Finite Math Examples

$\frac{x + 7}{\frac{2}{\frac{5 x - 1}{8}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(x + 7) \frac{\frac{5 x - 1}{8}}{2}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$(x + 7) (\frac{5 x - 1}{8} \cdot \frac{1}{2})$

Step 3

Multiply $\frac{5 x - 1}{8} \cdot \frac{1}{2}$ .

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Step 3.1

Multiply $\frac{5 x - 1}{8}$ by $\frac{1}{2}$ .

$(x + 7) \frac{5 x - 1}{8 \cdot 2}$

Step 3.2

Multiply $8$ by $2$ .

$(x + 7) \frac{5 x - 1}{16}$

Step 4

Simplify terms.

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Step 4.1

Apply the distributive property.

$x \frac{5 x - 1}{16} + 7 \frac{5 x - 1}{16}$

Step 4.2

Combine $x$ and $\frac{5 x - 1}{16}$ .

$\frac{x (5 x - 1)}{16} + 7 \frac{5 x - 1}{16}$

Step 4.3

Combine $7$ and $\frac{5 x - 1}{16}$ .

$\frac{x (5 x - 1)}{16} + \frac{7 (5 x - 1)}{16}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{x (5 x - 1) + 7 (5 x - 1)}{16}$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$\frac{x (5 x) + x \cdot - 1 + 7 (5 x - 1)}{16}$

Step 5.2

Rewrite using the commutative property of multiplication.

$\frac{5 x \cdot x + x \cdot - 1 + 7 (5 x - 1)}{16}$

Step 5.3

Move $- 1$ to the left of $x$ .

$\frac{5 x \cdot x - 1 \cdot x + 7 (5 x - 1)}{16}$

Step 5.4

Simplify each term.

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Step 5.4.1

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

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Step 5.4.1.1

Move $x$ .

$\frac{5 (x \cdot x) - 1 \cdot x + 7 (5 x - 1)}{16}$

Step 5.4.1.2

Multiply $x$ by $x$ .

$\frac{5 x^{2} - 1 \cdot x + 7 (5 x - 1)}{16}$

Step 5.4.2

Rewrite $- 1 x$ as $- x$ .

$\frac{5 x^{2} - x + 7 (5 x - 1)}{16}$

Step 5.5

Apply the distributive property.

$\frac{5 x^{2} - x + 7 (5 x) + 7 \cdot - 1}{16}$

Step 5.6

Multiply $5$ by $7$ .

$\frac{5 x^{2} - x + 35 x + 7 \cdot - 1}{16}$

Step 5.7

Multiply $7$ by $- 1$ .

$\frac{5 x^{2} - x + 35 x - 7}{16}$

Step 6

Add $- x$ and $35 x$ .

$\frac{5 x^{2} + 34 x - 7}{16}$

Step 7

Factor by grouping.

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Step 7.1

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 $a \cdot c = 5 \cdot - 7 = - 35$ and whose sum is $b = 34$ .

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Step 7.1.1

Factor $34$ out of $34 x$ .

$\frac{5 x^{2} + 34 (x) - 7}{16}$

Step 7.1.2

Rewrite $34$ as $- 1$ plus $35$

$\frac{5 x^{2} + (- 1 + 35) x - 7}{16}$

Step 7.1.3

Apply the distributive property.

$\frac{5 x^{2} - 1 x + 35 x - 7}{16}$

Step 7.2

Factor out the greatest common factor from each group.

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Step 7.2.1

Group the first two terms and the last two terms.

$\frac{(5 x^{2} - 1 x) + 35 x - 7}{16}$

Step 7.2.2

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

$\frac{x (5 x - 1) + 7 (5 x - 1)}{16}$

Step 7.3

Factor the polynomial by factoring out the greatest common factor, $5 x - 1$ .

$\frac{(5 x - 1) (x + 7)}{16}$