Precalculus Examples

$\frac{3 x + 5}{2 x - 1} + \frac{7 x + 3}{1 - 2 x}$

Step 1

Set the denominator in $\frac{3 x + 5}{2 x - 1}$ equal to $0$ to find where the expression is undefined.

$2 x - 1 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 3

Set the denominator in $\frac{7 x + 3}{1 - 2 x}$ equal to $0$ to find where the expression is undefined.

$1 - 2 x = 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 4.2

Divide each term in $- 2 x = - 1$ by $- 2$ and simplify.

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Step 4.2.1

Divide each term in $- 2 x = - 1$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

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.

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{2}$

Step 5

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

Interval Notation:

$(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{1}{2}}$

Step 6