Algebra Examples

$\frac{a u}{2} - \frac{b v}{3} > 10$

Step 1

Subtract $\frac{a u}{2}$ from both sides of the inequality.

$- \frac{b v}{3} > 10 - \frac{a u}{2}$

Step 2

Divide each term in $- \frac{b v}{3} > 10 - \frac{a u}{2}$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- \frac{b v}{3} > 10 - \frac{a u}{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{b v}{3}}{- 1} < \frac{10}{- 1} + \frac{- \frac{a u}{2}}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{b v}{3}}{1} < \frac{10}{- 1} + \frac{- \frac{a u}{2}}{- 1}$

Step 2.2.2

Divide $\frac{b v}{3}$ by $1$ .

$\frac{b v}{3} < \frac{10}{- 1} + \frac{- \frac{a u}{2}}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Divide $10$ by $- 1$ .

$\frac{b v}{3} < - 10 + \frac{- \frac{a u}{2}}{- 1}$

Step 2.3.1.2

Dividing two negative values results in a positive value.

$\frac{b v}{3} < - 10 + \frac{\frac{a u}{2}}{1}$

Step 2.3.1.3

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

$\frac{b v}{3} < - 10 + \frac{a u}{2}$

Step 3

Multiply both sides by $3$ .

$\frac{b v}{3} \cdot 3 < (- 10 + \frac{a u}{2}) \cdot 3$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{b v}{3} \cdot 3 < (- 10 + \frac{a u}{2}) \cdot 3$

Step 4.1.1.2

Rewrite the expression.

$b v < (- 10 + \frac{a u}{2}) \cdot 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $(- 10 + \frac{a u}{2}) \cdot 3$ .

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Step 4.2.1.1

Simplify terms.

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Step 4.2.1.1.1

Apply the distributive property.

$b v < - 10 \cdot 3 + \frac{a u}{2} \cdot 3$

Step 4.2.1.1.2

Multiply $- 10$ by $3$ .

$b v < - 30 + \frac{a u}{2} \cdot 3$

Step 4.2.1.1.3

Combine $\frac{a u}{2}$ and $3$ .

$b v < - 30 + \frac{a u \cdot 3}{2}$

Step 4.2.1.2

Move $3$ to the left of $a u$ .

$b v < - 30 + \frac{3 a u}{2}$

Step 4.2.1.3

Reorder $- 30$ and $\frac{3 a u}{2}$ .

$b v < \frac{3 a u}{2} - 30$

Step 5

Divide each term in $b v < \frac{3 a u}{2} - 30$ by $b$ and simplify.

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Step 5.1

Divide each term in $b v < \frac{3 a u}{2} - 30$ by $b$ .

$\frac{b v}{b} < \frac{\frac{3 a u}{2}}{b} + \frac{- 30}{b}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $b$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{b v}{b} < \frac{\frac{3 a u}{2}}{b} + \frac{- 30}{b}$

Step 5.2.1.2

Divide $v$ by $1$ .

$v < \frac{\frac{3 a u}{2}}{b} + \frac{- 30}{b}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$v < \frac{3 a u}{2} \cdot \frac{1}{b} + \frac{- 30}{b}$

Step 5.3.1.2

Multiply $\frac{3 a u}{2}$ by $\frac{1}{b}$ .

$v < \frac{3 a u}{2 b} + \frac{- 30}{b}$

Step 5.3.1.3

Move the negative in front of the fraction.

$v < \frac{3 a u}{2 b} - \frac{30}{b}$