.2.2.1

Divide each term in $15 x \geq - 19$ by $15$ .

$\frac{15 x}{15} \geq \frac{- 19}{15}$

Step 4.4.2.2.2

Simplify the left side.

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Step 4.4.2.2.2.1

Cancel the common factor of $15$ .

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Step 4.4.2.2.2.1.1

Cancel the common factor.

$\frac{15 x}{15} \geq \frac{- 19}{15}$

Step 4.4.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 19}{15}$

Step 4.4.2.2.3

Simplify the right side.

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Step 4.4.2.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{19}{15}$

Step 4.4.3

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

$[- \frac{19}{15}, \infty)$

Step 4.5

The solution consists of all of the true intervals.

$x \geq - \frac{19}{15}$

Step 5

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

Interval Notation:

$[- \frac{19}{15}, \infty)$

Set-Builder Notation:

${x | x \geq - \frac{19}{15}}$

Step 6