Finite Math Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{4 x}{9} - \frac{1}{4} = 1 - \frac{3}{4} x - \frac{3}{4} (- \frac{1}{2})$

Step 1.1.2

Combine $x$ and $\frac{3}{4}$ .

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

Step 1.1.3

Multiply $- \frac{3}{4} (- \frac{1}{2})$ .

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Step 1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.3.2

Multiply $\frac{3}{4}$ by $1$ .

$\frac{4 x}{9} - \frac{1}{4} = 1 - \frac{x \cdot 3}{4} + \frac{3}{4} \cdot \frac{1}{2}$

Step 1.1.3.3

Multiply $\frac{3}{4}$ by $\frac{1}{2}$ .

$\frac{4 x}{9} - \frac{1}{4} = 1 - \frac{x \cdot 3}{4} + \frac{3}{4 \cdot 2}$

Step 1.1.3.4

Multiply $4$ by $2$ .

$\frac{4 x}{9} - \frac{1}{4} = 1 - \frac{x \cdot 3}{4} + \frac{3}{8}$

Step 1.1.4

Move $3$ to the left of $x$ .

$\frac{4 x}{9} - \frac{1}{4} = 1 - \frac{3 x}{4} + \frac{3}{8}$

Step 1.2

Simplify the expression.

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

$\frac{4 x}{9} - \frac{1}{4} = - \frac{3 x}{4} + \frac{8}{8} + \frac{3}{8}$

Step 1.2.2

Combine the numerators over the common denominator.

$\frac{4 x}{9} - \frac{1}{4} = - \frac{3 x}{4} + \frac{8 + 3}{8}$

Step 1.2.3

Add $8$ and $3$ .

$\frac{4 x}{9} - \frac{1}{4} = - \frac{3 x}{4} + \frac{11}{8}$

Step 2

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

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Step 2.1

Add $\frac{3 x}{4}$ to both sides of the equation.

$\frac{4 x}{9} - \frac{1}{4} + \frac{3 x}{4} = \frac{11}{8}$

Step 2.2

To write $\frac{4 x}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4 x}{9} \cdot \frac{4}{4} + \frac{3 x}{4} - \frac{1}{4} = \frac{11}{8}$

Step 2.3

To write $\frac{3 x}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{4 x}{9} \cdot \frac{4}{4} + \frac{3 x}{4} \cdot \frac{9}{9} - \frac{1}{4} = \frac{11}{8}$

Step 2.4

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{4 x}{9}$ by $\frac{4}{4}$ .

$\frac{4 x \cdot 4}{9 \cdot 4} + \frac{3 x}{4} \cdot \frac{9}{9} - \frac{1}{4} = \frac{11}{8}$

Step 2.4.2

Multiply $9$ by $4$ .

$\frac{4 x \cdot 4}{36} + \frac{3 x}{4} \cdot \frac{9}{9} - \frac{1}{4} = \frac{11}{8}$

Step 2.4.3

Multiply $\frac{3 x}{4}$ by $\frac{9}{9}$ .

$\frac{4 x \cdot 4}{36} + \frac{3 x \cdot 9}{4 \cdot 9} - \frac{1}{4} = \frac{11}{8}$

Step 2.4.4

Multiply $4$ by $9$ .

$\frac{4 x \cdot 4}{36} + \frac{3 x \cdot 9}{36} - \frac{1}{4} = \frac{11}{8}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{4 x \cdot 4 + 3 x \cdot 9}{36} - \frac{1}{4} = \frac{11}{8}$

Step 2.6

Simplify each term.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Factor $x$ out of $4 x \cdot 4 + 3 x \cdot 9$ .

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Step 2.6.1.1.1

Factor $x$ out of $4 x \cdot 4$ .

$\frac{x (4 \cdot 4) + 3 x \cdot 9}{36} - \frac{1}{4} = \frac{11}{8}$

Step 2.6.1.1.2

Factor $x$ out of $3 x \cdot 9$ .

$\frac{x (4 \cdot 4) + x (3 \cdot 9)}{36} - \frac{1}{4} = \frac{11}{8}$

Step 2.6.1.1.3

Factor $x$ out of $x (4 \cdot 4) + x (3 \cdot 9)$ .

$\frac{x (4 \cdot 4 + 3 \cdot 9)}{36} - \frac{1}{4} = \frac{11}{8}$

Step 2.6.1.2

Multiply $4$ by $4$ .

$\frac{x (16 + 3 \cdot 9)}{36} - \frac{1}{4} = \frac{11}{8}$

Step 2.6.1.3

Multiply $3$ by $9$ .

$\frac{x (16 + 27)}{36} - \frac{1}{4} = \frac{11}{8}$

Step 2.6.1.4

Add $16$ and $27$ .

$\frac{x \cdot 43}{36} - \frac{1}{4} = \frac{11}{8}$

Step 2.6.2

Move $43$ to the left of $x$ .

$\frac{43 x}{36} - \frac{1}{4} = \frac{11}{8}$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Add $\frac{1}{4}$ to both sides of the equation.

$\frac{43 x}{36} = \frac{11}{8} + \frac{1}{4}$

Step 3.2

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

$\frac{43 x}{36} = \frac{11}{8} + \frac{1}{4} \cdot \frac{2}{2}$

Step 3.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{1}{4}$ by $\frac{2}{2}$ .

$\frac{43 x}{36} = \frac{11}{8} + \frac{2}{4 \cdot 2}$

Step 3.3.2

Multiply $4$ by $2$ .

$\frac{43 x}{36} = \frac{11}{8} + \frac{2}{8}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{43 x}{36} = \frac{11 + 2}{8}$

Step 3.5

Add $11$ and $2$ .

$\frac{43 x}{36} = \frac{13}{8}$

Step 4

Multiply both sides of the equation by $\frac{36}{43}$ .

$\frac{36}{43} \cdot \frac{43 x}{36} = \frac{36}{43} \cdot \frac{13}{8}$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\frac{36}{43} \cdot \frac{43 x}{36}$ .

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Step 5.1.1.1

Cancel the common factor of $36$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{36}{43} \cdot \frac{43 x}{36} = \frac{36}{43} \cdot \frac{13}{8}$

Step 5.1.1.1.2

Rewrite the expression.

$\frac{1}{43} (43 x) = \frac{36}{43} \cdot \frac{13}{8}$

Step 5.1.1.2

Cancel the common factor of $43$ .

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Step 5.1.1.2.1

Factor $43$ out of $43 x$ .

$\frac{1}{43} (43 (x)) = \frac{36}{43} \cdot \frac{13}{8}$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{1}{43} (43 x) = \frac{36}{43} \cdot \frac{13}{8}$

Step 5.1.1.2.3

Rewrite the expression.

$x = \frac{36}{43} \cdot \frac{13}{8}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{36}{43} \cdot \frac{13}{8}$ .

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Step 5.2.1.1

Cancel the common factor of $4$ .

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Step 5.2.1.1.1

Factor $4$ out of $36$ .

$x = \frac{4 (9)}{43} \cdot \frac{13}{8}$

Step 5.2.1.1.2

Factor $4$ out of $8$ .

$x = \frac{4 \cdot 9}{43} \cdot \frac{13}{4 \cdot 2}$

Step 5.2.1.1.3

Cancel the common factor.

$x = \frac{4 \cdot 9}{43} \cdot \frac{13}{4 \cdot 2}$

Step 5.2.1.1.4

Rewrite the expression.

$x = \frac{9}{43} \cdot \frac{13}{2}$

Step 5.2.1.2

Multiply $\frac{9}{43}$ by $\frac{13}{2}$ .

$x = \frac{9 \cdot 13}{43 \cdot 2}$

Step 5.2.1.3

Multiply.

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Step 5.2.1.3.1

Multiply $9$ by $13$ .

$x = \frac{117}{43 \cdot 2}$

Step 5.2.1.3.2

Multiply $43$ by $2$ .

$x = \frac{117}{86}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{117}{86}$

Decimal Form:

$x = 1.36046511 \dots$

Mixed Number Form:

$x = 1 \frac{31}{86}$