Finite Math Examples

$\frac{4 (x - 2)}{x - 3} + \frac{3}{x} = \frac{- 3}{x (x - 3)}$

Step 1

Simplify $\frac{4 (x - 2)}{x - 3} + \frac{3}{x}$ .

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Step 1.1

To write $\frac{4 (x - 2)}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{4 (x - 2)}{x - 3} \cdot \frac{x}{x} + \frac{3}{x} = \frac{- 3}{x (x - 3)}$

Step 1.2

To write $\frac{3}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{4 (x - 2)}{x - 3} \cdot \frac{x}{x} + \frac{3}{x} \cdot \frac{x - 3}{x - 3} = \frac{- 3}{x (x - 3)}$

Step 1.3

Write each expression with a common denominator of $(x - 3) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{4 (x - 2)}{x - 3}$ by $\frac{x}{x}$ .

$\frac{4 (x - 2) x}{(x - 3) x} + \frac{3}{x} \cdot \frac{x - 3}{x - 3} = \frac{- 3}{x (x - 3)}$

Step 1.3.2

Multiply $\frac{3}{x}$ by $\frac{x - 3}{x - 3}$ .

$\frac{4 (x - 2) x}{(x - 3) x} + \frac{3 (x - 3)}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.3.3

Reorder the factors of $(x - 3) x$ .

$\frac{4 (x - 2) x}{x (x - 3)} + \frac{3 (x - 3)}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{4 (x - 2) x + 3 (x - 3)}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{(4 x + 4 \cdot - 2) x + 3 (x - 3)}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.2

Multiply $4$ by $- 2$ .

$\frac{(4 x - 8) x + 3 (x - 3)}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.3

Apply the distributive property.

$\frac{4 x \cdot x - 8 x + 3 (x - 3)}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.4

Multiply $x$ by $x$ by adding the exponents.

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Step 1.5.4.1

Move $x$ .

$\frac{4 (x \cdot x) - 8 x + 3 (x - 3)}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.4.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} - 8 x + 3 (x - 3)}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.5

Apply the distributive property.

$\frac{4 x^{2} - 8 x + 3 x + 3 \cdot - 3}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.6

Multiply $3$ by $- 3$ .

$\frac{4 x^{2} - 8 x + 3 x - 9}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.7

Add $- 8 x$ and $3 x$ .

$\frac{4 x^{2} - 5 x - 9}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.8

Factor by grouping.

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Step 1.5.8.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 9 = - 36$ and whose sum is $b = - 5$ .

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Step 1.5.8.1.1

Factor $- 5$ out of $- 5 x$ .

$\frac{4 x^{2} - 5 (x) - 9}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.8.1.2

Rewrite $- 5$ as $4$ plus $- 9$

$\frac{4 x^{2} + (4 - 9) x - 9}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.8.1.3

Apply the distributive property.

$\frac{4 x^{2} + 4 x - 9 x - 9}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.8.2

Factor out the greatest common factor from each group.

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Step 1.5.8.2.1

Group the first two terms and the last two terms.

$\frac{(4 x^{2} + 4 x) - 9 x - 9}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.8.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{4 x (x + 1) - 9 (x + 1)}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 1.5.8.3

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$\frac{(x + 1) (4 x - 9)}{x (x - 3)} = \frac{- 3}{x (x - 3)}$

Step 2

Move the negative in front of the fraction.

$\frac{(x + 1) (4 x - 9)}{x (x - 3)} = - \frac{3}{x (x - 3)}$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = - \frac{3}{4}, 2$

Step 4