Precalculus Examples

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

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 $\frac{3 a}{a + b}$ 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 $\frac{2 a b}{a^{2} - b^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 6

Solve for $a$ .

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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 $a$ .

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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.

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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

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${a | a \in ℝ}$