Algebra Examples

$1 - \frac{4}{x} = - 3 x$

Step 1

Subtract $1$ from both sides of the equation.

$- \frac{4}{x} = - 3 x - 1$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1, 1$

Step 2.2

The LCM of one and any expression is the expression.

$x$

Step 3

Multiply each term in $- \frac{4}{x} = - 3 x - 1$ by $x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $- \frac{4}{x} = - 3 x - 1$ by $x$ .

$- \frac{4}{x} x = - 3 x \cdot x - x$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x$ .

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Step 3.2.1.1

Move the leading negative in $- \frac{4}{x}$ into the numerator.

$\frac{- 4}{x} x = - 3 x \cdot x - x$

Step 3.2.1.2

Cancel the common factor.

$\frac{- 4}{x} x = - 3 x \cdot x - x$

Step 3.2.1.3

Rewrite the expression.

$- 4 = - 3 x \cdot x - x$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Move $x$ .

$- 4 = - 3 (x \cdot x) - x$

Step 3.3.1.2

Multiply $x$ by $x$ .

$- 4 = - 3 x^{2} - x$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $- 3 x^{2} - x = - 4$ .

$- 3 x^{2} - x = - 4$

Step 4.2

Add $4$ to both sides of the equation.

$- 3 x^{2} - x + 4 = 0$

Step 4.3

Factor the left side of the equation.

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Step 4.3.1

Factor $- 1$ out of $- 3 x^{2} - x + 4$ .

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Step 4.3.1.1

Factor $- 1$ out of $- 3 x^{2}$ .

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

Step 4.3.1.2

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

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

Step 4.3.1.3

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

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

Step 4.3.1.4

Factor $- 1$ out of $- (3 x^{2}) - (x)$ .

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

Step 4.3.1.5

Factor $- 1$ out of $- (3 x^{2} + x) - 1 (- 4)$ .

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

Step 4.3.2

Factor.

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Step 4.3.2.1

Factor by grouping.

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Step 4.3.2.1.1

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 $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 1$ .

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Step 4.3.2.1.1.1

Multiply by $1$ .

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

Step 4.3.2.1.1.2

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

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

Step 4.3.2.1.1.3

Apply the distributive property.

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

Step 4.3.2.1.2

Factor out the greatest common factor from each group.

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Step 4.3.2.1.2.1

Group the first two terms and the last two terms.

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

Step 4.3.2.1.2.2

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

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

Step 4.3.2.1.3

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

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

Step 4.3.2.2

Remove unnecessary parentheses.

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

Step 4.4

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

$x - 1 = 0$

$3 x + 4 = 0$

Step 4.5

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

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Step 4.5.1

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

$x - 1 = 0$

Step 4.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.6

Set $3 x + 4$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $3 x + 4$ equal to $0$ .

$3 x + 4 = 0$

Step 4.6.2

Solve $3 x + 4 = 0$ for $x$ .

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Step 4.6.2.1

Subtract $4$ from both sides of the equation.

$3 x = - 4$

Step 4.6.2.2

Divide each term in $3 x = - 4$ by $3$ and simplify.

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Step 4.6.2.2.1

Divide each term in $3 x = - 4$ by $3$ .

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 4.6.2.2.2

Simplify the left side.

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Step 4.6.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.6.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 4.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{3}$

Step 4.6.2.2.3

Simplify the right side.

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Step 4.6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{4}{3}$

Step 4.7

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

$x = 1, - \frac{4}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 1, - \frac{4}{3}$

Decimal Form:

$x = 1, - 1.\overline{3}$

Mixed Number Form:

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