Linear Algebra Examples

$\frac{\frac{6 x + 5}{2 x + 7}}{\frac{4 x}{5} - \frac{3}{4}}$

Step 1

Set the denominator in $\frac{6 x + 5}{2 x + 7}$ equal to $0$ to find where the expression is undefined.

$2 x + 7 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $7$ from both sides of the equation.

$2 x = - 7$

Step 2.2

Divide each term in $2 x = - 7$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x = - 7$ by $2$ .

$\frac{2 x}{2} = \frac{- 7}{2}$

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.

$\frac{2 x}{2} = \frac{- 7}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{2}$

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.

$x = - \frac{7}{2}$

Step 3

Set the denominator in $\frac{\frac{6 x + 5}{2 x + 7}}{\frac{4 x}{5} - \frac{3}{4}}$ equal to $0$ to find where the expression is undefined.

$\frac{4 x}{5} - \frac{3}{4} = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $\frac{3}{4}$ to both sides of the equation.

$\frac{4 x}{5} = \frac{3}{4}$

Step 4.2

Multiply both sides of the equation by $\frac{5}{4}$ .

$\frac{5}{4} \cdot \frac{4 x}{5} = \frac{5}{4} \cdot \frac{3}{4}$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Simplify $\frac{5}{4} \cdot \frac{4 x}{5}$ .

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Step 4.3.1.1.1

Cancel the common factor of $5$ .

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Step 4.3.1.1.1.1

Cancel the common factor.

$\frac{5}{4} \cdot \frac{4 x}{5} = \frac{5}{4} \cdot \frac{3}{4}$

Step 4.3.1.1.1.2

Rewrite the expression.

$\frac{1}{4} (4 x) = \frac{5}{4} \cdot \frac{3}{4}$

Step 4.3.1.1.2

Cancel the common factor of $4$ .

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Step 4.3.1.1.2.1

Factor $4$ out of $4 x$ .

$\frac{1}{4} (4 (x)) = \frac{5}{4} \cdot \frac{3}{4}$

Step 4.3.1.1.2.2

Cancel the common factor.

$\frac{1}{4} (4 x) = \frac{5}{4} \cdot \frac{3}{4}$

Step 4.3.1.1.2.3

Rewrite the expression.

$x = \frac{5}{4} \cdot \frac{3}{4}$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Multiply $\frac{5}{4} \cdot \frac{3}{4}$ .

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Step 4.3.2.1.1

Multiply $\frac{5}{4}$ by $\frac{3}{4}$ .

$x = \frac{5 \cdot 3}{4 \cdot 4}$

Step 4.3.2.1.2

Multiply $5$ by $3$ .

$x = \frac{15}{4 \cdot 4}$

Step 4.3.2.1.3

Multiply $4$ by $4$ .

$x = \frac{15}{16}$

Step 5

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

Interval Notation:

$(- \infty, - \frac{7}{2}) \cup (- \frac{7}{2}, \frac{15}{16}) \cup (\frac{15}{16}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{15}{16}, - \frac{7}{2}}$

Step 6