Algebra Examples

$\frac{11 x}{2 x - 3} - \frac{x - 18}{3 - 2 x}$

Step 1

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{11 x}{2 x - 3} - \frac{x - 18}{- 1 (- 3) - 2 x}$

Step 2

Factor $- 1$ out of $- 2 x$ .

$\frac{11 x}{2 x - 3} - \frac{x - 18}{- 1 (- 3) - (2 x)}$

Step 3

Factor $- 1$ out of $- 1 (- 3) - (2 x)$ .

$\frac{11 x}{2 x - 3} - \frac{x - 18}{- 1 (- 3 + 2 x)}$

Step 4

Reorder terms.

$\frac{11 x}{2 x - 3} - \frac{x - 18}{- 1 (2 x - 3)}$

Step 5

To write $\frac{11 x}{2 x - 3}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{11 x}{2 x - 3} \cdot \frac{- 1}{- 1} - \frac{x - 18}{- 1 (2 x - 3)}$

Step 6

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

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Step 6.1

Multiply $\frac{11 x}{2 x - 3}$ by $\frac{- 1}{- 1}$ .

$\frac{11 x \cdot - 1}{(2 x - 3) \cdot - 1} - \frac{x - 18}{- 1 (2 x - 3)}$

Step 6.2

Reorder the factors of $(2 x - 3) \cdot - 1$ .

$\frac{11 x \cdot - 1}{- (2 x - 3)} - \frac{x - 18}{- 1 (2 x - 3)}$

Step 7

Combine the numerators over the common denominator.

$\frac{11 x \cdot - 1 - (x - 18)}{- (2 x - 3)}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $11$ .

$\frac{- 11 x - (x - 18)}{- (2 x - 3)}$

Step 8.2

Apply the distributive property.

$\frac{- 11 x - x - - 18}{- (2 x - 3)}$

Step 8.3

Multiply $- 1$ by $- 18$ .

$\frac{- 11 x - x + 18}{- (2 x - 3)}$

Step 8.4

Subtract $x$ from $- 11 x$ .

$\frac{- 12 x + 18}{- (2 x - 3)}$

Step 8.5

Factor $6$ out of $- 12 x + 18$ .

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Step 8.5.1

Factor $6$ out of $- 12 x$ .

$\frac{6 (- 2 x) + 18}{- (2 x - 3)}$

Step 8.5.2

Factor $6$ out of $18$ .

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

Step 8.5.3

Factor $6$ out of $6 (- 2 x) + 6 (3)$ .

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

Step 9

Reduce the expression by cancelling the common factors.

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Step 9.1

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

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Step 9.1.1

Factor $- 1$ out of $- 2 x$ .

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

Step 9.1.2

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{6 (- (2 x) - 1 (- 3))}{- (2 x - 3)}$

Step 9.1.3

Factor $- 1$ out of $- (2 x) - 1 (- 3)$ .

$\frac{6 (- (2 x - 3))}{- (2 x - 3)}$

Step 9.1.4

Rewrite $- (2 x - 3)$ as $- 1 (2 x - 3)$ .

$\frac{6 (- 1 (2 x - 3))}{- (2 x - 3)}$

Step 9.1.5

Cancel the common factor.

$\frac{6 (- 1 (2 x - 3))}{- (2 x - 3)}$

Step 9.1.6

Rewrite the expression.

$\frac{6 \cdot (- 1)}{- 1}$

Step 9.1.7

Move the negative one from the denominator of $\frac{6 \cdot (- 1)}{- 1}$ .

$- 1 \cdot (6 \cdot (- 1))$

Step 9.2

Rewrite $- 1 \cdot (6 \cdot (- 1))$ as $- (6 \cdot (- 1))$ .

$- (6 \cdot (- 1))$

Step 10

Multiply $- (6 \cdot (- 1))$ .

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Step 10.1

Multiply $6$ by $- 1$ .

$- - 6$

Step 10.2

Multiply $- 1$ by $- 6$ .

$6$