Finite Math Examples

$\frac{3 x - 17}{x - 4} = \frac{7 x - 7}{28}$

Step 1

Subtract $\frac{7 x - 7}{28}$ from both sides of the equation.

$\frac{3 x - 17}{x - 4} - \frac{7 x - 7}{28} = 0$

Step 2

Simplify $\frac{3 x - 17}{x - 4} - \frac{7 x - 7}{28}$ .

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Step 2.1

Cancel the common factor of $7 x - 7$ and $28$ .

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Step 2.1.1

Factor $7$ out of $7 x$ .

$\frac{3 x - 17}{x - 4} - \frac{7 (x) - 7}{28} = 0$

Step 2.1.2

Factor $7$ out of $- 7$ .

$\frac{3 x - 17}{x - 4} - \frac{7 (x) + 7 (- 1)}{28} = 0$

Step 2.1.3

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

$\frac{3 x - 17}{x - 4} - \frac{7 (x - 1)}{28} = 0$

Step 2.1.4

Cancel the common factors.

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Step 2.1.4.1

Factor $7$ out of $28$ .

$\frac{3 x - 17}{x - 4} - \frac{7 (x - 1)}{7 (4)} = 0$

Step 2.1.4.2

Cancel the common factor.

$\frac{3 x - 17}{x - 4} - \frac{7 (x - 1)}{7 \cdot 4} = 0$

Step 2.1.4.3

Rewrite the expression.

$\frac{3 x - 17}{x - 4} - \frac{x - 1}{4} = 0$

Step 2.2

To write $\frac{3 x - 17}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 x - 17}{x - 4} \cdot \frac{4}{4} - \frac{x - 1}{4} = 0$

Step 2.3

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

$\frac{3 x - 17}{x - 4} \cdot \frac{4}{4} - \frac{x - 1}{4} \cdot \frac{x - 4}{x - 4} = 0$

Step 2.4

Write each expression with a common denominator of $(x - 4) \cdot 4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{3 x - 17}{x - 4}$ by $\frac{4}{4}$ .

$\frac{(3 x - 17) \cdot 4}{(x - 4) \cdot 4} - \frac{x - 1}{4} \cdot \frac{x - 4}{x - 4} = 0$

Step 2.4.2

Multiply $\frac{x - 1}{4}$ by $\frac{x - 4}{x - 4}$ .

$\frac{(3 x - 17) \cdot 4}{(x - 4) \cdot 4} - \frac{(x - 1) (x - 4)}{4 (x - 4)} = 0$

Step 2.4.3

Reorder the factors of $(x - 4) \cdot 4$ .

$\frac{(3 x - 17) \cdot 4}{4 (x - 4)} - \frac{(x - 1) (x - 4)}{4 (x - 4)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{(3 x - 17) \cdot 4 - (x - 1) (x - 4)}{4 (x - 4)} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

$\frac{3 x \cdot 4 - 17 \cdot 4 - (x - 1) (x - 4)}{4 (x - 4)} = 0$

Step 2.6.2

Multiply $4$ by $3$ .

$\frac{12 x - 17 \cdot 4 - (x - 1) (x - 4)}{4 (x - 4)} = 0$

Step 2.6.3

Multiply $- 17$ by $4$ .

$\frac{12 x - 68 - (x - 1) (x - 4)}{4 (x - 4)} = 0$

Step 2.6.4

Apply the distributive property.

$\frac{12 x - 68 + (- x - - 1) (x - 4)}{4 (x - 4)} = 0$

Step 2.6.5

Multiply $- 1$ by $- 1$ .

$\frac{12 x - 68 + (- x + 1) (x - 4)}{4 (x - 4)} = 0$

Step 2.6.6

Expand $(- x + 1) (x - 4)$ using the FOIL Method.

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Step 2.6.6.1

Apply the distributive property.

$\frac{12 x - 68 - x (x - 4) + 1 (x - 4)}{4 (x - 4)} = 0$

Step 2.6.6.2

Apply the distributive property.

$\frac{12 x - 68 - x \cdot x - x \cdot - 4 + 1 (x - 4)}{4 (x - 4)} = 0$

Step 2.6.6.3

Apply the distributive property.

$\frac{12 x - 68 - x \cdot x - x \cdot - 4 + 1 x + 1 \cdot - 4}{4 (x - 4)} = 0$

Step 2.6.7

Simplify and combine like terms.

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Step 2.6.7.1

Simplify each term.

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Step 2.6.7.1.1

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

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Step 2.6.7.1.1.1

Move $x$ .

$\frac{12 x - 68 - (x \cdot x) - x \cdot - 4 + 1 x + 1 \cdot - 4}{4 (x - 4)} = 0$

Step 2.6.7.1.1.2

Multiply $x$ by $x$ .

$\frac{12 x - 68 - x^{2} - x \cdot - 4 + 1 x + 1 \cdot - 4}{4 (x - 4)} = 0$

Step 2.6.7.1.2

Multiply $- 4$ by $- 1$ .

$\frac{12 x - 68 - x^{2} + 4 x + 1 x + 1 \cdot - 4}{4 (x - 4)} = 0$

Step 2.6.7.1.3

Multiply $x$ by $1$ .

$\frac{12 x - 68 - x^{2} + 4 x + x + 1 \cdot - 4}{4 (x - 4)} = 0$

Step 2.6.7.1.4

Multiply $- 4$ by $1$ .

$\frac{12 x - 68 - x^{2} + 4 x + x - 4}{4 (x - 4)} = 0$

Step 2.6.7.2

Add $4 x$ and $x$ .

$\frac{12 x - 68 - x^{2} + 5 x - 4}{4 (x - 4)} = 0$

Step 2.6.8

Add $12 x$ and $5 x$ .

$\frac{17 x - 68 - x^{2} - 4}{4 (x - 4)} = 0$

Step 2.6.9

Subtract $4$ from $- 68$ .

$\frac{17 x - x^{2} - 72}{4 (x - 4)} = 0$

Step 2.6.10

Reorder terms.

$\frac{- x^{2} + 17 x - 72}{4 (x - 4)} = 0$

Step 2.6.11

Factor by grouping.

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Step 2.6.11.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 72 = 72$ and whose sum is $b = 17$ .

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Step 2.6.11.1.1

Factor $17$ out of $17 x$ .

$\frac{- x^{2} + 17 (x) - 72}{4 (x - 4)} = 0$

Step 2.6.11.1.2

Rewrite $17$ as $8$ plus $9$

$\frac{- x^{2} + (8 + 9) x - 72}{4 (x - 4)} = 0$

Step 2.6.11.1.3

Apply the distributive property.

$\frac{- x^{2} + 8 x + 9 x - 72}{4 (x - 4)} = 0$

Step 2.6.11.2

Factor out the greatest common factor from each group.

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Step 2.6.11.2.1

Group the first two terms and the last two terms.

$\frac{(- x^{2} + 8 x) + 9 x - 72}{4 (x - 4)} = 0$

Step 2.6.11.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- x + 8) - 9 (- x + 8)}{4 (x - 4)} = 0$

Step 2.6.11.3

Factor the polynomial by factoring out the greatest common factor, $- x + 8$ .

$\frac{(- x + 8) (x - 9)}{4 (x - 4)} = 0$

Step 2.7

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

$\frac{(- (x) + 8) (x - 9)}{4 (x - 4)} = 0$

Step 2.8

Rewrite $8$ as $- 1 (- 8)$ .

$\frac{(- (x) - 1 (- 8)) (x - 9)}{4 (x - 4)} = 0$

Step 2.9

Factor $- 1$ out of $- (x) - 1 (- 8)$ .

$\frac{- (x - 8) (x - 9)}{4 (x - 4)} = 0$

Step 2.10

Rewrite $- (x - 8)$ as $- 1 (x - 8)$ .

$\frac{- 1 (x - 8) (x - 9)}{4 (x - 4)} = 0$

Step 2.11

Move the negative in front of the fraction.

$- \frac{(x - 8) (x - 9)}{4 (x - 4)} = 0$

Step 3

Set the numerator equal to zero.

$(x - 8) (x - 9) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 8 = 0$

$x - 9 = 0$

Step 4.2

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 4.2.2

Add $8$ to both sides of the equation.

$x = 8$

Step 4.3

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 4.3.2

Add $9$ to both sides of the equation.

$x = 9$

Step 4.4

The final solution is all the values that make $(x - 8) (x - 9) = 0$ true.

$x = 8, 9$