Algebra Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Write each expression with a common denominator of $(2 x - 5) (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{x - 5}{2 x - 5}$ by $\frac{x - 5}{x - 5}$ .

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

Step 2.3.2

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

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

Step 2.3.3

Reorder the factors of $(x - 5) (2 x - 5)$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Rewrite $\frac{(x - 5) (x - 5) - (x + 2) (2 x - 5)}{(2 x - 5) (x - 5)}$ in a factored form.

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Step 2.5.1

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 2.5.1.1

Apply the distributive property.

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

Step 2.5.1.2

Apply the distributive property.

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

Step 2.5.1.3

Apply the distributive property.

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

Step 2.5.2

Simplify and combine like terms.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Multiply $x$ by $x$ .

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

Step 2.5.2.1.2

Move $- 5$ to the left of $x$ .

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

Step 2.5.2.1.3

Multiply $- 5$ by $- 5$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 5 x - 5 x + 25 - (x + 2) (2 x - 5)}{(2 x - 5) (x - 5)}}$

Step 2.5.2.2

Subtract $5 x$ from $- 5 x$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 - (x + 2) (2 x - 5)}{(2 x - 5) (x - 5)}}$

Step 2.5.3

Apply the distributive property.

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 + (- x - 1 \cdot 2) (2 x - 5)}{(2 x - 5) (x - 5)}}$

Step 2.5.4

Multiply $- 1$ by $2$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 + (- x - 2) (2 x - 5)}{(2 x - 5) (x - 5)}}$

Step 2.5.5

Expand $(- x - 2) (2 x - 5)$ using the FOIL Method.

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Step 2.5.5.1

Apply the distributive property.

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 - x (2 x - 5) - 2 (2 x - 5)}{(2 x - 5) (x - 5)}}$

Step 2.5.5.2

Apply the distributive property.

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 - x (2 x) - x \cdot - 5 - 2 (2 x - 5)}{(2 x - 5) (x - 5)}}$

Step 2.5.5.3

Apply the distributive property.

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 - x (2 x) - x \cdot - 5 - 2 (2 x) - 2 \cdot - 5}{(2 x - 5) (x - 5)}}$

Step 2.5.6

Simplify and combine like terms.

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Step 2.5.6.1

Simplify each term.

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Step 2.5.6.1.1

Rewrite using the commutative property of multiplication.

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 - 1 \cdot 2 x \cdot x - x \cdot - 5 - 2 (2 x) - 2 \cdot - 5}{(2 x - 5) (x - 5)}}$

Step 2.5.6.1.2

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

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Step 2.5.6.1.2.1

Move $x$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 - 1 \cdot 2 (x \cdot x) - x \cdot - 5 - 2 (2 x) - 2 \cdot - 5}{(2 x - 5) (x - 5)}}$

Step 2.5.6.1.2.2

Multiply $x$ by $x$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 - 1 \cdot 2 x^{2} - x \cdot - 5 - 2 (2 x) - 2 \cdot - 5}{(2 x - 5) (x - 5)}}$

Step 2.5.6.1.3

Multiply $- 1$ by $2$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 - 2 x^{2} - x \cdot - 5 - 2 (2 x) - 2 \cdot - 5}{(2 x - 5) (x - 5)}}$

Step 2.5.6.1.4

Multiply $- 5$ by $- 1$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 - 2 x^{2} + 5 x - 2 (2 x) - 2 \cdot - 5}{(2 x - 5) (x - 5)}}$

Step 2.5.6.1.5

Multiply $2$ by $- 2$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 - 2 x^{2} + 5 x - 4 x - 2 \cdot - 5}{(2 x - 5) (x - 5)}}$

Step 2.5.6.1.6

Multiply $- 2$ by $- 5$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 - 2 x^{2} + 5 x - 4 x + 10}{(2 x - 5) (x - 5)}}$

Step 2.5.6.2

Subtract $4 x$ from $5 x$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{x^{2} - 10 x + 25 - 2 x^{2} + x + 10}{(2 x - 5) (x - 5)}}$

Step 2.5.7

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

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{- x^{2} - 10 x + 25 + x + 10}{(2 x - 5) (x - 5)}}$

Step 2.5.8

Add $- 10 x$ and $x$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{- x^{2} - 9 x + 25 + 10}{(2 x - 5) (x - 5)}}$

Step 2.5.9

Add $25$ and $10$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{1}{\frac{- x^{2} - 9 x + 35}{(2 x - 5) (x - 5)}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{x - 5}{2 x - 5} (1 \frac{(2 x - 5) (x - 5)}{- x^{2} - 9 x + 35})$

Step 4

Simplify terms.

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Step 4.1

Multiply $\frac{(2 x - 5) (x - 5)}{- x^{2} - 9 x + 35}$ by $1$ .

$\frac{x - 5}{2 x - 5} \cdot \frac{(2 x - 5) (x - 5)}{- x^{2} - 9 x + 35}$

Step 4.2

Cancel the common factor of $2 x - 5$ .

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Step 4.2.1

Cancel the common factor.

$\frac{x - 5}{2 x - 5} \cdot \frac{(2 x - 5) (x - 5)}{- x^{2} - 9 x + 35}$

Step 4.2.2

Rewrite the expression.

$(x - 5) \frac{x - 5}{- x^{2} - 9 x + 35}$

Step 4.3

Multiply $x - 5$ by $\frac{x - 5}{- x^{2} - 9 x + 35}$ .

$\frac{(x - 5) (x - 5)}{- x^{2} - 9 x + 35}$

Step 5

Simplify the numerator.

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Step 5.1

Raise $x - 5$ to the power of $1$ .

$\frac{{(x - 5)}^{1} (x - 5)}{- x^{2} - 9 x + 35}$

Step 5.2

Raise $x - 5$ to the power of $1$ .

$\frac{{(x - 5)}^{1} {(x - 5)}^{1}}{- x^{2} - 9 x + 35}$

Step 5.3

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

$\frac{{(x - 5)}^{1 + 1}}{- x^{2} - 9 x + 35}$

Step 5.4

Add $1$ and $1$ .

$\frac{{(x - 5)}^{2}}{- x^{2} - 9 x + 35}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- x^{2}$ .

$\frac{{(x - 5)}^{2}}{- (x^{2}) - 9 x + 35}$

Step 6.2

Factor $- 1$ out of $- 9 x$ .

$\frac{{(x - 5)}^{2}}{- (x^{2}) - (9 x) + 35}$

Step 6.3

Factor $- 1$ out of $- (x^{2}) - (9 x)$ .

$\frac{{(x - 5)}^{2}}{- (x^{2} + 9 x) + 35}$

Step 6.4

Rewrite $35$ as $- 1 (- 35)$ .

$\frac{{(x - 5)}^{2}}{- (x^{2} + 9 x) - 1 (- 35)}$

Step 6.5

Factor $- 1$ out of $- (x^{2} + 9 x) - 1 (- 35)$ .

$\frac{{(x - 5)}^{2}}{- (x^{2} + 9 x - 35)}$

Step 6.6

Rewrite negatives.

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Step 6.6.1

Rewrite $- (x^{2} + 9 x - 35)$ as $- 1 (x^{2} + 9 x - 35)$ .

$\frac{{(x - 5)}^{2}}{- 1 (x^{2} + 9 x - 35)}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{{(x - 5)}^{2}}{x^{2} + 9 x - 35}$