Pre-Algebra Examples

$\frac{x + 3}{\frac{x - 1}{\frac{x + 3}{x - 5} + 4}} - 5$

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator by the reciprocal of the denominator.

$(x + 3) \frac{\frac{x + 3}{x - 5} + 4}{x - 1} - 5$

Step 1.2

Simplify the numerator.

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Step 1.2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$(x + 3) \frac{\frac{x + 3}{x - 5} + \frac{4 (x - 5)}{x - 5}}{x - 1} - 5$

Step 1.2.2

Combine the numerators over the common denominator.

$(x + 3) \frac{\frac{x + 3 + 4 (x - 5)}{x - 5}}{x - 1} - 5$

Step 1.2.3

Reorder terms.

$(x + 3) \frac{\frac{x + 4 (x - 5) + 3}{x - 5}}{x - 1} - 5$

Step 1.2.4

Rewrite $\frac{x + 4 (x - 5) + 3}{x - 5}$ in a factored form.

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Step 1.2.4.1

Apply the distributive property.

$(x + 3) \frac{\frac{x + 4 x + 4 \cdot - 5 + 3}{x - 5}}{x - 1} - 5$

Step 1.2.4.2

Multiply $4$ by $- 5$ .

$(x + 3) \frac{\frac{x + 4 x - 20 + 3}{x - 5}}{x - 1} - 5$

Step 1.2.4.3

Add $x$ and $4 x$ .

$(x + 3) \frac{\frac{5 x - 20 + 3}{x - 5}}{x - 1} - 5$

Step 1.2.4.4

Add $- 20$ and $3$ .

$(x + 3) \frac{\frac{5 x - 17}{x - 5}}{x - 1} - 5$

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

$(x + 3) (\frac{5 x - 17}{x - 5} \cdot \frac{1}{x - 1}) - 5$

Step 1.4

Multiply $\frac{5 x - 17}{x - 5}$ by $\frac{1}{x - 1}$ .

$(x + 3) \frac{5 x - 17}{(x - 5) (x - 1)} - 5$

Step 1.5

Multiply $x + 3$ by $\frac{5 x - 17}{(x - 5) (x - 1)}$ .

$\frac{(x + 3) (5 x - 17)}{(x - 5) (x - 1)} - 5$

Step 2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{(x - 5) (x - 1)}{(x - 5) (x - 1)}$ .

$\frac{(x + 3) (5 x - 17)}{(x - 5) (x - 1)} - 5 \cdot \frac{(x - 5) (x - 1)}{(x - 5) (x - 1)}$

Step 3

Simplify terms.

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Step 3.1

Combine $- 5$ and $\frac{(x - 5) (x - 1)}{(x - 5) (x - 1)}$ .

$\frac{(x + 3) (5 x - 17)}{(x - 5) (x - 1)} + \frac{- 5 ((x - 5) (x - 1))}{(x - 5) (x - 1)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{(x + 3) (5 x - 17) - 5 ((x - 5) (x - 1))}{(x - 5) (x - 1)}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(x + 3) (5 x - 17)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{x (5 x - 17) + 3 (5 x - 17) - 5 (x - 5) (x - 1)}{(x - 5) (x - 1)}$

Step 4.1.2

Apply the distributive property.

$\frac{x (5 x) + x \cdot - 17 + 3 (5 x - 17) - 5 (x - 5) (x - 1)}{(x - 5) (x - 1)}$

Step 4.1.3

Apply the distributive property.

$\frac{x (5 x) + x \cdot - 17 + 3 (5 x) + 3 \cdot - 17 - 5 (x - 5) (x - 1)}{(x - 5) (x - 1)}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{5 x \cdot x + x \cdot - 17 + 3 (5 x) + 3 \cdot - 17 - 5 (x - 5) (x - 1)}{(x - 5) (x - 1)}$

Step 4.2.1.2

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

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Step 4.2.1.2.1

Move $x$ .

$\frac{5 (x \cdot x) + x \cdot - 17 + 3 (5 x) + 3 \cdot - 17 - 5 (x - 5) (x - 1)}{(x - 5) (x - 1)}$

Step 4.2.1.2.2

Multiply $x$ by $x$ .

$\frac{5 x^{2} + x \cdot - 17 + 3 (5 x) + 3 \cdot - 17 - 5 (x - 5) (x - 1)}{(x - 5) (x - 1)}$

Step 4.2.1.3

Move $- 17$ to the left of $x$ .

$\frac{5 x^{2} - 17 \cdot x + 3 (5 x) + 3 \cdot - 17 - 5 (x - 5) (x - 1)}{(x - 5) (x - 1)}$

Step 4.2.1.4

Multiply $5$ by $3$ .

$\frac{5 x^{2} - 17 x + 15 x + 3 \cdot - 17 - 5 (x - 5) (x - 1)}{(x - 5) (x - 1)}$

Step 4.2.1.5

Multiply $3$ by $- 17$ .

$\frac{5 x^{2} - 17 x + 15 x - 51 - 5 (x - 5) (x - 1)}{(x - 5) (x - 1)}$

Step 4.2.2

Add $- 17 x$ and $15 x$ .

$\frac{5 x^{2} - 2 x - 51 - 5 (x - 5) (x - 1)}{(x - 5) (x - 1)}$

Step 4.3

Apply the distributive property.

$\frac{5 x^{2} - 2 x - 51 + (- 5 x - 5 \cdot - 5) (x - 1)}{(x - 5) (x - 1)}$

Step 4.4

Multiply $- 5$ by $- 5$ .

$\frac{5 x^{2} - 2 x - 51 + (- 5 x + 25) (x - 1)}{(x - 5) (x - 1)}$

Step 4.5

Expand $(- 5 x + 25) (x - 1)$ using the FOIL Method.

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Step 4.5.1

Apply the distributive property.

$\frac{5 x^{2} - 2 x - 51 - 5 x (x - 1) + 25 (x - 1)}{(x - 5) (x - 1)}$

Step 4.5.2

Apply the distributive property.

$\frac{5 x^{2} - 2 x - 51 - 5 x \cdot x - 5 x \cdot - 1 + 25 (x - 1)}{(x - 5) (x - 1)}$

Step 4.5.3

Apply the distributive property.

$\frac{5 x^{2} - 2 x - 51 - 5 x \cdot x - 5 x \cdot - 1 + 25 x + 25 \cdot - 1}{(x - 5) (x - 1)}$

Step 4.6

Simplify and combine like terms.

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Step 4.6.1

Simplify each term.

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Step 4.6.1.1

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

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Step 4.6.1.1.1

Move $x$ .

$\frac{5 x^{2} - 2 x - 51 - 5 (x \cdot x) - 5 x \cdot - 1 + 25 x + 25 \cdot - 1}{(x - 5) (x - 1)}$

Step 4.6.1.1.2

Multiply $x$ by $x$ .

$\frac{5 x^{2} - 2 x - 51 - 5 x^{2} - 5 x \cdot - 1 + 25 x + 25 \cdot - 1}{(x - 5) (x - 1)}$

Step 4.6.1.2

Multiply $- 1$ by $- 5$ .

$\frac{5 x^{2} - 2 x - 51 - 5 x^{2} + 5 x + 25 x + 25 \cdot - 1}{(x - 5) (x - 1)}$

Step 4.6.1.3

Multiply $25$ by $- 1$ .

$\frac{5 x^{2} - 2 x - 51 - 5 x^{2} + 5 x + 25 x - 25}{(x - 5) (x - 1)}$

Step 4.6.2

Add $5 x$ and $25 x$ .

$\frac{5 x^{2} - 2 x - 51 - 5 x^{2} + 30 x - 25}{(x - 5) (x - 1)}$

Step 4.7

Subtract $5 x^{2}$ from $5 x^{2}$ .

$\frac{- 2 x - 51 + 0 + 30 x - 25}{(x - 5) (x - 1)}$

Step 4.8

Add $- 2 x$ and $0$ .

$\frac{- 2 x - 51 + 30 x - 25}{(x - 5) (x - 1)}$

Step 4.9

Add $- 2 x$ and $30 x$ .

$\frac{28 x - 51 - 25}{(x - 5) (x - 1)}$

Step 4.10

Subtract $25$ from $- 51$ .

$\frac{28 x - 76}{(x - 5) (x - 1)}$

Step 4.11

Factor $4$ out of $28 x - 76$ .

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Step 4.11.1

Factor $4$ out of $28 x$ .

$\frac{4 (7 x) - 76}{(x - 5) (x - 1)}$

Step 4.11.2

Factor $4$ out of $- 76$ .

$\frac{4 (7 x) + 4 (- 19)}{(x - 5) (x - 1)}$

Step 4.11.3

Factor $4$ out of $4 (7 x) + 4 (- 19)$ .

$\frac{4 (7 x - 19)}{(x - 5) (x - 1)}$