Pre-Algebra Examples

$\frac{2 x - 5}{6 x + 9} - \frac{4}{2 x^{2} + 3 x}$

Step 1

Simplify each term.

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Step 1.1

Factor $3$ out of $6 x + 9$ .

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Step 1.1.1

Factor $3$ out of $6 x$ .

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

Step 1.1.2

Factor $3$ out of $9$ .

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

Step 1.1.3

Factor $3$ out of $3 (2 x) + 3 (3)$ .

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

Step 1.2

Factor $x$ out of $2 x^{2} + 3 x$ .

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Step 1.2.1

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

$\frac{2 x - 5}{3 (2 x + 3)} - \frac{4}{x (2 x) + 3 x}$

Step 1.2.2

Factor $x$ out of $3 x$ .

$\frac{2 x - 5}{3 (2 x + 3)} - \frac{4}{x (2 x) + x \cdot 3}$

Step 1.2.3

Factor $x$ out of $x (2 x) + x \cdot 3$ .

$\frac{2 x - 5}{3 (2 x + 3)} - \frac{4}{x (2 x + 3)}$

Step 2

To write $\frac{2 x - 5}{3 (2 x + 3)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2 x - 5}{3 (2 x + 3)} \cdot \frac{x}{x} - \frac{4}{x (2 x + 3)}$

Step 3

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

$\frac{2 x - 5}{3 (2 x + 3)} \cdot \frac{x}{x} - \frac{4}{x (2 x + 3)} \cdot \frac{3}{3}$

Step 4

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

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Step 4.1

Multiply $\frac{2 x - 5}{3 (2 x + 3)}$ by $\frac{x}{x}$ .

$\frac{(2 x - 5) x}{3 (2 x + 3) x} - \frac{4}{x (2 x + 3)} \cdot \frac{3}{3}$

Step 4.2

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

$\frac{(2 x - 5) x}{3 (2 x + 3) x} - \frac{4 \cdot 3}{x (2 x + 3) \cdot 3}$

Step 4.3

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

$\frac{(2 x - 5) x}{3 x (2 x + 3)} - \frac{4 \cdot 3}{x (2 x + 3) \cdot 3}$

Step 4.4

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

$\frac{(2 x - 5) x}{3 x (2 x + 3)} - \frac{4 \cdot 3}{3 x (2 x + 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(2 x - 5) x - 4 \cdot 3}{3 x (2 x + 3)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{2 x \cdot x - 5 x - 4 \cdot 3}{3 x (2 x + 3)}$

Step 6.2

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

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Step 6.2.1

Move $x$ .

$\frac{2 (x \cdot x) - 5 x - 4 \cdot 3}{3 x (2 x + 3)}$

Step 6.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 5 x - 4 \cdot 3}{3 x (2 x + 3)}$

Step 6.3

Multiply $- 4$ by $3$ .

$\frac{2 x^{2} - 5 x - 12}{3 x (2 x + 3)}$

Step 6.4

Factor by grouping.

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Step 6.4.1

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 $a \cdot c = 2 \cdot - 12 = - 24$ and whose sum is $b = - 5$ .

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Step 6.4.1.1

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

$\frac{2 x^{2} - 5 (x) - 12}{3 x (2 x + 3)}$

Step 6.4.1.2

Rewrite $- 5$ as $3$ plus $- 8$

$\frac{2 x^{2} + (3 - 8) x - 12}{3 x (2 x + 3)}$

Step 6.4.1.3

Apply the distributive property.

$\frac{2 x^{2} + 3 x - 8 x - 12}{3 x (2 x + 3)}$

Step 6.4.2

Factor out the greatest common factor from each group.

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Step 6.4.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} + 3 x) - 8 x - 12}{3 x (2 x + 3)}$

Step 6.4.2.2

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

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

Step 6.4.3

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

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

Step 7

Cancel the common factor of $2 x + 3$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

$\frac{x - 4}{3 x}$