Pre-Algebra Examples

$\frac{2 x + 4}{x - 2} - \frac{3 x - 6}{2 x + 3} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1

Simplify $\frac{2 x + 4}{x - 2} - \frac{3 x - 6}{2 x + 3}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Factor $2$ out of $2 x + 4$ .

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Step 1.1.1.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 4}{x - 2} - \frac{3 x - 6}{2 x + 3} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.1.1.2

Factor $2$ out of $4$ .

$\frac{2 x + 2 \cdot 2}{x - 2} - \frac{3 x - 6}{2 x + 3} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.1.1.3

Factor $2$ out of $2 x + 2 \cdot 2$ .

$\frac{2 (x + 2)}{x - 2} - \frac{3 x - 6}{2 x + 3} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.1.2

Factor $3$ out of $3 x - 6$ .

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Step 1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{2 (x + 2)}{x - 2} - \frac{3 (x) - 6}{2 x + 3} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.1.2.2

Factor $3$ out of $- 6$ .

$\frac{2 (x + 2)}{x - 2} - \frac{3 x + 3 \cdot - 2}{2 x + 3} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.1.2.3

Factor $3$ out of $3 x + 3 \cdot - 2$ .

$\frac{2 (x + 2)}{x - 2} - \frac{3 (x - 2)}{2 x + 3} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.2

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

$\frac{2 (x + 2)}{x - 2} \cdot \frac{2 x + 3}{2 x + 3} - \frac{3 (x - 2)}{2 x + 3} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.3

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

$\frac{2 (x + 2)}{x - 2} \cdot \frac{2 x + 3}{2 x + 3} - \frac{3 (x - 2)}{2 x + 3} \cdot \frac{x - 2}{x - 2} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.4

Write each expression with a common denominator of $(x - 2) (2 x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

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

$\frac{2 (x + 2) (2 x + 3)}{(x - 2) (2 x + 3)} - \frac{3 (x - 2)}{2 x + 3} \cdot \frac{x - 2}{x - 2} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.4.2

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

$\frac{2 (x + 2) (2 x + 3)}{(x - 2) (2 x + 3)} - \frac{3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.4.3

Reorder the factors of $(x - 2) (2 x + 3)$ .

$\frac{2 (x + 2) (2 x + 3)}{(2 x + 3) (x - 2)} - \frac{3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{2 (x + 2) (2 x + 3) - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Apply the distributive property.

$\frac{(2 x + 2 \cdot 2) (2 x + 3) - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.2

Multiply $2$ by $2$ .

$\frac{(2 x + 4) (2 x + 3) - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.3

Expand $(2 x + 4) (2 x + 3)$ using the FOIL Method.

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Step 1.6.3.1

Apply the distributive property.

$\frac{2 x (2 x + 3) + 4 (2 x + 3) - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.3.2

Apply the distributive property.

$\frac{2 x (2 x) + 2 x \cdot 3 + 4 (2 x + 3) - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.3.3

Apply the distributive property.

$\frac{2 x (2 x) + 2 x \cdot 3 + 4 (2 x) + 4 \cdot 3 - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.4

Simplify and combine like terms.

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Step 1.6.4.1

Simplify each term.

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Step 1.6.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 \cdot 2 x \cdot x + 2 x \cdot 3 + 4 (2 x) + 4 \cdot 3 - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.4.1.2

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

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Step 1.6.4.1.2.1

Move $x$ .

$\frac{2 \cdot 2 (x \cdot x) + 2 x \cdot 3 + 4 (2 x) + 4 \cdot 3 - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.4.1.2.2

Multiply $x$ by $x$ .

$\frac{2 \cdot 2 x^{2} + 2 x \cdot 3 + 4 (2 x) + 4 \cdot 3 - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.4.1.3

Multiply $2$ by $2$ .

$\frac{4 x^{2} + 2 x \cdot 3 + 4 (2 x) + 4 \cdot 3 - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.4.1.4

Multiply $3$ by $2$ .

$\frac{4 x^{2} + 6 x + 4 (2 x) + 4 \cdot 3 - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.4.1.5

Multiply $2$ by $4$ .

$\frac{4 x^{2} + 6 x + 8 x + 4 \cdot 3 - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.4.1.6

Multiply $4$ by $3$ .

$\frac{4 x^{2} + 6 x + 8 x + 12 - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.4.2

Add $6 x$ and $8 x$ .

$\frac{4 x^{2} + 14 x + 12 - 3 (x - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.5

Apply the distributive property.

$\frac{4 x^{2} + 14 x + 12 + (- 3 x - 3 \cdot - 2) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.6

Multiply $- 3$ by $- 2$ .

$\frac{4 x^{2} + 14 x + 12 + (- 3 x + 6) (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.7

Expand $(- 3 x + 6) (x - 2)$ using the FOIL Method.

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Step 1.6.7.1

Apply the distributive property.

$\frac{4 x^{2} + 14 x + 12 - 3 x (x - 2) + 6 (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.7.2

Apply the distributive property.

$\frac{4 x^{2} + 14 x + 12 - 3 x \cdot x - 3 x \cdot - 2 + 6 (x - 2)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.7.3

Apply the distributive property.

$\frac{4 x^{2} + 14 x + 12 - 3 x \cdot x - 3 x \cdot - 2 + 6 x + 6 \cdot - 2}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.8

Simplify and combine like terms.

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Step 1.6.8.1

Simplify each term.

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Step 1.6.8.1.1

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

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Step 1.6.8.1.1.1

Move $x$ .

$\frac{4 x^{2} + 14 x + 12 - 3 (x \cdot x) - 3 x \cdot - 2 + 6 x + 6 \cdot - 2}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.8.1.1.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} + 14 x + 12 - 3 x^{2} - 3 x \cdot - 2 + 6 x + 6 \cdot - 2}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.8.1.2

Multiply $- 2$ by $- 3$ .

$\frac{4 x^{2} + 14 x + 12 - 3 x^{2} + 6 x + 6 x + 6 \cdot - 2}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.8.1.3

Multiply $6$ by $- 2$ .

$\frac{4 x^{2} + 14 x + 12 - 3 x^{2} + 6 x + 6 x - 12}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.8.2

Add $6 x$ and $6 x$ .

$\frac{4 x^{2} + 14 x + 12 - 3 x^{2} + 12 x - 12}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.9

Subtract $3 x^{2}$ from $4 x^{2}$ .

$\frac{x^{2} + 14 x + 12 + 12 x - 12}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.10

Add $14 x$ and $12 x$ .

$\frac{x^{2} + 26 x + 12 - 12}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.11

Subtract $12$ from $12$ .

$\frac{x^{2} + 26 x + 0}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.12

Add $x^{2} + 26 x$ and $0$ .

$\frac{x^{2} + 26 x}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.13

Factor $x$ out of $x^{2} + 26 x$ .

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Step 1.6.13.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x + 26 x}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.13.2

Factor $x$ out of $26 x$ .

$\frac{x \cdot x + x \cdot 26}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 1.6.13.3

Factor $x$ out of $x \cdot x + x \cdot 26$ .

$\frac{x (x + 26)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - x - 6}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = - 1$ .

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Step 2.1.1

Factor $- 1$ out of $- x$ .

$\frac{x (x + 26)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} - (x) - 6}$

Step 2.1.2

Rewrite $- 1$ as $3$ plus $- 4$

$\frac{x (x + 26)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} + (3 - 4) x - 6}$

Step 2.1.3

Apply the distributive property.

$\frac{x (x + 26)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{2 x^{2} + 3 x - 4 x - 6}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{x (x + 26)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{(2 x^{2} + 3 x) - 4 x - 6}$

Step 2.2.2

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

$\frac{x (x + 26)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{x (2 x + 3) - 2 (2 x + 3)}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

$\frac{x (x + 26)}{(2 x + 3) (x - 2)} = \frac{x^{2} + 78}{(2 x + 3) (x - 2)}$

Step 3

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

$x = 3$

Step 4