Finite Math Examples

$\frac{1}{5} b + \frac{3}{4} b - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1

Simplify $\frac{1}{5} b + \frac{3}{4} b - \frac{1}{8}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{5}$ and $b$ .

$\frac{b}{5} + \frac{3}{4} b - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.1.2

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

$\frac{b}{5} + \frac{3 b}{4} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.2

To write $\frac{b}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{b}{5} \cdot \frac{4}{4} + \frac{3 b}{4} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.3

To write $\frac{3 b}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{b}{5} \cdot \frac{4}{4} + \frac{3 b}{4} \cdot \frac{5}{5} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{b}{5}$ by $\frac{4}{4}$ .

$\frac{b \cdot 4}{5 \cdot 4} + \frac{3 b}{4} \cdot \frac{5}{5} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.4.2

Multiply $5$ by $4$ .

$\frac{b \cdot 4}{20} + \frac{3 b}{4} \cdot \frac{5}{5} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.4.3

Multiply $\frac{3 b}{4}$ by $\frac{5}{5}$ .

$\frac{b \cdot 4}{20} + \frac{3 b \cdot 5}{4 \cdot 5} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.4.4

Multiply $4$ by $5$ .

$\frac{b \cdot 4}{20} + \frac{3 b \cdot 5}{20} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.5

Combine the numerators over the common denominator.

$\frac{b \cdot 4 + 3 b \cdot 5}{20} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.6

Simplify each term.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Factor $b$ out of $b \cdot 4 + 3 b \cdot 5$ .

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Step 1.6.1.1.1

Factor $b$ out of $b \cdot 4$ .

$\frac{b (4) + 3 b \cdot 5}{20} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.6.1.1.2

Factor $b$ out of $3 b \cdot 5$ .

$\frac{b (4) + b (3 \cdot 5)}{20} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.6.1.1.3

Factor $b$ out of $b (4) + b (3 \cdot 5)$ .

$\frac{b (4 + 3 \cdot 5)}{20} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.6.1.2

Multiply $3$ by $5$ .

$\frac{b (4 + 15)}{20} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.6.1.3

Add $4$ and $15$ .

$\frac{b \cdot 19}{20} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 1.6.2

Move $19$ to the left of $b$ .

$\frac{19 b}{20} - \frac{1}{8} > \frac{1}{8} \cdot (b + 1) + b$

Step 2

Simplify $\frac{1}{8} \cdot (b + 1) + b$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$\frac{19 b}{20} - \frac{1}{8} > \frac{1}{8} b + \frac{1}{8} \cdot 1 + b$

Step 2.1.2

Combine $\frac{1}{8}$ and $b$ .

$\frac{19 b}{20} - \frac{1}{8} > \frac{b}{8} + \frac{1}{8} \cdot 1 + b$

Step 2.1.3

Multiply $\frac{1}{8}$ by $1$ .

$\frac{19 b}{20} - \frac{1}{8} > \frac{b}{8} + \frac{1}{8} + b$

Step 2.2

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

$\frac{19 b}{20} - \frac{1}{8} > \frac{b}{8} + b \cdot \frac{8}{8} + \frac{1}{8}$

Step 2.3

Simplify terms.

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Step 2.3.1

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

$\frac{19 b}{20} - \frac{1}{8} > \frac{b}{8} + \frac{b \cdot 8}{8} + \frac{1}{8}$

Step 2.3.2

Combine the numerators over the common denominator.

$\frac{19 b}{20} - \frac{1}{8} > \frac{b + b \cdot 8}{8} + \frac{1}{8}$

Step 2.3.3

Combine the numerators over the common denominator.

$\frac{19 b}{20} - \frac{1}{8} > \frac{b + b \cdot 8 + 1}{8}$

Step 2.4

Move $8$ to the left of $b$ .

$\frac{19 b}{20} - \frac{1}{8} > \frac{b + 8 b + 1}{8}$

Step 2.5

Add $b$ and $8 b$ .

$\frac{19 b}{20} - \frac{1}{8} > \frac{9 b + 1}{8}$

Step 3

Move all terms containing $b$ to the left side of the inequality.

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Step 3.1

Subtract $\frac{9 b + 1}{8}$ from both sides of the inequality.

$\frac{19 b}{20} - \frac{1}{8} - \frac{9 b + 1}{8} > 0$

Step 3.2

Combine the numerators over the common denominator.

$\frac{19 b}{20} + \frac{- 1 - (9 b + 1)}{8} > 0$

Step 3.3

Simplify each term.

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Step 3.3.1

Apply the distributive property.

$\frac{19 b}{20} + \frac{- 1 - (9 b) - 1 \cdot 1}{8} > 0$

Step 3.3.2

Multiply $9$ by $- 1$ .

$\frac{19 b}{20} + \frac{- 1 - 9 b - 1 \cdot 1}{8} > 0$

Step 3.3.3

Multiply $- 1$ by $1$ .

$\frac{19 b}{20} + \frac{- 1 - 9 b - 1}{8} > 0$

Step 3.4

Subtract $1$ from $- 1$ .

$\frac{19 b}{20} + \frac{- 9 b - 2}{8} > 0$

Step 3.5

To write $\frac{19 b}{20}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{19 b}{20} \cdot \frac{2}{2} + \frac{- 9 b - 2}{8} > 0$

Step 3.6

To write $\frac{- 9 b - 2}{8}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{19 b}{20} \cdot \frac{2}{2} + \frac{- 9 b - 2}{8} \cdot \frac{5}{5} > 0$

Step 3.7

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.7.1

Multiply $\frac{19 b}{20}$ by $\frac{2}{2}$ .

$\frac{19 b \cdot 2}{20 \cdot 2} + \frac{- 9 b - 2}{8} \cdot \frac{5}{5} > 0$

Step 3.7.2

Multiply $20$ by $2$ .

$\frac{19 b \cdot 2}{40} + \frac{- 9 b - 2}{8} \cdot \frac{5}{5} > 0$

Step 3.7.3

Multiply $\frac{- 9 b - 2}{8}$ by $\frac{5}{5}$ .

$\frac{19 b \cdot 2}{40} + \frac{(- 9 b - 2) \cdot 5}{8 \cdot 5} > 0$

Step 3.7.4

Multiply $8$ by $5$ .

$\frac{19 b \cdot 2}{40} + \frac{(- 9 b - 2) \cdot 5}{40} > 0$

Step 3.8

Combine the numerators over the common denominator.

$\frac{19 b \cdot 2 + (- 9 b - 2) \cdot 5}{40} > 0$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Multiply $2$ by $19$ .

$\frac{38 b + (- 9 b - 2) \cdot 5}{40} > 0$

Step 3.9.2

Apply the distributive property.

$\frac{38 b - 9 b \cdot 5 - 2 \cdot 5}{40} > 0$

Step 3.9.3

Multiply $5$ by $- 9$ .

$\frac{38 b - 45 b - 2 \cdot 5}{40} > 0$

Step 3.9.4

Multiply $- 2$ by $5$ .

$\frac{38 b - 45 b - 10}{40} > 0$

Step 3.9.5

Subtract $45 b$ from $38 b$ .

$\frac{- 7 b - 10}{40} > 0$

Step 3.10

Factor $- 1$ out of $- 7 b$ .

$\frac{- (7 b) - 10}{40} > 0$

Step 3.11

Rewrite $- 10$ as $- 1 (10)$ .

$\frac{- (7 b) - 1 (10)}{40} > 0$

Step 3.12

Factor $- 1$ out of $- (7 b) - 1 (10)$ .

$\frac{- (7 b + 10)}{40} > 0$

Step 3.13

Rewrite $- (7 b + 10)$ as $- 1 (7 b + 10)$ .

$\frac{- 1 (7 b + 10)}{40} > 0$

Step 3.14

Move the negative in front of the fraction.

$- \frac{7 b + 10}{40} > 0$

Step 4

Divide each term in $- \frac{7 b + 10}{40} > 0$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \frac{7 b + 10}{40} > 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{7 b + 10}{40}}{- 1} < \frac{0}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{7 b + 10}{40}}{1} < \frac{0}{- 1}$

Step 4.2.2

Divide $\frac{7 b + 10}{40}$ by $1$ .

$\frac{7 b + 10}{40} < \frac{0}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 1$ .

$\frac{7 b + 10}{40} < 0$

Step 5

Multiply both sides by $40$ .

$\frac{7 b + 10}{40} \cdot 40 < 0 \cdot 40$

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $40$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{7 b + 10}{40} \cdot 40 < 0 \cdot 40$

Step 6.1.1.2

Rewrite the expression.

$7 b + 10 < 0 \cdot 40$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $0$ by $40$ .

$7 b + 10 < 0$

Step 7

Solve for $b$ .

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Step 7.1

Subtract $10$ from both sides of the inequality.

$7 b < - 10$

Step 7.2

Divide each term in $7 b < - 10$ by $7$ and simplify.

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Step 7.2.1

Divide each term in $7 b < - 10$ by $7$ .

$\frac{7 b}{7} < \frac{- 10}{7}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $7$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{7 b}{7} < \frac{- 10}{7}$

Step 7.2.2.1.2

Divide $b$ by $1$ .

$b < \frac{- 10}{7}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Move the negative in front of the fraction.

$b < - \frac{10}{7}$

Step 8

Convert the inequality to interval notation.

$(- \infty, - \frac{10}{7})$

Step 9