Precalculus Examples

$\frac{(x - 1) | x - 4 |}{\sqrt{x + 3}} > 0$

Step 1

Multiply both sides by $\sqrt{x + 3}$ .

$\frac{(x - 1) | x - 4 |}{\sqrt{x + 3}} \sqrt{x + 3} = 0 \sqrt{x + 3}$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

Simplify $\frac{(x - 1) | x - 4 |}{\sqrt{x + 3}} \sqrt{x + 3}$ .

Tap for more steps...

Step 2.1.1.1

Cancel the common factor of $\sqrt{x + 3}$ .

Tap for more steps...

Step 2.1.1.1.1

Cancel the common factor.

$\frac{(x - 1) | x - 4 |}{\sqrt{x + 3}} \sqrt{x + 3} = 0 \sqrt{x + 3}$

Step 2.1.1.1.2

Rewrite the expression.

$(x - 1) | x - 4 | = 0 \sqrt{x + 3}$

Step 2.1.1.2

Apply the distributive property.

$x | x - 4 | - 1 | x - 4 | = 0 \sqrt{x + 3}$

Step 2.1.1.3

Rewrite $- 1 | x - 4 |$ as $- | x - 4 |$ .

$x | x - 4 | - | x - 4 | = 0 \sqrt{x + 3}$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

Multiply $0$ by $\sqrt{x + 3}$ .

$x | x - 4 | - | x - 4 | = 0$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Factor $| x - 4 |$ out of $x | x - 4 | - | x - 4 |$ .

Tap for more steps...

Step 3.1.1

Factor $| x - 4 |$ out of $x | x - 4 |$ .

$| x - 4 | x - | x - 4 | = 0$

Step 3.1.2

Factor $| x - 4 |$ out of $- | x - 4 |$ .

$| x - 4 | x + | x - 4 | \cdot - 1 = 0$

Step 3.1.3

Factor $| x - 4 |$ out of $| x - 4 | x + | x - 4 | \cdot - 1$ .

$| x - 4 | (x - 1) = 0$

Step 3.2

Divide each term in $| x - 4 | (x - 1) = 0$ by $x - 1$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $| x - 4 | (x - 1) = 0$ by $x - 1$ .

$\frac{| x - 4 | (x - 1)}{x - 1} = \frac{0}{x - 1}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{| x - 4 | (x - 1)}{x - 1} = \frac{0}{x - 1}$

Step 3.2.2.1.2

Divide $| x - 4 |$ by $1$ .

$| x - 4 | = \frac{0}{x - 1}$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Divide $0$ by $x - 1$ .

$| x - 4 | = 0$

Step 3.3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x - 4 = \pm 0$

Step 3.4

Plus or minus $0$ is $0$ .

$x - 4 = 0$

Step 3.5

Add $4$ to both sides of the equation.

$x = 4$

Step 3.6

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x - 4 = \pm 0$

Step 3.7

Plus or minus $0$ is $0$ .

$x - 4 = 0$

Step 3.8

Add $4$ to both sides of the equation.

$x = 4$

Step 4

Find the domain of $\frac{(x - 1) | x - 4 |}{\sqrt{x + 3}}$ .

Tap for more steps...

Step 4.1

Set the radicand in $\sqrt{x + 3}$ greater than or equal to $0$ to find where the expression is defined.

$x + 3 \geq 0$

Step 4.2

Subtract $3$ from both sides of the inequality.

$x \geq - 3$

Step 4.3

Set the denominator in $\frac{(x - 1) | x - 4 |}{\sqrt{x + 3}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x + 3} = 0$

Step 4.4

Solve for $x$ .

Tap for more steps...

Step 4.4.1

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

${\sqrt{x + 3}}^{2} = 0^{2}$

Step 4.4.2

Simplify each side of the equation.

Tap for more steps...

Step 4.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

${({(x + 3)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.4.2.2

Simplify the left side.

Tap for more steps...

Step 4.4.2.2.1

Simplify ${({(x + 3)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 4.4.2.2.1.1

Multiply the exponents in ${({(x + 3)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 4.4.2.2.1.1.1

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

${(x + 3)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.4.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.2.2.1.1.2.1

Cancel the common factor.

${(x + 3)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.4.2.2.1.1.2.2

Rewrite the expression.

${(x + 3)}^{1} = 0^{2}$

Step 4.4.2.2.1.2

Simplify.

$x + 3 = 0^{2}$

Step 4.4.2.3

Simplify the right side.

Tap for more steps...

Step 4.4.2.3.1

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

$x + 3 = 0$

Step 4.4.3

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4.5

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

$(- 3, \infty)$

Step 5

The solution consists of all of the true intervals.

$x > 4$

Step 6

Convert the inequality to interval notation.

$(4, \infty)$

Step 7