Algebra Examples

$\frac{1}{4} b - \frac{1}{5} b > \frac{3}{10} b - \frac{1}{2}$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{4}$ and $b$ .

$\frac{b}{4} - \frac{1}{5} b > \frac{3}{10} b - \frac{1}{2}$

Step 1.1.2

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

$\frac{b}{4} - \frac{b}{5} > \frac{3}{10} b - \frac{1}{2}$

Step 1.2

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

$\frac{b}{4} \cdot \frac{5}{5} - \frac{b}{5} > \frac{3}{10} b - \frac{1}{2}$

Step 1.3

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

$\frac{b}{4} \cdot \frac{5}{5} - \frac{b}{5} \cdot \frac{4}{4} > \frac{3}{10} b - \frac{1}{2}$

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

$\frac{b \cdot 5}{4 \cdot 5} - \frac{b}{5} \cdot \frac{4}{4} > \frac{3}{10} b - \frac{1}{2}$

Step 1.4.2

Multiply $4$ by $5$ .

$\frac{b \cdot 5}{20} - \frac{b}{5} \cdot \frac{4}{4} > \frac{3}{10} b - \frac{1}{2}$

Step 1.4.3

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

$\frac{b \cdot 5}{20} - \frac{b \cdot 4}{5 \cdot 4} > \frac{3}{10} b - \frac{1}{2}$

Step 1.4.4

Multiply $5$ by $4$ .

$\frac{b \cdot 5}{20} - \frac{b \cdot 4}{20} > \frac{3}{10} b - \frac{1}{2}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{b \cdot 5 - b \cdot 4}{20} > \frac{3}{10} b - \frac{1}{2}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Factor $b$ out of $b \cdot 5 - b \cdot 4$ .

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Step 1.6.1.1

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

$\frac{b (5) - b \cdot 4}{20} > \frac{3}{10} b - \frac{1}{2}$

Step 1.6.1.2

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

$\frac{b (5) + b (- 1 \cdot 4)}{20} > \frac{3}{10} b - \frac{1}{2}$

Step 1.6.1.3

Factor $b$ out of $b (5) + b (- 1 \cdot 4)$ .

$\frac{b (5 - 1 \cdot 4)}{20} > \frac{3}{10} b - \frac{1}{2}$

Step 1.6.2

Multiply $- 1$ by $4$ .

$\frac{b (5 - 4)}{20} > \frac{3}{10} b - \frac{1}{2}$

Step 1.6.3

Subtract $4$ from $5$ .

$\frac{b \cdot 1}{20} > \frac{3}{10} b - \frac{1}{2}$

Step 1.7

Multiply $b$ by $1$ .

$\frac{b}{20} > \frac{3}{10} b - \frac{1}{2}$

Step 2

Combine $\frac{3}{10}$ and $b$ .

$\frac{b}{20} > \frac{3 b}{10} - \frac{1}{2}$

Step 3

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

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Step 3.1

Subtract $\frac{3 b}{10}$ from both sides of the inequality.

$\frac{b}{20} - \frac{3 b}{10} > - \frac{1}{2}$

Step 3.2

To write $- \frac{3 b}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{b}{20} - \frac{3 b}{10} \cdot \frac{2}{2} > - \frac{1}{2}$

Step 3.3

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

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Step 3.3.1

Multiply $\frac{3 b}{10}$ by $\frac{2}{2}$ .

$\frac{b}{20} - \frac{3 b \cdot 2}{10 \cdot 2} > - \frac{1}{2}$

Step 3.3.2

Multiply $10$ by $2$ .

$\frac{b}{20} - \frac{3 b \cdot 2}{20} > - \frac{1}{2}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{b - 3 b \cdot 2}{20} > - \frac{1}{2}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Factor $b$ out of $b - 3 b \cdot 2$ .

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Step 3.5.1.1

Raise $b$ to the power of $1$ .

$\frac{b^{1} - 3 b \cdot 2}{20} > - \frac{1}{2}$

Step 3.5.1.2

Factor $b$ out of $b^{1}$ .

$\frac{b \cdot 1 - 3 b \cdot 2}{20} > - \frac{1}{2}$

Step 3.5.1.3

Factor $b$ out of $- 3 b \cdot 2$ .

$\frac{b \cdot 1 + b (- 3 \cdot 2)}{20} > - \frac{1}{2}$

Step 3.5.1.4

Factor $b$ out of $b \cdot 1 + b (- 3 \cdot 2)$ .

$\frac{b (1 - 3 \cdot 2)}{20} > - \frac{1}{2}$

Step 3.5.2

Multiply $- 3$ by $2$ .

$\frac{b (1 - 6)}{20} > - \frac{1}{2}$

Step 3.5.3

Subtract $6$ from $1$ .

$\frac{b \cdot - 5}{20} > - \frac{1}{2}$

Step 3.6

Cancel the common factor of $- 5$ and $20$ .

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Step 3.6.1

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

$\frac{5 (b \cdot - 1)}{20} > - \frac{1}{2}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $5$ out of $20$ .

$\frac{5 (b \cdot - 1)}{5 (4)} > - \frac{1}{2}$

Step 3.6.2.2

Cancel the common factor.

$\frac{5 (b \cdot - 1)}{5 \cdot 4} > - \frac{1}{2}$

Step 3.6.2.3

Rewrite the expression.

$\frac{b \cdot - 1}{4} > - \frac{1}{2}$

Step 3.7

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

$\frac{- 1 \cdot b}{4} > - \frac{1}{2}$

Step 3.8

Move the negative in front of the fraction.

$- \frac{b}{4} > - \frac{1}{2}$

Step 4

Divide each term in $- \frac{b}{4} > - \frac{1}{2}$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \frac{b}{4} > - \frac{1}{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{b}{4}}{- 1} < \frac{- \frac{1}{2}}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{b}{4}}{1} < \frac{- \frac{1}{2}}{- 1}$

Step 4.2.2

Divide $\frac{b}{4}$ by $1$ .

$\frac{b}{4} < \frac{- \frac{1}{2}}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Dividing two negative values results in a positive value.

$\frac{b}{4} < \frac{\frac{1}{2}}{1}$

Step 4.3.2

Divide $\frac{1}{2}$ by $1$ .

$\frac{b}{4} < \frac{1}{2}$

Step 5

Multiply both sides by $4$ .

$\frac{b}{4} \cdot 4 < \frac{1}{2} \cdot 4$

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 $4$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{b}{4} \cdot 4 < \frac{1}{2} \cdot 4$

Step 6.1.1.2

Rewrite the expression.

$b < \frac{1}{2} \cdot 4$

Step 6.2

Simplify the right side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Factor $2$ out of $4$ .

$b < \frac{1}{2} \cdot (2 (2))$

Step 6.2.1.2

Cancel the common factor.

$b < \frac{1}{2} \cdot (2 \cdot 2)$

Step 6.2.1.3

Rewrite the expression.

$b < 2$

Step 7

Convert the inequality to interval notation.

$(- \infty, 2)$

Step 8