Pre-Algebra Examples

$\frac{3 x - 22}{x - 5} - \frac{13 - 4 x}{x - 59}$

Step 1

To write $\frac{3 x - 22}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x - 59}{x - 59}$ .

$\frac{3 x - 22}{x - 5} \cdot \frac{x - 59}{x - 59} - \frac{13 - 4 x}{x - 59}$

Step 2

To write $- \frac{13 - 4 x}{x - 59}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{3 x - 22}{x - 5} \cdot \frac{x - 59}{x - 59} - \frac{13 - 4 x}{x - 59} \cdot \frac{x - 5}{x - 5}$

Step 3

Write each expression with a common denominator of $(x - 5) (x - 59)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{3 x - 22}{x - 5}$ by $\frac{x - 59}{x - 59}$ .

$\frac{(3 x - 22) (x - 59)}{(x - 5) (x - 59)} - \frac{13 - 4 x}{x - 59} \cdot \frac{x - 5}{x - 5}$

Step 3.2

Multiply $\frac{13 - 4 x}{x - 59}$ by $\frac{x - 5}{x - 5}$ .

$\frac{(3 x - 22) (x - 59)}{(x - 5) (x - 59)} - \frac{(13 - 4 x) (x - 5)}{(x - 59) (x - 5)}$

Step 3.3

Reorder the factors of $(x - 59) (x - 5)$ .

$\frac{(3 x - 22) (x - 59)}{(x - 5) (x - 59)} - \frac{(13 - 4 x) (x - 5)}{(x - 5) (x - 59)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(3 x - 22) (x - 59) - (13 - 4 x) (x - 5)}{(x - 5) (x - 59)}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(3 x - 22) (x - 59)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{3 x (x - 59) - 22 (x - 59) - (13 - 4 x) (x - 5)}{(x - 5) (x - 59)}$

Step 5.1.2

Apply the distributive property.

$\frac{3 x \cdot x + 3 x \cdot - 59 - 22 (x - 59) - (13 - 4 x) (x - 5)}{(x - 5) (x - 59)}$

Step 5.1.3

Apply the distributive property.

$\frac{3 x \cdot x + 3 x \cdot - 59 - 22 x - 22 \cdot - 59 - (13 - 4 x) (x - 5)}{(x - 5) (x - 59)}$

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

Move $x$ .

$\frac{3 (x \cdot x) + 3 x \cdot - 59 - 22 x - 22 \cdot - 59 - (13 - 4 x) (x - 5)}{(x - 5) (x - 59)}$

Step 5.2.1.1.2

Multiply $x$ by $x$ .

$\frac{3 x^{2} + 3 x \cdot - 59 - 22 x - 22 \cdot - 59 - (13 - 4 x) (x - 5)}{(x - 5) (x - 59)}$

Step 5.2.1.2

Multiply $- 59$ by $3$ .

$\frac{3 x^{2} - 177 x - 22 x - 22 \cdot - 59 - (13 - 4 x) (x - 5)}{(x - 5) (x - 59)}$

Step 5.2.1.3

Multiply $- 22$ by $- 59$ .

$\frac{3 x^{2} - 177 x - 22 x + 1298 - (13 - 4 x) (x - 5)}{(x - 5) (x - 59)}$

Step 5.2.2

Subtract $22 x$ from $- 177 x$ .

$\frac{3 x^{2} - 199 x + 1298 - (13 - 4 x) (x - 5)}{(x - 5) (x - 59)}$

Step 5.3

Apply the distributive property.

$\frac{3 x^{2} - 199 x + 1298 + (- 1 \cdot 13 - (- 4 x)) (x - 5)}{(x - 5) (x - 59)}$

Step 5.4

Multiply $- 1$ by $13$ .

$\frac{3 x^{2} - 199 x + 1298 + (- 13 - (- 4 x)) (x - 5)}{(x - 5) (x - 59)}$

Step 5.5

Multiply $- 4$ by $- 1$ .

$\frac{3 x^{2} - 199 x + 1298 + (- 13 + 4 x) (x - 5)}{(x - 5) (x - 59)}$

Step 5.6

Expand $(- 13 + 4 x) (x - 5)$ using the FOIL Method.

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Step 5.6.1

Apply the distributive property.

$\frac{3 x^{2} - 199 x + 1298 - 13 (x - 5) + 4 x (x - 5)}{(x - 5) (x - 59)}$

Step 5.6.2

Apply the distributive property.

$\frac{3 x^{2} - 199 x + 1298 - 13 x - 13 \cdot - 5 + 4 x (x - 5)}{(x - 5) (x - 59)}$

Step 5.6.3

Apply the distributive property.

$\frac{3 x^{2} - 199 x + 1298 - 13 x - 13 \cdot - 5 + 4 x \cdot x + 4 x \cdot - 5}{(x - 5) (x - 59)}$

Step 5.7

Simplify and combine like terms.

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Step 5.7.1

Simplify each term.

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Step 5.7.1.1

Multiply $- 13$ by $- 5$ .

$\frac{3 x^{2} - 199 x + 1298 - 13 x + 65 + 4 x \cdot x + 4 x \cdot - 5}{(x - 5) (x - 59)}$

Step 5.7.1.2

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

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Step 5.7.1.2.1

Move $x$ .

$\frac{3 x^{2} - 199 x + 1298 - 13 x + 65 + 4 (x \cdot x) + 4 x \cdot - 5}{(x - 5) (x - 59)}$

Step 5.7.1.2.2

Multiply $x$ by $x$ .

$\frac{3 x^{2} - 199 x + 1298 - 13 x + 65 + 4 x^{2} + 4 x \cdot - 5}{(x - 5) (x - 59)}$

Step 5.7.1.3

Multiply $- 5$ by $4$ .

$\frac{3 x^{2} - 199 x + 1298 - 13 x + 65 + 4 x^{2} - 20 x}{(x - 5) (x - 59)}$

Step 5.7.2

Subtract $20 x$ from $- 13 x$ .

$\frac{3 x^{2} - 199 x + 1298 - 33 x + 65 + 4 x^{2}}{(x - 5) (x - 59)}$

Step 5.8

Add $3 x^{2}$ and $4 x^{2}$ .

$\frac{7 x^{2} - 199 x + 1298 - 33 x + 65}{(x - 5) (x - 59)}$

Step 5.9

Subtract $33 x$ from $- 199 x$ .

$\frac{7 x^{2} - 232 x + 1298 + 65}{(x - 5) (x - 59)}$

Step 5.10

Add $1298$ and $65$ .

$\frac{7 x^{2} - 232 x + 1363}{(x - 5) (x - 59)}$