Finite Math Examples

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

Step 1

Simplify both sides.

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Step 1.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 1.1.1

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

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

Step 1.1.2

Combine.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Simplify by cancelling.

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Step 1.3.1

Cancel the common factor of $2$ .

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Step 1.3.1.1

Cancel the common factor.

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

Step 1.3.1.2

Rewrite the expression.

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

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.4

Factor $2$ out of $2 x + 2 \cdot - 4$ .

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Step 1.4.1

Factor $2$ out of $2 x$ .

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

Step 1.4.2

Factor $2$ out of $2 \cdot - 4$ .

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

Step 1.4.3

Factor $2$ out of $2 (x) + 2 (- 4)$ .

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

Step 1.5

Multiply the numerator and denominator of the fraction by $2$ .

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Step 1.5.1

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

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

Step 1.5.2

Combine.

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

Step 1.6

Apply the distributive property.

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

Step 1.7

Cancel the common factor of $2$ .

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Step 1.7.1

Move the leading negative in $- \frac{x}{2}$ into the numerator.

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

Step 1.7.2

Cancel the common factor.

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

Step 1.7.3

Rewrite the expression.

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

Step 1.8

Factor $2$ out of $2 \cdot 5 + 2 (- x)$ .

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Step 1.8.1

Factor $2$ out of $2 \cdot 5$ .

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

Step 1.8.2

Factor $2$ out of $2 (5) + 2 (- x)$ .

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

Step 1.9

Multiply $2$ by $2$ .

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

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$2 (x - 4) (4 - x) = (x - 3) (2 (5 - x))$

Step 3

Solve the equation for $x$ .

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Step 3.1

Simplify $2 (x - 4) (4 - x)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 2 (x - 4) (4 - x) = (x - 3) \cdot (2 (5 - x))$

Step 3.1.2

Simplify by multiplying through.

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Step 3.1.2.1

Apply the distributive property.

$(2 x + 2 \cdot - 4) (4 - x) = (x - 3) \cdot (2 (5 - x))$

Step 3.1.2.2

Multiply $2$ by $- 4$ .

$(2 x - 8) (4 - x) = (x - 3) \cdot (2 (5 - x))$

Step 3.1.3

Expand $(2 x - 8) (4 - x)$ using the FOIL Method.

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Step 3.1.3.1

Apply the distributive property.

$2 x (4 - x) - 8 (4 - x) = (x - 3) \cdot (2 (5 - x))$

Step 3.1.3.2

Apply the distributive property.

$2 x \cdot 4 + 2 x (- x) - 8 (4 - x) = (x - 3) \cdot (2 (5 - x))$

Step 3.1.3.3

Apply the distributive property.

$2 x \cdot 4 + 2 x (- x) - 8 \cdot 4 - 8 (- x) = (x - 3) \cdot (2 (5 - x))$

Step 3.1.4

Simplify and combine like terms.

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Step 3.1.4.1

Simplify each term.

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Step 3.1.4.1.1

Multiply $4$ by $2$ .

$8 x + 2 x (- x) - 8 \cdot 4 - 8 (- x) = (x - 3) \cdot (2 (5 - x))$

Step 3.1.4.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.4.1.3

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

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Step 3.1.4.1.3.1

Move $x$ .

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

Step 3.1.4.1.3.2

Multiply $x$ by $x$ .

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

Step 3.1.4.1.4

Multiply $2$ by $- 1$ .

$8 x - 2 x^{2} - 8 \cdot 4 - 8 (- x) = (x - 3) \cdot (2 (5 - x))$

Step 3.1.4.1.5

Multiply $- 8$ by $4$ .

$8 x - 2 x^{2} - 32 - 8 (- x) = (x - 3) \cdot (2 (5 - x))$

Step 3.1.4.1.6

Multiply $- 1$ by $- 8$ .

$8 x - 2 x^{2} - 32 + 8 x = (x - 3) \cdot (2 (5 - x))$

Step 3.1.4.2

Add $8 x$ and $8 x$ .

$16 x - 2 x^{2} - 32 = (x - 3) \cdot (2 (5 - x))$

Step 3.2

Simplify $(x - 3) \cdot (2 (5 - x))$ .

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Step 3.2.1

Simplify by multiplying through.

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Step 3.2.1.1

Apply the distributive property.

$16 x - 2 x^{2} - 32 = (x - 3) \cdot (2 \cdot 5 + 2 (- x))$

Step 3.2.1.2

Multiply.

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Step 3.2.1.2.1

Multiply $2$ by $5$ .

$16 x - 2 x^{2} - 32 = (x - 3) \cdot (10 + 2 (- x))$

Step 3.2.1.2.2

Multiply $- 1$ by $2$ .

$16 x - 2 x^{2} - 32 = (x - 3) \cdot (10 - 2 x)$

Step 3.2.2

Expand $(x - 3) (10 - 2 x)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

$16 x - 2 x^{2} - 32 = x (10 - 2 x) - 3 (10 - 2 x)$

Step 3.2.2.2

Apply the distributive property.

$16 x - 2 x^{2} - 32 = x \cdot 10 + x (- 2 x) - 3 (10 - 2 x)$

Step 3.2.2.3

Apply the distributive property.

$16 x - 2 x^{2} - 32 = x \cdot 10 + x (- 2 x) - 3 \cdot 10 - 3 (- 2 x)$

Step 3.2.3

Simplify and combine like terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Move $10$ to the left of $x$ .

$16 x - 2 x^{2} - 32 = 10 \cdot x + x (- 2 x) - 3 \cdot 10 - 3 (- 2 x)$

Step 3.2.3.1.2

Rewrite using the commutative property of multiplication.

$16 x - 2 x^{2} - 32 = 10 x - 2 x \cdot x - 3 \cdot 10 - 3 (- 2 x)$

Step 3.2.3.1.3

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

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Step 3.2.3.1.3.1

Move $x$ .

$16 x - 2 x^{2} - 32 = 10 x - 2 (x \cdot x) - 3 \cdot 10 - 3 (- 2 x)$

Step 3.2.3.1.3.2

Multiply $x$ by $x$ .

$16 x - 2 x^{2} - 32 = 10 x - 2 x^{2} - 3 \cdot 10 - 3 (- 2 x)$

Step 3.2.3.1.4

Multiply $- 3$ by $10$ .

$16 x - 2 x^{2} - 32 = 10 x - 2 x^{2} - 30 - 3 (- 2 x)$

Step 3.2.3.1.5

Multiply $- 2$ by $- 3$ .

$16 x - 2 x^{2} - 32 = 10 x - 2 x^{2} - 30 + 6 x$

Step 3.2.3.2

Add $10 x$ and $6 x$ .

$16 x - 2 x^{2} - 32 = 16 x - 2 x^{2} - 30$

Step 3.3

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

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Step 3.3.1

Subtract $16 x$ from both sides of the equation.

$16 x - 2 x^{2} - 32 - 16 x = - 2 x^{2} - 30$

Step 3.3.2

Add $2 x^{2}$ to both sides of the equation.

$16 x - 2 x^{2} - 32 - 16 x + 2 x^{2} = - 30$

Step 3.3.3

Combine the opposite terms in $16 x - 2 x^{2} - 32 - 16 x + 2 x^{2}$ .

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Step 3.3.3.1

Subtract $16 x$ from $16 x$ .

$- 2 x^{2} - 32 + 0 + 2 x^{2} = - 30$

Step 3.3.3.2

Add $- 2 x^{2} - 32$ and $0$ .

$- 2 x^{2} - 32 + 2 x^{2} = - 30$

Step 3.3.3.3

Add $- 2 x^{2}$ and $2 x^{2}$ .

$0 - 32 = - 30$

Step 3.3.3.4

Subtract $32$ from $0$ .

$- 32 = - 30$

Step 3.4

Since $- 32 \neq - 30$ , there are no solutions.

No solution