Algebra Examples

$\frac{3}{2 x} - \frac{9}{2} = 6 x$

Step 1

Subtract $6 x$ from both sides of the equation.

$\frac{3}{2 x} - \frac{9}{2} - 6 x = 0$

Step 2

Reorder terms.

$- 6 x + \frac{3}{2 x} - \frac{9}{2} = 0$

Step 3

Factor $3$ out of $- 6 x + \frac{3}{2 x} - \frac{9}{2}$ .

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Step 3.1

Factor $3$ out of $- 6 x$ .

$3 (- 2 x) + \frac{3}{2 x} - \frac{9}{2} = 0$

Step 3.2

Factor $3$ out of $\frac{3}{2 x}$ .

$3 (- 2 x) + 3 (\frac{1}{2 x}) - \frac{9}{2} = 0$

Step 3.3

Factor $3$ out of $- \frac{9}{2}$ .

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

Step 3.4

Factor $3$ out of $3 (- 2 x) + 3 \frac{1}{2 x}$ .

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

Step 3.5

Factor $3$ out of $3 (- 2 x + \frac{1}{2 x}) + 3 (- \frac{3}{2})$ .

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

Step 4

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

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

Step 5

Combine $- 2 x$ and $\frac{2 x}{2 x}$ .

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Rewrite $1$ as $1^{2}$ .

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

Step 7.2

Rewrite $2 x (2 x)$ as ${(2 x)}^{2}$ .

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

Step 7.3

Reorder $- {(2 x)}^{2}$ and $1^{2}$ .

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

Step 7.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 2 x$ .

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

Step 7.5

Multiply $2$ by $- 1$ .

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

Step 8

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

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

Step 9

Multiply $\frac{3}{2}$ by $\frac{x}{x}$ .

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Simplify the numerator.

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Step 11.1

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

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Step 11.1.1

Apply the distributive property.

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

Step 11.1.2

Apply the distributive property.

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

Step 11.1.3

Apply the distributive property.

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

Step 11.2

Simplify and combine like terms.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Multiply $1$ by $1$ .

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

Step 11.2.1.2

Multiply $- 2 x$ by $1$ .

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

Step 11.2.1.3

Multiply $2$ by $1$ .

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

Step 11.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 11.2.1.5

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

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Step 11.2.1.5.1

Move $x$ .

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

Step 11.2.1.5.2

Multiply $x$ by $x$ .

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

Step 11.2.1.6

Multiply $2$ by $- 2$ .

$3 (\frac{1 - 2 x + 2 x - 4 x^{2} - 3 x}{2 x}) = 0$

Step 11.2.2

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

$3 (\frac{1 + 0 - 4 x^{2} - 3 x}{2 x}) = 0$

Step 11.2.3

Add $1$ and $0$ .

$3 (\frac{1 - 4 x^{2} - 3 x}{2 x}) = 0$

Step 11.3

Reorder terms.

$3 (\frac{- 4 x^{2} - 3 x + 1}{2 x}) = 0$

Step 11.4

Factor by grouping.

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Step 11.4.1

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 11.4.1.1

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

$3 (\frac{- 4 x^{2} - 3 x + 1}{2 x}) = 0$

Step 11.4.1.2

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

$3 (\frac{- 4 x^{2} + (1 - 4) x + 1}{2 x}) = 0$

Step 11.4.1.3

Apply the distributive property.

$3 (\frac{- 4 x^{2} + 1 x - 4 x + 1}{2 x}) = 0$

Step 11.4.1.4

Multiply $x$ by $1$ .

$3 (\frac{- 4 x^{2} + x - 4 x + 1}{2 x}) = 0$

Step 11.4.2

Factor out the greatest common factor from each group.

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Step 11.4.2.1

Group the first two terms and the last two terms.

$3 (\frac{(- 4 x^{2} + x) - 4 x + 1}{2 x}) = 0$

Step 11.4.2.2

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

$3 (\frac{x (- 4 x + 1) + 1 (- 4 x + 1)}{2 x}) = 0$

Step 11.4.3

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

$3 (\frac{(- 4 x + 1) (x + 1)}{2 x}) = 0$

Step 12

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

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Step 12.1

Combine $3$ and $\frac{(- 4 x + 1) (x + 1)}{2 x}$ .

$\frac{3 ((- 4 x + 1) (x + 1))}{2 x} = 0$

Step 12.2

Factor $- 1$ out of $- 4 x$ .

$\frac{3 (- (4 x) + 1) (x + 1)}{2 x} = 0$

Step 12.3

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

$\frac{3 (- (4 x) - 1 (- 1)) (x + 1)}{2 x} = 0$

Step 12.4

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

$\frac{3 (- (4 x - 1)) (x + 1)}{2 x} = 0$

Step 12.5

Simplify the expression.

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Step 12.5.1

Rewrite $- (4 x - 1)$ as $- 1 (4 x - 1)$ .

$\frac{3 (- 1 (4 x - 1)) (x + 1)}{2 x} = 0$

Step 12.5.2

Move the negative in front of the fraction.

$- \frac{(3 (4 x - 1)) (x + 1)}{2 x} = 0$

Step 12.5.3

Reorder factors in $- \frac{(3 (4 x - 1)) (x + 1)}{2 x}$ .

$- \frac{3 (4 x - 1) (x + 1)}{2 x} = 0$

Step 13

Set the numerator equal to zero.

$3 (4 x - 1) (x + 1) = 0$

Step 14

Solve the equation for $x$ .

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Step 14.1

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

$4 x - 1 = 0$

$x + 1 = 0$

Step 14.2

Set $4 x - 1$ equal to $0$ and solve for $x$ .

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Step 14.2.1

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Step 14.2.2

Solve $4 x - 1 = 0$ for $x$ .

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Step 14.2.2.1

Add $1$ to both sides of the equation.

$4 x = 1$

Step 14.2.2.2

Divide each term in $4 x = 1$ by $4$ and simplify.

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Step 14.2.2.2.1

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Step 14.2.2.2.2

Simplify the left side.

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Step 14.2.2.2.2.1

Cancel the common factor of $4$ .

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Step 14.2.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Step 14.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 14.3

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

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Step 14.3.1

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

$x + 1 = 0$

Step 14.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 14.4

The final solution is all the values that make $3 (4 x - 1) (x + 1) = 0$ true.

$x = \frac{1}{4}, - 1$