Finite Math Examples

$\frac{1}{4} \cdot (3 x - 2) - 8 = \frac{3}{8} \cdot (x - 5) + 2$

Step 1

Simplify $\frac{1}{4} \cdot (3 x - 2) - 8$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{1}{4} (3 x) + \frac{1}{4} \cdot - 2 - 8 = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.1.2

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

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Step 1.1.2.1

Combine $3$ and $\frac{1}{4}$ .

$\frac{3}{4} x + \frac{1}{4} \cdot - 2 - 8 = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.1.2.2

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

$\frac{3 x}{4} + \frac{1}{4} \cdot - 2 - 8 = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.1

Factor $2$ out of $4$ .

$\frac{3 x}{4} + \frac{1}{2 (2)} \cdot - 2 - 8 = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.1.3.2

Factor $2$ out of $- 2$ .

$\frac{3 x}{4} + \frac{1}{2 \cdot 2} \cdot (2 \cdot - 1) - 8 = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.1.3.3

Cancel the common factor.

$\frac{3 x}{4} + \frac{1}{2 \cdot 2} \cdot (2 \cdot - 1) - 8 = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.1.3.4

Rewrite the expression.

$\frac{3 x}{4} + \frac{1}{2} \cdot - 1 - 8 = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.1.4

Combine $\frac{1}{2}$ and $- 1$ .

$\frac{3 x}{4} + \frac{- 1}{2} - 8 = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.1.5

Move the negative in front of the fraction.

$\frac{3 x}{4} - \frac{1}{2} - 8 = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x}{4} - \frac{1}{2} - 8 \cdot \frac{2}{2} = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.3

Combine $- 8$ and $\frac{2}{2}$ .

$\frac{3 x}{4} - \frac{1}{2} + \frac{- 8 \cdot 2}{2} = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.4

Combine the numerators over the common denominator.

$\frac{3 x}{4} + \frac{- 1 - 8 \cdot 2}{2} = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 8$ by $2$ .

$\frac{3 x}{4} + \frac{- 1 - 16}{2} = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.5.2

Subtract $16$ from $- 1$ .

$\frac{3 x}{4} + \frac{- 17}{2} = \frac{3}{8} \cdot (x - 5) + 2$

Step 1.6

Move the negative in front of the fraction.

$\frac{3 x}{4} - \frac{17}{2} = \frac{3}{8} \cdot (x - 5) + 2$

Step 2

Simplify $\frac{3}{8} \cdot (x - 5) + 2$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$\frac{3 x}{4} - \frac{17}{2} = \frac{3}{8} x + \frac{3}{8} \cdot - 5 + 2$

Step 2.1.2

Combine $\frac{3}{8}$ and $x$ .

$\frac{3 x}{4} - \frac{17}{2} = \frac{3 x}{8} + \frac{3}{8} \cdot - 5 + 2$

Step 2.1.3

Multiply $\frac{3}{8} \cdot - 5$ .

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Step 2.1.3.1

Combine $\frac{3}{8}$ and $- 5$ .

$\frac{3 x}{4} - \frac{17}{2} = \frac{3 x}{8} + \frac{3 \cdot - 5}{8} + 2$

Step 2.1.3.2

Multiply $3$ by $- 5$ .

$\frac{3 x}{4} - \frac{17}{2} = \frac{3 x}{8} + \frac{- 15}{8} + 2$

Step 2.1.4

Move the negative in front of the fraction.

$\frac{3 x}{4} - \frac{17}{2} = \frac{3 x}{8} - \frac{15}{8} + 2$

Step 2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{3 x}{4} - \frac{17}{2} = \frac{3 x}{8} - \frac{15}{8} + 2 \cdot \frac{8}{8}$

Step 2.3

Combine $2$ and $\frac{8}{8}$ .

$\frac{3 x}{4} - \frac{17}{2} = \frac{3 x}{8} - \frac{15}{8} + \frac{2 \cdot 8}{8}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 x}{4} - \frac{17}{2} = \frac{3 x}{8} + \frac{- 15 + 2 \cdot 8}{8}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $2$ by $8$ .

$\frac{3 x}{4} - \frac{17}{2} = \frac{3 x}{8} + \frac{- 15 + 16}{8}$

Step 2.5.2

Add $- 15$ and $16$ .

$\frac{3 x}{4} - \frac{17}{2} = \frac{3 x}{8} + \frac{1}{8}$

Step 3

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

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Step 3.1

Subtract $\frac{3 x}{8}$ from both sides of the equation.

$\frac{3 x}{4} - \frac{17}{2} - \frac{3 x}{8} = \frac{1}{8}$

Step 3.2

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

$\frac{3 x}{4} \cdot \frac{2}{2} - \frac{3 x}{8} - \frac{17}{2} = \frac{1}{8}$

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 $\frac{3 x}{4}$ by $\frac{2}{2}$ .

$\frac{3 x \cdot 2}{4 \cdot 2} - \frac{3 x}{8} - \frac{17}{2} = \frac{1}{8}$

Step 3.3.2

Multiply $4$ by $2$ .

$\frac{3 x \cdot 2}{8} - \frac{3 x}{8} - \frac{17}{2} = \frac{1}{8}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{3 x \cdot 2 - 3 x}{8} - \frac{17}{2} = \frac{1}{8}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Factor $3 x$ out of $3 x \cdot 2 - 3 x$ .

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Step 3.5.1.1

Factor $3 x$ out of $3 x \cdot 2$ .

$\frac{3 x (2) - 3 x}{8} - \frac{17}{2} = \frac{1}{8}$

Step 3.5.1.2

Factor $3 x$ out of $- 3 x$ .

$\frac{3 x (2) + 3 x (- 1)}{8} - \frac{17}{2} = \frac{1}{8}$

Step 3.5.1.3

Factor $3 x$ out of $3 x (2) + 3 x (- 1)$ .

$\frac{3 x (2 - 1)}{8} - \frac{17}{2} = \frac{1}{8}$

Step 3.5.2

Subtract $1$ from $2$ .

$\frac{3 x \cdot 1}{8} - \frac{17}{2} = \frac{1}{8}$

Step 3.5.3

Multiply $3$ by $1$ .

$\frac{3 x}{8} - \frac{17}{2} = \frac{1}{8}$

Step 4

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

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Step 4.1

Add $\frac{17}{2}$ to both sides of the equation.

$\frac{3 x}{8} = \frac{1}{8} + \frac{17}{2}$

Step 4.2

To write $\frac{17}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 x}{8} = \frac{1}{8} + \frac{17}{2} \cdot \frac{4}{4}$

Step 4.3

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

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Step 4.3.1

Multiply $\frac{17}{2}$ by $\frac{4}{4}$ .

$\frac{3 x}{8} = \frac{1}{8} + \frac{17 \cdot 4}{2 \cdot 4}$

Step 4.3.2

Multiply $2$ by $4$ .

$\frac{3 x}{8} = \frac{1}{8} + \frac{17 \cdot 4}{8}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{3 x}{8} = \frac{1 + 17 \cdot 4}{8}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $17$ by $4$ .

$\frac{3 x}{8} = \frac{1 + 68}{8}$

Step 4.5.2

Add $1$ and $68$ .

$\frac{3 x}{8} = \frac{69}{8}$

Step 5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$3 x = 69$

Step 6

Divide each term in $3 x = 69$ by $3$ and simplify.

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Step 6.1

Divide each term in $3 x = 69$ by $3$ .

$\frac{3 x}{3} = \frac{69}{3}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $3$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{69}{3}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{69}{3}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $69$ by $3$ .

$x = 23$