Algebra Examples

$\frac{2 m}{2 m + 3} - \frac{2 m}{2 m - 3} = 1$

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Write each expression with a common denominator of $(2 m + 3) (2 m - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Reorder the factors of $(2 m - 3) (2 m + 3)$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Factor $2 m$ out of $2 m (2 m - 3) - 2 m (2 m + 3)$ .

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Step 2.5.1.1.1

Factor $2 m$ out of $- 2 m (2 m + 3)$ .

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

Step 2.5.1.1.2

Factor $2 m$ out of $2 m (2 m - 3) + 2 m (- (2 m + 3))$ .

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

Step 2.5.1.2

Apply the distributive property.

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

Step 2.5.1.3

Multiply $2$ by $- 1$ .

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

Step 2.5.1.4

Multiply $- 1$ by $3$ .

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

Step 2.5.1.5

Subtract $2 m$ from $2 m$ .

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

Step 2.5.1.6

Subtract $3$ from $0$ .

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

Step 2.5.1.7

Subtract $3$ from $- 3$ .

$\frac{2 m \cdot - 6}{(2 m + 3) (2 m - 3)} - 1 = 0$

Step 2.5.1.8

Multiply $- 6$ by $2$ .

$\frac{- 12 m}{(2 m + 3) (2 m - 3)} - 1 = 0$

Step 2.5.2

Move the negative in front of the fraction.

$- \frac{12 m}{(2 m + 3) (2 m - 3)} - 1 = 0$

Step 3

Set the denominator in $\frac{12 m}{(2 m + 3) (2 m - 3)}$ equal to $0$ to find where the expression is undefined.

$(2 m + 3) (2 m - 3) = 0$

Step 4

Solve for $m$ .

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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 $0$ .

$2 m + 3 = 0$

$2 m - 3 = 0$

Step 4.2

Set $2 m + 3$ equal to $0$ and solve for $m$ .

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Step 4.2.1

Set $2 m + 3$ equal to $0$ .

$2 m + 3 = 0$

Step 4.2.2

Solve $2 m + 3 = 0$ for $m$ .

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Step 4.2.2.1

Subtract $3$ from both sides of the equation.

$2 m = - 3$

Step 4.2.2.2

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

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Step 4.2.2.2.1

Divide each term in $2 m = - 3$ by $2$ .

$\frac{2 m}{2} = \frac{- 3}{2}$

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 $2$ .

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Step 4.2.2.2.2.1.1

Cancel the common factor.

$\frac{2 m}{2} = \frac{- 3}{2}$

Step 4.2.2.2.2.1.2

Divide $m$ by $1$ .

$m = \frac{- 3}{2}$

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.

$m = - \frac{3}{2}$

Step 4.3

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

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Step 4.3.1

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

$2 m - 3 = 0$

Step 4.3.2

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

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Step 4.3.2.1

Add $3$ to both sides of the equation.

$2 m = 3$

Step 4.3.2.2

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

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Step 4.3.2.2.1

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

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

Step 4.3.2.2.2

Simplify the left side.

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Step 4.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.2.2.1.1

Cancel the common factor.

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

Step 4.3.2.2.2.1.2

Divide $m$ by $1$ .

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

Step 4.4

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

$m = - \frac{3}{2}, \frac{3}{2}$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$m = - \frac{3}{2}, m = \frac{3}{2}$

Step 6