Pre-Algebra Examples

$\frac{3 x + 5}{x + 5} - \frac{x + 1}{2 - x} - \frac{4 x^{2} - 3 x - 1}{x^{2} + 3 x - 10}$

Step 1

Simplify each term.

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Step 1.1

Factor by grouping.

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Step 1.1.1

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 $a \cdot c = 4 \cdot - 1 = - 4$ and whose sum is $b = - 3$ .

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Step 1.1.1.1

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

$\frac{3 x + 5}{x + 5} - \frac{x + 1}{2 - x} - \frac{4 x^{2} - 3 (x) - 1}{x^{2} + 3 x - 10}$

Step 1.1.1.2

Rewrite $- 3$ as $1$ plus $- 4$

$\frac{3 x + 5}{x + 5} - \frac{x + 1}{2 - x} - \frac{4 x^{2} + (1 - 4) x - 1}{x^{2} + 3 x - 10}$

Step 1.1.1.3

Apply the distributive property.

$\frac{3 x + 5}{x + 5} - \frac{x + 1}{2 - x} - \frac{4 x^{2} + 1 x - 4 x - 1}{x^{2} + 3 x - 10}$

Step 1.1.1.4

Multiply $x$ by $1$ .

$\frac{3 x + 5}{x + 5} - \frac{x + 1}{2 - x} - \frac{4 x^{2} + x - 4 x - 1}{x^{2} + 3 x - 10}$

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

$\frac{3 x + 5}{x + 5} - \frac{x + 1}{2 - x} - \frac{(4 x^{2} + x) - 4 x - 1}{x^{2} + 3 x - 10}$

Step 1.1.2.2

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

$\frac{3 x + 5}{x + 5} - \frac{x + 1}{2 - x} - \frac{x (4 x + 1) - (4 x + 1)}{x^{2} + 3 x - 10}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$\frac{3 x + 5}{x + 5} - \frac{x + 1}{2 - x} - \frac{(4 x + 1) (x - 1)}{x^{2} + 3 x - 10}$

Step 1.2

Factor $x^{2} + 3 x - 10$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 10$ and whose sum is $3$ .

$- 2, 5$

Step 1.2.2

Write the factored form using these integers.

$\frac{3 x + 5}{x + 5} - \frac{x + 1}{2 - x} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 2

To write $\frac{3 x + 5}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{2 - x}{2 - x}$ .

$\frac{3 x + 5}{x + 5} \cdot \frac{2 - x}{2 - x} - \frac{x + 1}{2 - x} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 3

To write $- \frac{x + 1}{2 - x}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{3 x + 5}{x + 5} \cdot \frac{2 - x}{2 - x} - \frac{x + 1}{2 - x} \cdot \frac{x + 5}{x + 5} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 4

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

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Step 4.1

Multiply $\frac{3 x + 5}{x + 5}$ by $\frac{2 - x}{2 - x}$ .

$\frac{(3 x + 5) (2 - x)}{(x + 5) (2 - x)} - \frac{x + 1}{2 - x} \cdot \frac{x + 5}{x + 5} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 4.2

Multiply $\frac{x + 1}{2 - x}$ by $\frac{x + 5}{x + 5}$ .

$\frac{(3 x + 5) (2 - x)}{(x + 5) (2 - x)} - \frac{(x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 4.3

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

$\frac{(3 x + 5) (2 - x)}{(2 - x) (x + 5)} - \frac{(x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(3 x + 5) (2 - x) - (x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(3 x + 5) (2 - x)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{3 x (2 - x) + 5 (2 - x) - (x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.1.2

Apply the distributive property.

$\frac{3 x \cdot 2 + 3 x (- x) + 5 (2 - x) - (x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.1.3

Apply the distributive property.

$\frac{3 x \cdot 2 + 3 x (- x) + 5 \cdot 2 + 5 (- x) - (x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $2$ by $3$ .

$\frac{6 x + 3 x (- x) + 5 \cdot 2 + 5 (- x) - (x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.2.1.2

Rewrite using the commutative property of multiplication.

$\frac{6 x + 3 \cdot - 1 x \cdot x + 5 \cdot 2 + 5 (- x) - (x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.2.1.3

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

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Step 6.2.1.3.1

Move $x$ .

$\frac{6 x + 3 \cdot - 1 (x \cdot x) + 5 \cdot 2 + 5 (- x) - (x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.2.1.3.2

Multiply $x$ by $x$ .

$\frac{6 x + 3 \cdot - 1 x^{2} + 5 \cdot 2 + 5 (- x) - (x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.2.1.4

Multiply $3$ by $- 1$ .

$\frac{6 x - 3 x^{2} + 5 \cdot 2 + 5 (- x) - (x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.2.1.5

Multiply $5$ by $2$ .

$\frac{6 x - 3 x^{2} + 10 + 5 (- x) - (x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.2.1.6

Multiply $- 1$ by $5$ .

$\frac{6 x - 3 x^{2} + 10 - 5 x - (x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.2.2

Subtract $5 x$ from $6 x$ .

$\frac{x - 3 x^{2} + 10 - (x + 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.3

Apply the distributive property.

$\frac{x - 3 x^{2} + 10 + (- x - 1 \cdot 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.4

Multiply $- 1$ by $1$ .

$\frac{x - 3 x^{2} + 10 + (- x - 1) (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.5

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

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Step 6.5.1

Apply the distributive property.

$\frac{x - 3 x^{2} + 10 - x (x + 5) - 1 (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.5.2

Apply the distributive property.

$\frac{x - 3 x^{2} + 10 - x \cdot x - x \cdot 5 - 1 (x + 5)}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.5.3

Apply the distributive property.

$\frac{x - 3 x^{2} + 10 - x \cdot x - x \cdot 5 - 1 x - 1 \cdot 5}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.6

Simplify and combine like terms.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

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

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Step 6.6.1.1.1

Move $x$ .

$\frac{x - 3 x^{2} + 10 - (x \cdot x) - x \cdot 5 - 1 x - 1 \cdot 5}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.6.1.1.2

Multiply $x$ by $x$ .

$\frac{x - 3 x^{2} + 10 - x^{2} - x \cdot 5 - 1 x - 1 \cdot 5}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.6.1.2

Multiply $5$ by $- 1$ .

$\frac{x - 3 x^{2} + 10 - x^{2} - 5 x - 1 x - 1 \cdot 5}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.6.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{x - 3 x^{2} + 10 - x^{2} - 5 x - x - 1 \cdot 5}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.6.1.4

Multiply $- 1$ by $5$ .

$\frac{x - 3 x^{2} + 10 - x^{2} - 5 x - x - 5}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.6.2

Subtract $x$ from $- 5 x$ .

$\frac{x - 3 x^{2} + 10 - x^{2} - 6 x - 5}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.7

Subtract $6 x$ from $x$ .

$\frac{- 3 x^{2} + 10 - x^{2} - 5 x - 5}{(2 - x) (x + 5)} - \frac{(4 x + 1) (x - 1)}{(x - 2) (x + 5)}$

Step 6.8

Subtract $x^{2}$ from $- 3 x^{2}$ .

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

Step 6.9

Subtract $5$ from $10$ .

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

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $x$ .

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Reorder terms.

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

Step 8

To write $\frac{- 4 x^{2} - 5 x + 5}{(- x + 2) (x + 5)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

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

Step 9

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

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Step 9.1

Multiply $\frac{- 4 x^{2} - 5 x + 5}{(- x + 2) (x + 5)}$ by $\frac{- 1}{- 1}$ .

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

Step 9.2

Reorder the factors of $(- x + 2) (x + 5) \cdot - 1$ .

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Simplify the numerator.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Simplify.

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Step 11.2.1

Multiply $- 1$ by $- 4$ .

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

Step 11.2.2

Multiply $- 1$ by $- 5$ .

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

Step 11.2.3

Multiply $5$ by $- 1$ .

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

Step 11.3

Apply the distributive property.

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

Step 11.4

Multiply $4$ by $- 1$ .

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

Step 11.5

Multiply $- 1$ by $1$ .

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

Step 11.6

Expand $(- 4 x - 1) (x - 1)$ using the FOIL Method.

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Step 11.6.1

Apply the distributive property.

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

Step 11.6.2

Apply the distributive property.

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

Step 11.6.3

Apply the distributive property.

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

Step 11.7

Simplify and combine like terms.

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Step 11.7.1

Simplify each term.

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Step 11.7.1.1

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

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Step 11.7.1.1.1

Move $x$ .

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

Step 11.7.1.1.2

Multiply $x$ by $x$ .

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

Step 11.7.1.2

Multiply $- 1$ by $- 4$ .

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

Step 11.7.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 11.7.1.4

Multiply $- 1$ by $- 1$ .

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

Step 11.7.2

Subtract $x$ from $4 x$ .

$\frac{4 x^{2} + 5 x - 5 - 4 x^{2} + 3 x + 1}{- (- x + 2) (x + 5)}$

Step 11.8

Subtract $4 x^{2}$ from $4 x^{2}$ .

$\frac{5 x - 5 + 0 + 3 x + 1}{- (- x + 2) (x + 5)}$

Step 11.9

Add $5 x$ and $0$ .

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

Step 11.10

Add $5 x$ and $3 x$ .

$\frac{8 x - 5 + 1}{- (- x + 2) (x + 5)}$

Step 11.11

Add $- 5$ and $1$ .

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

Step 11.12

Factor $4$ out of $8 x - 4$ .

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Step 11.12.1

Factor $4$ out of $8 x$ .

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

Step 11.12.2

Factor $4$ out of $- 4$ .

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

Step 11.12.3

Factor $4$ out of $4 (2 x) + 4 (- 1)$ .

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

Step 12

Simplify with factoring out.

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Step 12.1

Move the negative in front of the fraction.

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

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

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

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

Step 12.5

Simplify the expression.

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Step 12.5.1

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

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

Step 12.5.2

Move the negative in front of the fraction.

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

Step 12.5.3

Multiply $- 1$ by $- 1$ .

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

Step 12.5.4

Multiply $\frac{4 (2 x - 1)}{(x - 2) (x + 5)}$ by $1$ .

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