Pre-Algebra Examples

$4 a - 3 y > 6$ , $3 a + 3 y < - 1$

Step 1

Solve $4 a - 3 y > 6$ for $y$ .

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Step 1.1

Subtract $4 a$ from both sides of the inequality.

$- 3 y > 6 - 4 a$

Step 1.2

Divide each term in $- 3 y > 6 - 4 a$ by $- 3$ and simplify.

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Step 1.2.1

Divide each term in $- 3 y > 6 - 4 a$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 y}{- 3} < \frac{6}{- 3} + \frac{- 4 a}{- 3}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} < \frac{6}{- 3} + \frac{- 4 a}{- 3}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y < \frac{6}{- 3} + \frac{- 4 a}{- 3}$

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 $6$ by $- 3$ .

$y < - 2 + \frac{- 4 a}{- 3}$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$y < - 2 + \frac{4 a}{3}$

Step 2

Solve $3 a + 3 y < - 1$ for $y$ .

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Step 2.1

Subtract $3 a$ from both sides of the inequality.

$3 y < - 1 - 3 a$

Step 2.2

Divide each term in $3 y < - 1 - 3 a$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 y < - 1 - 3 a$ by $3$ .

$\frac{3 y}{3} < \frac{- 1}{3} + \frac{- 3 a}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} < \frac{- 1}{3} + \frac{- 3 a}{3}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y < \frac{- 1}{3} + \frac{- 3 a}{3}$

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

Move the negative in front of the fraction.

$y < - \frac{1}{3} + \frac{- 3 a}{3}$

Step 2.2.3.1.2

Cancel the common factor of $- 3$ and $3$ .

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Step 2.2.3.1.2.1

Factor $3$ out of $- 3 a$ .

$y < - \frac{1}{3} + \frac{3 (- a)}{3}$

Step 2.2.3.1.2.2

Cancel the common factors.

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Step 2.2.3.1.2.2.1

Factor $3$ out of $3$ .

$y < - \frac{1}{3} + \frac{3 (- a)}{3 (1)}$

Step 2.2.3.1.2.2.2

Cancel the common factor.

$y < - \frac{1}{3} + \frac{3 (- a)}{3 \cdot 1}$

Step 2.2.3.1.2.2.3

Rewrite the expression.

$y < - \frac{1}{3} + \frac{- a}{1}$

Step 2.2.3.1.2.2.4

Divide $- a$ by $1$ .

$y < - \frac{1}{3} - a$

Step 3

Find the intersection of $y < - 2 + \frac{4 a}{3}$ and $y < - \frac{1}{3} - a$ .

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

Step 4