Finite Math Examples

$\frac{4 x + 2}{9} - \frac{3 x - 5}{5} = x$

Step 1

Simplify $\frac{4 x + 2}{9} - \frac{3 x - 5}{5}$ .

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Step 1.1

Factor $2$ out of $4 x + 2$ .

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Step 1.1.1

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) + 2}{9} - \frac{3 x - 5}{5} = x$

Step 1.1.2

Factor $2$ out of $2$ .

$\frac{2 (2 x) + 2 (1)}{9} - \frac{3 x - 5}{5} = x$

Step 1.1.3

Factor $2$ out of $2 (2 x) + 2 (1)$ .

$\frac{2 (2 x + 1)}{9} - \frac{3 x - 5}{5} = x$

Step 1.2

To write $\frac{2 (2 x + 1)}{9}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2 (2 x + 1)}{9} \cdot \frac{5}{5} - \frac{3 x - 5}{5} = x$

Step 1.3

To write $- \frac{3 x - 5}{5}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{2 (2 x + 1)}{9} \cdot \frac{5}{5} - \frac{3 x - 5}{5} \cdot \frac{9}{9} = x$

Step 1.4

Write each expression with a common denominator of $45$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{2 (2 x + 1)}{9}$ by $\frac{5}{5}$ .

$\frac{2 (2 x + 1) \cdot 5}{9 \cdot 5} - \frac{3 x - 5}{5} \cdot \frac{9}{9} = x$

Step 1.4.2

Multiply $9$ by $5$ .

$\frac{2 (2 x + 1) \cdot 5}{45} - \frac{3 x - 5}{5} \cdot \frac{9}{9} = x$

Step 1.4.3

Multiply $\frac{3 x - 5}{5}$ by $\frac{9}{9}$ .

$\frac{2 (2 x + 1) \cdot 5}{45} - \frac{(3 x - 5) \cdot 9}{5 \cdot 9} = x$

Step 1.4.4

Multiply $5$ by $9$ .

$\frac{2 (2 x + 1) \cdot 5}{45} - \frac{(3 x - 5) \cdot 9}{45} = x$

Step 1.5

Combine the numerators over the common denominator.

$\frac{2 (2 x + 1) \cdot 5 - (3 x - 5) \cdot 9}{45} = x$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Apply the distributive property.

$\frac{(2 (2 x) + 2 \cdot 1) \cdot 5 - (3 x - 5) \cdot 9}{45} = x$

Step 1.6.2

Multiply $2$ by $2$ .

$\frac{(4 x + 2 \cdot 1) \cdot 5 - (3 x - 5) \cdot 9}{45} = x$

Step 1.6.3

Multiply $2$ by $1$ .

$\frac{(4 x + 2) \cdot 5 - (3 x - 5) \cdot 9}{45} = x$

Step 1.6.4

Apply the distributive property.

$\frac{4 x \cdot 5 + 2 \cdot 5 - (3 x - 5) \cdot 9}{45} = x$

Step 1.6.5

Multiply $5$ by $4$ .

$\frac{20 x + 2 \cdot 5 - (3 x - 5) \cdot 9}{45} = x$

Step 1.6.6

Multiply $2$ by $5$ .

$\frac{20 x + 10 - (3 x - 5) \cdot 9}{45} = x$

Step 1.6.7

Apply the distributive property.

$\frac{20 x + 10 + (- (3 x) - - 5) \cdot 9}{45} = x$

Step 1.6.8

Multiply $3$ by $- 1$ .

$\frac{20 x + 10 + (- 3 x - - 5) \cdot 9}{45} = x$

Step 1.6.9

Multiply $- 1$ by $- 5$ .

$\frac{20 x + 10 + (- 3 x + 5) \cdot 9}{45} = x$

Step 1.6.10

Apply the distributive property.

$\frac{20 x + 10 - 3 x \cdot 9 + 5 \cdot 9}{45} = x$

Step 1.6.11

Multiply $9$ by $- 3$ .

$\frac{20 x + 10 - 27 x + 5 \cdot 9}{45} = x$

Step 1.6.12

Multiply $5$ by $9$ .

$\frac{20 x + 10 - 27 x + 45}{45} = x$

Step 1.6.13

Subtract $27 x$ from $20 x$ .

$\frac{- 7 x + 10 + 45}{45} = x$

Step 1.6.14

Add $10$ and $45$ .

$\frac{- 7 x + 55}{45} = x$

Step 1.7

Simplify with factoring out.

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Step 1.7.1

Factor $- 1$ out of $- 7 x$ .

$\frac{- (7 x) + 55}{45} = x$

Step 1.7.2

Rewrite $55$ as $- 1 (- 55)$ .

$\frac{- (7 x) - 1 (- 55)}{45} = x$

Step 1.7.3

Factor $- 1$ out of $- (7 x) - 1 (- 55)$ .

$\frac{- (7 x - 55)}{45} = x$

Step 1.7.4

Simplify the expression.

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Step 1.7.4.1

Rewrite $- (7 x - 55)$ as $- 1 (7 x - 55)$ .

$\frac{- 1 (7 x - 55)}{45} = x$

Step 1.7.4.2

Move the negative in front of the fraction.

$- \frac{7 x - 55}{45} = x$

Step 2

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

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Step 2.1

Subtract $x$ from both sides of the equation.

$- \frac{7 x - 55}{45} - x = 0$

Step 2.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{45}{45}$ .

$- \frac{7 x - 55}{45} - x \cdot \frac{45}{45} = 0$

Step 2.3

Combine $- x$ and $\frac{45}{45}$ .

$- \frac{7 x - 55}{45} + \frac{- x \cdot 45}{45} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{- (7 x - 55) - x \cdot 45}{45} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{- (7 x) - - 55 - x \cdot 45}{45} = 0$

Step 2.5.2

Multiply $7$ by $- 1$ .

$\frac{- 7 x - - 55 - x \cdot 45}{45} = 0$

Step 2.5.3

Multiply $- 1$ by $- 55$ .

$\frac{- 7 x + 55 - x \cdot 45}{45} = 0$

Step 2.5.4

Multiply $45$ by $- 1$ .

$\frac{- 7 x + 55 - 45 x}{45} = 0$

Step 2.5.5

Subtract $45 x$ from $- 7 x$ .

$\frac{- 52 x + 55}{45} = 0$

Step 2.6

Factor $- 1$ out of $- 52 x$ .

$\frac{- (52 x) + 55}{45} = 0$

Step 2.7

Rewrite $55$ as $- 1 (- 55)$ .

$\frac{- (52 x) - 1 (- 55)}{45} = 0$

Step 2.8

Factor $- 1$ out of $- (52 x) - 1 (- 55)$ .

$\frac{- (52 x - 55)}{45} = 0$

Step 2.9

Rewrite $- (52 x - 55)$ as $- 1 (52 x - 55)$ .

$\frac{- 1 (52 x - 55)}{45} = 0$

Step 2.10

Move the negative in front of the fraction.

$- \frac{52 x - 55}{45} = 0$

Step 3

Set the numerator equal to zero.

$52 x - 55 = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Add $55$ to both sides of the equation.

$52 x = 55$

Step 4.2

Divide each term in $52 x = 55$ by $52$ and simplify.

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Step 4.2.1

Divide each term in $52 x = 55$ by $52$ .

$\frac{52 x}{52} = \frac{55}{52}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $52$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{52 x}{52} = \frac{55}{52}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{55}{52}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{55}{52}$

Decimal Form:

$x = 1.05769230 \dots$

Mixed Number Form:

$x = 1 \frac{3}{52}$