Pre-Algebra Examples

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

Step 1

Rewrite the division as a fraction.

$\frac{\frac{4}{2 - x} + \frac{5}{x - 2}}{\frac{4}{x} + \frac{2}{x - 2}}$

Step 2

Multiply the numerator and denominator of the fraction by $(2 - x) (x - 2) x$ .

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Step 2.1

Multiply $\frac{\frac{4}{2 - x} + \frac{5}{x - 2}}{\frac{4}{x} + \frac{2}{x - 2}}$ by $\frac{(2 - x) (x - 2) x}{(2 - x) (x - 2) x}$ .

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify by cancelling.

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Step 4.1

Cancel the common factor of $2 - x$ .

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x - 2$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.3

Cancel the common factor of $x$ .

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Step 4.3.1

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

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

Step 4.3.2

Cancel the common factor.

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

Step 4.3.3

Rewrite the expression.

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

Step 4.4

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

$\frac{(x - 2) x \cdot 4 + (2 - x) x \cdot 5}{(- 1 (- 2) - x) (x - 2) \cdot 4 + (2 - x) (x - 2) x \frac{2}{x - 2}}$

Step 4.5

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

$\frac{(x - 2) x \cdot 4 + (2 - x) x \cdot 5}{(- 1 (- 2) - (x)) (x - 2) \cdot 4 + (2 - x) (x - 2) x \frac{2}{x - 2}}$

Step 4.6

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

$\frac{(x - 2) x \cdot 4 + (2 - x) x \cdot 5}{- 1 (- 2 + x) (x - 2) \cdot 4 + (2 - x) (x - 2) x \frac{2}{x - 2}}$

Step 4.7

Reorder terms.

$\frac{(x - 2) x \cdot 4 + (2 - x) x \cdot 5}{- 1 (x - 2) (x - 2) \cdot 4 + (2 - x) (x - 2) x \frac{2}{x - 2}}$

Step 4.8

Raise $x - 2$ to the power of $1$ .

$\frac{(x - 2) x \cdot 4 + (2 - x) x \cdot 5}{- 1 ({(x - 2)}^{1} (x - 2)) \cdot 4 + (2 - x) (x - 2) x \frac{2}{x - 2}}$

Step 4.9

Raise $x - 2$ to the power of $1$ .

$\frac{(x - 2) x \cdot 4 + (2 - x) x \cdot 5}{- 1 ({(x - 2)}^{1} {(x - 2)}^{1}) \cdot 4 + (2 - x) (x - 2) x \frac{2}{x - 2}}$

Step 4.10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(x - 2) x \cdot 4 + (2 - x) x \cdot 5}{- 1 {(x - 2)}^{1 + 1} \cdot 4 + (2 - x) (x - 2) x \frac{2}{x - 2}}$

Step 4.11

Add $1$ and $1$ .

$\frac{(x - 2) x \cdot 4 + (2 - x) x \cdot 5}{- 1 {(x - 2)}^{2} \cdot 4 + (2 - x) (x - 2) x \frac{2}{x - 2}}$

Step 4.12

Multiply $4$ by $- 1$ .

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

Step 4.13

Cancel the common factor of $x - 2$ .

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Step 4.13.1

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

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

Step 4.13.2

Cancel the common factor.

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

Step 4.13.3

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Factor $x$ out of $(x - 2) x \cdot 4 + (2 - x) x \cdot 5$ .

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Step 5.1.1

Factor $x$ out of $(x - 2) x \cdot 4$ .

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

Step 5.1.2

Factor $x$ out of $(2 - x) x \cdot 5$ .

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

Step 5.1.3

Factor $x$ out of $x ((x - 2) \cdot 4) + x ((2 - x) \cdot 5)$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Move $4$ to the left of $x$ .

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

Step 5.4

Multiply $- 2$ by $4$ .

$\frac{x (4 \cdot x - 8 + (2 - x) \cdot 5)}{- 4 {(x - 2)}^{2} + (2 - x) x \cdot 2}$

Step 5.5

Apply the distributive property.

$\frac{x (4 x - 8 + 2 \cdot 5 - x \cdot 5)}{- 4 {(x - 2)}^{2} + (2 - x) x \cdot 2}$

Step 5.6

Multiply $2$ by $5$ .

$\frac{x (4 x - 8 + 10 - x \cdot 5)}{- 4 {(x - 2)}^{2} + (2 - x) x \cdot 2}$

Step 5.7

Multiply $5$ by $- 1$ .

$\frac{x (4 x - 8 + 10 - 5 x)}{- 4 {(x - 2)}^{2} + (2 - x) x \cdot 2}$

Step 5.8

Subtract $5 x$ from $4 x$ .

$\frac{x (- x - 8 + 10)}{- 4 {(x - 2)}^{2} + (2 - x) x \cdot 2}$

Step 5.9

Add $- 8$ and $10$ .

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

Step 6

Simplify the denominator.

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Step 6.1

Factor $2$ out of $- 4 {(x - 2)}^{2} + (2 - x) x \cdot 2$ .

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Step 6.1.1

Factor $2$ out of $- 4 {(x - 2)}^{2}$ .

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

Step 6.1.2

Factor $2$ out of $(2 - x) x \cdot 2$ .

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

Step 6.1.3

Factor $2$ out of $2 (- 2 {(x - 2)}^{2}) + 2 \cdot ((2 - x) x)$ .

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

Step 6.2

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

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

Step 6.3

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

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Step 6.3.1

Apply the distributive property.

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

Step 6.3.2

Apply the distributive property.

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

Step 6.3.3

Apply the distributive property.

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

Step 6.4

Simplify and combine like terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Multiply $x$ by $x$ .

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

Step 6.4.1.2

Move $- 2$ to the left of $x$ .

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

Step 6.4.1.3

Multiply $- 2$ by $- 2$ .

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

Step 6.4.2

Subtract $2 x$ from $- 2 x$ .

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

Step 6.5

Apply the distributive property.

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

Step 6.6

Simplify.

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Step 6.6.1

Multiply $- 4$ by $- 2$ .

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

Step 6.6.2

Multiply $- 2$ by $4$ .

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

Step 6.7

Apply the distributive property.

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

Step 6.8

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

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Step 6.8.1

Move $x$ .

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

Step 6.8.2

Multiply $x$ by $x$ .

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

Step 6.9

Subtract $x^{2}$ from $- 2 x^{2}$ .

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

Step 6.10

Add $8 x$ and $2 x$ .

$\frac{x (- x + 2)}{2 (- 3 x^{2} + 10 x - 8)}$

Step 6.11

Factor by grouping.

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Step 6.11.1

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 $a \cdot c = - 3 \cdot - 8 = 24$ and whose sum is $b = 10$ .

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Step 6.11.1.1

Factor $10$ out of $10 x$ .

$\frac{x (- x + 2)}{2 (- 3 x^{2} + 10 (x) - 8)}$

Step 6.11.1.2

Rewrite $10$ as $4$ plus $6$

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

Step 6.11.1.3

Apply the distributive property.

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

Step 6.11.2

Factor out the greatest common factor from each group.

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Step 6.11.2.1

Group the first two terms and the last two terms.

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

Step 6.11.2.2

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

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

Step 6.11.3

Factor the polynomial by factoring out the greatest common factor, $- 3 x + 4$ .

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

$\frac{x (- (x) - 1 (- 2))}{2 (- 3 x + 4) (x - 2)}$

Step 7.3

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

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

Step 7.4

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

$\frac{x (- 1 (x - 2))}{2 (- 3 x + 4) (x - 2)}$

Step 7.5

Cancel the common factor.

$\frac{x (- 1 (x - 2))}{2 (- 3 x + 4) (x - 2)}$

Step 7.6

Rewrite the expression.

$\frac{x \cdot (- 1)}{2 (- 3 x + 4)}$

Step 8

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{2 (- 3 x + 4)}$

Step 9

Move the negative in front of the fraction.

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

Step 10

Factor $- 1$ out of $- 3 x$ .

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

Step 11

Rewrite $4$ as $- 1 (- 4)$ .

$- \frac{x}{2 (- (3 x) - 1 (- 4))}$

Step 12

Factor $- 1$ out of $- (3 x) - 1 (- 4)$ .

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

Step 13

Rewrite $- (3 x - 4)$ as $- 1 (3 x - 4)$ .

$- \frac{x}{2 (- 1 (3 x - 4))}$

Step 14

Move the negative in front of the fraction.

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

Step 15

Multiply $- 1$ by $- 1$ .

$1 \frac{x}{2 (3 x - 4)}$

Step 16

Multiply $\frac{x}{2 (3 x - 4)}$ by $1$ .

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