Finite Math Examples

$20 (\frac{x - 5}{11} - \frac{x - 11}{5}) = 1$

Step 1

Subtract $1$ from both sides of the equation.

$20 (\frac{x - 5}{11} - \frac{x - 11}{5}) - 1 = 0$

Step 2

Simplify $20 (\frac{x - 5}{11} - \frac{x - 11}{5}) - 1$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

To write $\frac{x - 5}{11}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$20 (\frac{x - 5}{11} \cdot \frac{5}{5} - \frac{x - 11}{5}) - 1 = 0$

Step 2.1.2

To write $- \frac{x - 11}{5}$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$20 (\frac{x - 5}{11} \cdot \frac{5}{5} - \frac{x - 11}{5} \cdot \frac{11}{11}) - 1 = 0$

Step 2.1.3

Write each expression with a common denominator of $55$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.3.1

Multiply $\frac{x - 5}{11}$ by $\frac{5}{5}$ .

$20 (\frac{(x - 5) \cdot 5}{11 \cdot 5} - \frac{x - 11}{5} \cdot \frac{11}{11}) - 1 = 0$

Step 2.1.3.2

Multiply $11$ by $5$ .

$20 (\frac{(x - 5) \cdot 5}{55} - \frac{x - 11}{5} \cdot \frac{11}{11}) - 1 = 0$

Step 2.1.3.3

Multiply $\frac{x - 11}{5}$ by $\frac{11}{11}$ .

$20 (\frac{(x - 5) \cdot 5}{55} - \frac{(x - 11) \cdot 11}{5 \cdot 11}) - 1 = 0$

Step 2.1.3.4

Multiply $5$ by $11$ .

$20 (\frac{(x - 5) \cdot 5}{55} - \frac{(x - 11) \cdot 11}{55}) - 1 = 0$

Step 2.1.4

Combine the numerators over the common denominator.

$20 \frac{(x - 5) \cdot 5 - (x - 11) \cdot 11}{55} - 1 = 0$

Step 2.1.5

Simplify the numerator.

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Step 2.1.5.1

Apply the distributive property.

$20 \frac{x \cdot 5 - 5 \cdot 5 - (x - 11) \cdot 11}{55} - 1 = 0$

Step 2.1.5.2

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

$20 \frac{5 \cdot x - 5 \cdot 5 - (x - 11) \cdot 11}{55} - 1 = 0$

Step 2.1.5.3

Multiply $- 5$ by $5$ .

$20 \frac{5 \cdot x - 25 - (x - 11) \cdot 11}{55} - 1 = 0$

Step 2.1.5.4

Apply the distributive property.

$20 \frac{5 x - 25 + (- x - - 11) \cdot 11}{55} - 1 = 0$

Step 2.1.5.5

Multiply $- 1$ by $- 11$ .

$20 \frac{5 x - 25 + (- x + 11) \cdot 11}{55} - 1 = 0$

Step 2.1.5.6

Apply the distributive property.

$20 \frac{5 x - 25 - x \cdot 11 + 11 \cdot 11}{55} - 1 = 0$

Step 2.1.5.7

Multiply $11$ by $- 1$ .

$20 \frac{5 x - 25 - 11 x + 11 \cdot 11}{55} - 1 = 0$

Step 2.1.5.8

Multiply $11$ by $11$ .

$20 \frac{5 x - 25 - 11 x + 121}{55} - 1 = 0$

Step 2.1.5.9

Subtract $11 x$ from $5 x$ .

$20 \frac{- 6 x - 25 + 121}{55} - 1 = 0$

Step 2.1.5.10

Add $- 25$ and $121$ .

$20 \frac{- 6 x + 96}{55} - 1 = 0$

Step 2.1.5.11

Factor $6$ out of $- 6 x + 96$ .

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Step 2.1.5.11.1

Factor $6$ out of $- 6 x$ .

$20 \frac{6 (- x) + 96}{55} - 1 = 0$

Step 2.1.5.11.2

Factor $6$ out of $96$ .

$20 \frac{6 (- x) + 6 (16)}{55} - 1 = 0$

Step 2.1.5.11.3

Factor $6$ out of $6 (- x) + 6 (16)$ .

$20 \frac{6 (- x + 16)}{55} - 1 = 0$

Step 2.1.6

Cancel the common factor of $5$ .

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Step 2.1.6.1

Factor $5$ out of $20$ .

$5 (4) \frac{6 (- x + 16)}{55} - 1 = 0$

Step 2.1.6.2

Factor $5$ out of $55$ .

$5 \cdot 4 \frac{6 (- x + 16)}{5 \cdot 11} - 1 = 0$

Step 2.1.6.3

Cancel the common factor.

$5 \cdot 4 \frac{6 (- x + 16)}{5 \cdot 11} - 1 = 0$

Step 2.1.6.4

Rewrite the expression.

$4 \frac{6 (- x + 16)}{11} - 1 = 0$

Step 2.1.7

Combine $4$ and $\frac{6 (- x + 16)}{11}$ .

$\frac{4 (6 (- x + 16))}{11} - 1 = 0$

Step 2.1.8

Multiply $6$ by $4$ .

$\frac{24 (- x + 16)}{11} - 1 = 0$

Step 2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$\frac{24 (- x + 16)}{11} - 1 \cdot \frac{11}{11} = 0$

Step 2.3

Combine $- 1$ and $\frac{11}{11}$ .

$\frac{24 (- x + 16)}{11} + \frac{- 1 \cdot 11}{11} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{24 (- x + 16) - 1 \cdot 11}{11} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{24 (- x) + 24 \cdot 16 - 1 \cdot 11}{11} = 0$

Step 2.5.2

Multiply $- 1$ by $24$ .

$\frac{- 24 x + 24 \cdot 16 - 1 \cdot 11}{11} = 0$

Step 2.5.3

Multiply $24$ by $16$ .

$\frac{- 24 x + 384 - 1 \cdot 11}{11} = 0$

Step 2.5.4

Multiply $- 1$ by $11$ .

$\frac{- 24 x + 384 - 11}{11} = 0$

Step 2.5.5

Subtract $11$ from $384$ .

$\frac{- 24 x + 373}{11} = 0$

Step 2.6

Factor $- 1$ out of $- 24 x$ .

$\frac{- (24 x) + 373}{11} = 0$

Step 2.7

Rewrite $373$ as $- 1 (- 373)$ .

$\frac{- (24 x) - 1 (- 373)}{11} = 0$

Step 2.8

Factor $- 1$ out of $- (24 x) - 1 (- 373)$ .

$\frac{- (24 x - 373)}{11} = 0$

Step 2.9

Rewrite $- (24 x - 373)$ as $- 1 (24 x - 373)$ .

$\frac{- 1 (24 x - 373)}{11} = 0$

Step 2.10

Move the negative in front of the fraction.

$- \frac{24 x - 373}{11} = 0$

Step 3

Set the numerator equal to zero.

$24 x - 373 = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Add $373$ to both sides of the equation.

$24 x = 373$

Step 4.2

Divide each term in $24 x = 373$ by $24$ and simplify.

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Step 4.2.1

Divide each term in $24 x = 373$ by $24$ .

$\frac{24 x}{24} = \frac{373}{24}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $24$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{24 x}{24} = \frac{373}{24}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{373}{24}$