Algebra Examples

$10 x + 6 y < 30$ , $6 x - 10 y > 30$

Step 1

Solve $10 x + 6 y < 30$ for $y$ .

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Step 1.1

Subtract $10 x$ from both sides of the inequality.

$6 y < 30 - 10 x$

Step 1.2

Divide each term in $6 y < 30 - 10 x$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 y < 30 - 10 x$ by $6$ .

$\frac{6 y}{6} < \frac{30}{6} + \frac{- 10 x}{6}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{6 y}{6} < \frac{30}{6} + \frac{- 10 x}{6}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y < \frac{30}{6} + \frac{- 10 x}{6}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $30$ by $6$ .

$y < 5 + \frac{- 10 x}{6}$

Step 1.2.3.1.2

Cancel the common factor of $- 10$ and $6$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $- 10 x$ .

$y < 5 + \frac{2 (- 5 x)}{6}$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $6$ .

$y < 5 + \frac{2 (- 5 x)}{2 (3)}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y < 5 + \frac{2 (- 5 x)}{2 \cdot 3}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y < 5 + \frac{- 5 x}{3}$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$y < 5 - \frac{5 x}{3}$

Step 2

Solve $6 x - 10 y > 30$ for $y$ .

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Step 2.1

Subtract $6 x$ from both sides of the inequality.

$- 10 y > 30 - 6 x$

Step 2.2

Divide each term in $- 10 y > 30 - 6 x$ by $- 10$ and simplify.

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Step 2.2.1

Divide each term in $- 10 y > 30 - 6 x$ by $- 10$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 10 y}{- 10} < \frac{30}{- 10} + \frac{- 6 x}{- 10}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 10$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 10 y}{- 10} < \frac{30}{- 10} + \frac{- 6 x}{- 10}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y < \frac{30}{- 10} + \frac{- 6 x}{- 10}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Divide $30$ by $- 10$ .

$y < - 3 + \frac{- 6 x}{- 10}$

Step 2.2.3.1.2

Cancel the common factor of $- 6$ and $- 10$ .

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Step 2.2.3.1.2.1

Factor $- 2$ out of $- 6 x$ .

$y < - 3 + \frac{- 2 (3 x)}{- 10}$

Step 2.2.3.1.2.2

Cancel the common factors.

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Step 2.2.3.1.2.2.1

Factor $- 2$ out of $- 10$ .

$y < - 3 + \frac{- 2 (3 x)}{- 2 (5)}$

Step 2.2.3.1.2.2.2

Cancel the common factor.

$y < - 3 + \frac{- 2 (3 x)}{- 2 \cdot 5}$

Step 2.2.3.1.2.2.3

Rewrite the expression.

$y < - 3 + \frac{3 x}{5}$

Step 3

Find the intersection of $y < 5 - \frac{5 x}{3}$ and $y < - 3 + \frac{3 x}{5}$ .

$y < N o (Min i m u m)$

Step 4