Algebra Examples

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

Step 1

Subtract $5$ from both sides of the equation.

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

Step 2

Simplify $\frac{x + 3}{x - 1} + \frac{x - 2}{x} - 5$ .

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Step 2.1

To write $\frac{x + 3}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x + 3}{x - 1} \cdot \frac{x}{x} + \frac{x - 2}{x} - 5 = 0$

Step 2.2

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

$\frac{x + 3}{x - 1} \cdot \frac{x}{x} + \frac{x - 2}{x} \cdot \frac{x - 1}{x - 1} - 5 = 0$

Step 2.3

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

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Step 2.3.1

Multiply $\frac{x + 3}{x - 1}$ by $\frac{x}{x}$ .

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

Step 2.3.2

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

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

Step 2.3.3

Reorder the factors of $(x - 1) x$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Multiply $x$ by $x$ .

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

Step 2.5.3

Expand $(x - 2) (x - 1)$ using the FOIL Method.

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Step 2.5.3.1

Apply the distributive property.

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

Step 2.5.3.2

Apply the distributive property.

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

Step 2.5.3.3

Apply the distributive property.

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

Step 2.5.4

Simplify and combine like terms.

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Step 2.5.4.1

Simplify each term.

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Step 2.5.4.1.1

Multiply $x$ by $x$ .

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

Step 2.5.4.1.2

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

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

Step 2.5.4.1.3

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

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

Step 2.5.4.1.4

Multiply $- 2$ by $- 1$ .

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

Step 2.5.4.2

Subtract $2 x$ from $- x$ .

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

Step 2.5.5

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

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

Step 2.5.6

Subtract $3 x$ from $3 x$ .

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

Step 2.5.7

Add $2 x^{2}$ and $0$ .

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

Step 2.5.8

Factor $2$ out of $2 x^{2} + 2$ .

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Step 2.5.8.1

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

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

Step 2.5.8.2

Factor $2$ out of $2$ .

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

Step 2.5.8.3

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

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

Step 2.6

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

$\frac{2 (x^{2} + 1)}{x (x - 1)} - 5 \cdot \frac{x (x - 1)}{x (x - 1)} = 0$

Step 2.7

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

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

$\frac{2 x^{2} + 2 \cdot 1 - 5 x (x - 1)}{x (x - 1)} = 0$

Step 2.9.2

Multiply $2$ by $1$ .

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

Step 2.9.3

Apply the distributive property.

$\frac{2 x^{2} + 2 - 5 x \cdot x - 5 x \cdot - 1}{x (x - 1)} = 0$

Step 2.9.4

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

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Step 2.9.4.1

Move $x$ .

$\frac{2 x^{2} + 2 - 5 (x \cdot x) - 5 x \cdot - 1}{x (x - 1)} = 0$

Step 2.9.4.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 - 5 x^{2} - 5 x \cdot - 1}{x (x - 1)} = 0$

Step 2.9.5

Multiply $- 1$ by $- 5$ .

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

Step 2.9.6

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

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

Step 2.9.7

Reorder terms.

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

Step 2.9.8

Factor by grouping.

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Step 2.9.8.1

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 $a \cdot c = - 3 \cdot 2 = - 6$ and whose sum is $b = 5$ .

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Step 2.9.8.1.1

Factor $5$ out of $5 x$ .

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

Step 2.9.8.1.2

Rewrite $5$ as $- 1$ plus $6$

$\frac{- 3 x^{2} + (- 1 + 6) x + 2}{x (x - 1)} = 0$

Step 2.9.8.1.3

Apply the distributive property.

$\frac{- 3 x^{2} - 1 x + 6 x + 2}{x (x - 1)} = 0$

Step 2.9.8.2

Factor out the greatest common factor from each group.

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Step 2.9.8.2.1

Group the first two terms and the last two terms.

$\frac{(- 3 x^{2} - 1 x) + 6 x + 2}{x (x - 1)} = 0$

Step 2.9.8.2.2

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

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

Step 2.9.8.3

Factor the polynomial by factoring out the greatest common factor, $- 3 x - 1$ .

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

Move the negative in front of the fraction.

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

Step 3

Set the numerator equal to zero.

$(3 x + 1) (x - 2) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 x + 1 = 0$

$x - 2 = 0$

Step 4.2

Set $3 x + 1$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Step 4.2.2

Solve $3 x + 1 = 0$ for $x$ .

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Step 4.2.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 4.2.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 4.2.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 4.2.2.2.2

Simplify the left side.

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Step 4.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 4.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 4.2.2.2.3

Simplify the right side.

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Step 4.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 4.3

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.4

The final solution is all the values that make $(3 x + 1) (x - 2) = 0$ true.

$x = - \frac{1}{3}, 2$