Precalculus Examples

$(\frac{a}{a - b} - \frac{a - b}{a + b} + \frac{b^{2}}{a^{2} - b^{2}}) \cdot \frac{a - b}{3 a b}$

Step 1

Set the denominator in $\frac{a}{a - b}$ equal to $0$ to find where the expression is undefined.

$a - b = 0$

Step 2

Add $b$ to both sides of the equation.

$a = b$

Step 3

Set the denominator in $\frac{a - b}{a + b}$ equal to $0$ to find where the expression is undefined.

$a + b = 0$

Step 4

Subtract $b$ from both sides of the equation.

$a = - b$

Step 5

Set the denominator in $\frac{b^{2}}{a^{2} - b^{2}}$ equal to $0$ to find where the expression is undefined.

$a^{2} - b^{2} = 0$

Step 6

Solve for $a$ .

Tap for more steps...

Step 6.1

Add $b^{2}$ to both sides of the equation.

$a^{2} = b^{2}$

Step 6.2

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| a | = | b |$

Step 6.3

Solve for $a$ .

Tap for more steps...

Step 6.3.1

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

$a = \pm | b |$

Step 6.3.2

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 6.3.2.1

First, use the positive value of the $\pm$ to find the first solution.

$a = | b |$

Step 6.3.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$a = - | b |$

Step 6.3.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$a = | b |$

$a = - | b |$

$a = | b |$

$a = - | b |$

$a = | b |$

$a = - | b |$

$a = | b |$

$a = - | b |$

Step 7

Set the denominator in $\frac{a - b}{3 a b}$ equal to $0$ to find where the expression is undefined.

$3 a b = 0$

Step 8

Divide each term in $3 a b = 0$ by $3 b$ and simplify.

Tap for more steps...

Step 8.1

Divide each term in $3 a b = 0$ by $3 b$ .

$\frac{3 a b}{3 b} = \frac{0}{3 b}$

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.2.1.1

Cancel the common factor.

$\frac{3 a b}{3 b} = \frac{0}{3 b}$

Step 8.2.1.2

Rewrite the expression.

$\frac{a b}{b} = \frac{0}{3 b}$

Step 8.2.2

Cancel the common factor of $b$ .

Tap for more steps...

Step 8.2.2.1

Cancel the common factor.

$\frac{a b}{b} = \frac{0}{3 b}$

Step 8.2.2.2

Divide $a$ by $1$ .

$a = \frac{0}{3 b}$

Step 8.3

Simplify the right side.

Tap for more steps...

Step 8.3.1

Cancel the common factor of $0$ and $3$ .

Tap for more steps...

Step 8.3.1.1

Factor $3$ out of $0$ .

$a = \frac{3 (0)}{3 b}$

Step 8.3.1.2

Cancel the common factors.

Tap for more steps...

Step 8.3.1.2.1

Factor $3$ out of $3 b$ .

$a = \frac{3 (0)}{3 (b)}$

Step 8.3.1.2.2

Cancel the common factor.

$a = \frac{3 \cdot 0}{3 b}$

Step 8.3.1.2.3

Rewrite the expression.

$a = \frac{0}{b}$

Step 8.3.2

Divide $0$ by $b$ .

$a = 0$

Step 9

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${a | a \neq 0}$