Algebra Examples

$3 x - \frac{1}{2 x - 1} = 4$

Step 1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 1.1

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

$1, 2 x - 1, 1$

Step 1.2

Remove parentheses.

$1, 2 x - 1, 1$

Step 1.3

The LCM of one and any expression is the expression.

$2 x - 1$

Step 2

Multiply each term in $3 x - \frac{1}{2 x - 1} = 4$ by $2 x - 1$ to eliminate the fractions.

Tap for more steps...

Step 2.1

Multiply each term in $3 x - \frac{1}{2 x - 1} = 4$ by $2 x - 1$ .

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.2.1.3

Multiply $- 1$ by $3$ .

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

Step 2.2.1.4

Simplify each term.

Tap for more steps...

Step 2.2.1.4.1

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

Tap for more steps...

Step 2.2.1.4.1.1

Move $x$ .

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

Step 2.2.1.4.1.2

Multiply $x$ by $x$ .

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

Step 2.2.1.4.2

Multiply $3$ by $2$ .

$6 x^{2} - 3 x - \frac{1}{2 x - 1} (2 x - 1) = 4 (2 x - 1)$

Step 2.2.1.5

Cancel the common factor of $2 x - 1$ .

Tap for more steps...

Step 2.2.1.5.1

Move the leading negative in $- \frac{1}{2 x - 1}$ into the numerator.

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

Step 2.2.1.5.2

Cancel the common factor.

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

Step 2.2.1.5.3

Rewrite the expression.

$6 x^{2} - 3 x - 1 = 4 (2 x - 1)$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Multiply.

Tap for more steps...

Step 2.3.2.1

Multiply $2$ by $4$ .

$6 x^{2} - 3 x - 1 = 8 x + 4 \cdot - 1$

Step 2.3.2.2

Multiply $4$ by $- 1$ .

$6 x^{2} - 3 x - 1 = 8 x - 4$

Step 3

Solve the equation.

Tap for more steps...

Step 3.1

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 3.1.1

Subtract $8 x$ from both sides of the equation.

$6 x^{2} - 3 x - 1 - 8 x = - 4$

Step 3.1.2

Subtract $8 x$ from $- 3 x$ .

$6 x^{2} - 11 x - 1 = - 4$

Step 3.2

Add $4$ to both sides of the equation.

$6 x^{2} - 11 x - 1 + 4 = 0$

Step 3.3

Add $- 1$ and $4$ .

$6 x^{2} - 11 x + 3 = 0$

Step 3.4

Factor by grouping.

Tap for more steps...

Step 3.4.1

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 $a \cdot c = 6 \cdot 3 = 18$ and whose sum is $b = - 11$ .

Tap for more steps...

Step 3.4.1.1

Factor $- 11$ out of $- 11 x$ .

$6 x^{2} - 11 x + 3 = 0$

Step 3.4.1.2

Rewrite $- 11$ as $- 2$ plus $- 9$

$6 x^{2} + (- 2 - 9) x + 3 = 0$

Step 3.4.1.3

Apply the distributive property.

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

Step 3.4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.4.2.1

Group the first two terms and the last two terms.

$(6 x^{2} - 2 x) - 9 x + 3 = 0$

Step 3.4.2.2

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

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

Step 3.4.3

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

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

Step 3.5

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

$3 x - 1 = 0$

$2 x - 3 = 0$

Step 3.6

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

Tap for more steps...

Step 3.6.1

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

$3 x - 1 = 0$

Step 3.6.2

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

Tap for more steps...

Step 3.6.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 3.6.2.2

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

Tap for more steps...

Step 3.6.2.2.1

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

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

Step 3.6.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.6.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.6.2.2.2.1.1

Cancel the common factor.

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

Step 3.6.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.7

Set $2 x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.7.1

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 3.7.2

Solve $2 x - 3 = 0$ for $x$ .

Tap for more steps...

Step 3.7.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 3.7.2.2

Divide each term in $2 x = 3$ by $2$ and simplify.

Tap for more steps...

Step 3.7.2.2.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Step 3.7.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.7.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.7.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Step 3.7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 3.8

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{3}, 1.5$

Mixed Number Form:

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