Finite Math Examples

$\frac{x - 3}{3 x - 1} = \frac{x + 4}{2 x + 5}$

Step 1

Subtract $\frac{x + 4}{2 x + 5}$ from both sides of the equation.

$\frac{x - 3}{3 x - 1} - \frac{x + 4}{2 x + 5} = 0$

Step 2

Simplify $\frac{x - 3}{3 x - 1} - \frac{x + 4}{2 x + 5}$ .

Tap for more steps...

Step 2.1

To write $\frac{x - 3}{3 x - 1}$ as a fraction with a common denominator, multiply by $\frac{2 x + 5}{2 x + 5}$ .

$\frac{x - 3}{3 x - 1} \cdot \frac{2 x + 5}{2 x + 5} - \frac{x + 4}{2 x + 5} = 0$

Step 2.2

To write $- \frac{x + 4}{2 x + 5}$ as a fraction with a common denominator, multiply by $\frac{3 x - 1}{3 x - 1}$ .

$\frac{x - 3}{3 x - 1} \cdot \frac{2 x + 5}{2 x + 5} - \frac{x + 4}{2 x + 5} \cdot \frac{3 x - 1}{3 x - 1} = 0$

Step 2.3

Write each expression with a common denominator of $(3 x - 1) (2 x + 5)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.3.1

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

$\frac{(x - 3) (2 x + 5)}{(3 x - 1) (2 x + 5)} - \frac{x + 4}{2 x + 5} \cdot \frac{3 x - 1}{3 x - 1} = 0$

Step 2.3.2

Multiply $\frac{x + 4}{2 x + 5}$ by $\frac{3 x - 1}{3 x - 1}$ .

$\frac{(x - 3) (2 x + 5)}{(3 x - 1) (2 x + 5)} - \frac{(x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.3.3

Reorder the factors of $(3 x - 1) (2 x + 5)$ .

$\frac{(x - 3) (2 x + 5)}{(2 x + 5) (3 x - 1)} - \frac{(x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{(x - 3) (2 x + 5) - (x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5

Simplify the numerator.

Tap for more steps...

Step 2.5.1

Expand $(x - 3) (2 x + 5)$ using the FOIL Method.

Tap for more steps...

Step 2.5.1.1

Apply the distributive property.

$\frac{x (2 x + 5) - 3 (2 x + 5) - (x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.1.2

Apply the distributive property.

$\frac{x (2 x) + x \cdot 5 - 3 (2 x + 5) - (x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.1.3

Apply the distributive property.

$\frac{x (2 x) + x \cdot 5 - 3 (2 x) - 3 \cdot 5 - (x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.2

Simplify and combine like terms.

Tap for more steps...

Step 2.5.2.1

Simplify each term.

Tap for more steps...

Step 2.5.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x \cdot x + x \cdot 5 - 3 (2 x) - 3 \cdot 5 - (x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.2.1.2

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

Tap for more steps...

Step 2.5.2.1.2.1

Move $x$ .

$\frac{2 (x \cdot x) + x \cdot 5 - 3 (2 x) - 3 \cdot 5 - (x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.2.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x \cdot 5 - 3 (2 x) - 3 \cdot 5 - (x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.2.1.3

Move $5$ to the left of $x$ .

$\frac{2 x^{2} + 5 \cdot x - 3 (2 x) - 3 \cdot 5 - (x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.2.1.4

Multiply $2$ by $- 3$ .

$\frac{2 x^{2} + 5 x - 6 x - 3 \cdot 5 - (x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.2.1.5

Multiply $- 3$ by $5$ .

$\frac{2 x^{2} + 5 x - 6 x - 15 - (x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.2.2

Subtract $6 x$ from $5 x$ .

$\frac{2 x^{2} - x - 15 - (x + 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.3

Apply the distributive property.

$\frac{2 x^{2} - x - 15 + (- x - 1 \cdot 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.4

Multiply $- 1$ by $4$ .

$\frac{2 x^{2} - x - 15 + (- x - 4) (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.5

Expand $(- x - 4) (3 x - 1)$ using the FOIL Method.

Tap for more steps...

Step 2.5.5.1

Apply the distributive property.

$\frac{2 x^{2} - x - 15 - x (3 x - 1) - 4 (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.5.2

Apply the distributive property.

$\frac{2 x^{2} - x - 15 - x (3 x) - x \cdot - 1 - 4 (3 x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.5.3

Apply the distributive property.

$\frac{2 x^{2} - x - 15 - x (3 x) - x \cdot - 1 - 4 (3 x) - 4 \cdot - 1}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.6

Simplify and combine like terms.

Tap for more steps...

Step 2.5.6.1

Simplify each term.

Tap for more steps...

Step 2.5.6.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} - x - 15 - 1 \cdot 3 x \cdot x - x \cdot - 1 - 4 (3 x) - 4 \cdot - 1}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.6.1.2

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

Tap for more steps...

Step 2.5.6.1.2.1

Move $x$ .

$\frac{2 x^{2} - x - 15 - 1 \cdot 3 (x \cdot x) - x \cdot - 1 - 4 (3 x) - 4 \cdot - 1}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.6.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} - x - 15 - 1 \cdot 3 x^{2} - x \cdot - 1 - 4 (3 x) - 4 \cdot - 1}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.6.1.3

Multiply $- 1$ by $3$ .

$\frac{2 x^{2} - x - 15 - 3 x^{2} - x \cdot - 1 - 4 (3 x) - 4 \cdot - 1}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.6.1.4

Multiply $- x \cdot - 1$ .

Tap for more steps...

Step 2.5.6.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{2} - x - 15 - 3 x^{2} + 1 x - 4 (3 x) - 4 \cdot - 1}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.6.1.4.2

Multiply $x$ by $1$ .

$\frac{2 x^{2} - x - 15 - 3 x^{2} + x - 4 (3 x) - 4 \cdot - 1}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.6.1.5

Multiply $3$ by $- 4$ .

$\frac{2 x^{2} - x - 15 - 3 x^{2} + x - 12 x - 4 \cdot - 1}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.6.1.6

Multiply $- 4$ by $- 1$ .

$\frac{2 x^{2} - x - 15 - 3 x^{2} + x - 12 x + 4}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.6.2

Subtract $12 x$ from $x$ .

$\frac{2 x^{2} - x - 15 - 3 x^{2} - 11 x + 4}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.7

Subtract $3 x^{2}$ from $2 x^{2}$ .

$\frac{- x^{2} - x - 15 - 11 x + 4}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.8

Subtract $11 x$ from $- x$ .

$\frac{- x^{2} - 12 x - 15 + 4}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.9

Add $- 15$ and $4$ .

$\frac{- x^{2} - 12 x - 11}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.10

Factor by grouping.

Tap for more steps...

Step 2.5.10.1

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 $a \cdot c = - 1 \cdot - 11 = 11$ and whose sum is $b = - 12$ .

Tap for more steps...

Step 2.5.10.1.1

Factor $- 12$ out of $- 12 x$ .

$\frac{- x^{2} - 12 (x) - 11}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.10.1.2

Rewrite $- 12$ as $- 1$ plus $- 11$

$\frac{- x^{2} + (- 1 - 11) x - 11}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.10.1.3

Apply the distributive property.

$\frac{- x^{2} - 1 x - 11 x - 11}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.10.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.5.10.2.1

Group the first two terms and the last two terms.

$\frac{(- x^{2} - 1 x) - 11 x - 11}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.10.2.2

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

$\frac{x (- x - 1) + 11 (- x - 1)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.5.10.3

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

$\frac{(- x - 1) (x + 11)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.6

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

$\frac{(- (x) - 1) (x + 11)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.7

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

$\frac{(- (x) - 1 (1)) (x + 11)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.8

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

$\frac{- (x + 1) (x + 11)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.9

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

$\frac{- 1 (x + 1) (x + 11)}{(2 x + 5) (3 x - 1)} = 0$

Step 2.10

Move the negative in front of the fraction.

$- \frac{(x + 1) (x + 11)}{(2 x + 5) (3 x - 1)} = 0$

Step 3

Set the denominator in $\frac{(x + 1) (x + 11)}{(2 x + 5) (3 x - 1)}$ equal to $0$ to find where the expression is undefined.

$(2 x + 5) (3 x - 1) = 0$

Step 4

Solve for $x$ .

Tap for more steps...

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 $0$ .

$2 x + 5 = 0$

$3 x - 1 = 0$

Step 4.2

Set $2 x + 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.2.1

Set $2 x + 5$ equal to $0$ .

$2 x + 5 = 0$

Step 4.2.2

Solve $2 x + 5 = 0$ for $x$ .

Tap for more steps...

Step 4.2.2.1

Subtract $5$ from both sides of the equation.

$2 x = - 5$

Step 4.2.2.2

Divide each term in $2 x = - 5$ by $2$ and simplify.

Tap for more steps...

Step 4.2.2.2.1

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Step 4.2.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Step 4.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Step 4.2.2.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Step 4.3

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

Tap for more steps...

Step 4.3.1

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

$3 x - 1 = 0$

Step 4.3.2

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

Tap for more steps...

Step 4.3.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 4.3.2.2

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

Tap for more steps...

Step 4.3.2.2.1

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

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

Step 4.3.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.2.2.2.1.1

Cancel the common factor.

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

Step 4.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.4

The final solution is all the values that make $(2 x + 5) (3 x - 1) = 0$ true.

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

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 $0$ .

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

Step 6