Algebra Examples

$y > 3 x + 8 y \leq - \frac{1}{2} x + 4$

Step 1

Solve $y > 3 x + 8 y$ for $y$ .

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Step 1.1

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

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Step 1.1.1

Subtract $8 y$ from both sides of the inequality.

$y - 8 y > 3 x$

Step 1.1.2

Subtract $8 y$ from $y$ .

$- 7 y > 3 x$

Step 1.2

Divide each term in $- 7 y > 3 x$ by $- 7$ and simplify.

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Step 1.2.1

Divide each term in $- 7 y > 3 x$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 y}{- 7} < \frac{3 x}{- 7}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 7$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 7 y}{- 7} < \frac{3 x}{- 7}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y < \frac{3 x}{- 7}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$y < - \frac{3 x}{7}$

Step 2

Solve $3 x + 8 y \leq - \frac{1}{2} x + 4$ for $y$ .

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Step 2.1

Combine $x$ and $\frac{1}{2}$ .

$3 x + 8 y \leq - \frac{x}{2} + 4$

Step 2.2

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

$8 y \leq - \frac{x}{2} + 4 - 3 x$

Step 2.3

Divide each term in $8 y \leq - \frac{x}{2} + 4 - 3 x$ by $8$ and simplify.

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Step 2.3.1

Divide each term in $8 y \leq - \frac{x}{2} + 4 - 3 x$ by $8$ .

$\frac{8 y}{8} \leq \frac{- \frac{x}{2}}{8} + \frac{4}{8} + \frac{- 3 x}{8}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $8$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{8 y}{8} \leq \frac{- \frac{x}{2}}{8} + \frac{4}{8} + \frac{- 3 x}{8}$

Step 2.3.2.1.2

Divide $y$ by $1$ .

$y \leq \frac{- \frac{x}{2}}{8} + \frac{4}{8} + \frac{- 3 x}{8}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Combine the numerators over the common denominator.

$y \leq \frac{- \frac{x}{2} + 4 - 3 x}{8}$

Step 2.3.3.2

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

$y \leq \frac{- \frac{x}{2} - 3 x \cdot \frac{2}{2} + 4}{8}$

Step 2.3.3.3

Simplify terms.

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Step 2.3.3.3.1

Combine $- 3 x$ and $\frac{2}{2}$ .

$y \leq \frac{- \frac{x}{2} + \frac{- 3 x \cdot 2}{2} + 4}{8}$

Step 2.3.3.3.2

Combine the numerators over the common denominator.

$y \leq \frac{\frac{- x - 3 x \cdot 2}{2} + 4}{8}$

Step 2.3.3.3.3

Simplify each term.

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Step 2.3.3.3.3.1

Simplify the numerator.

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Step 2.3.3.3.3.1.1

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

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Step 2.3.3.3.3.1.1.1

Factor $x$ out of $- x$ .

$y \leq \frac{\frac{x \cdot - 1 - 3 x \cdot 2}{2} + 4}{8}$

Step 2.3.3.3.3.1.1.2

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

$y \leq \frac{\frac{x \cdot - 1 + x (- 3 \cdot 2)}{2} + 4}{8}$

Step 2.3.3.3.3.1.1.3

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

$y \leq \frac{\frac{x (- 1 - 3 \cdot 2)}{2} + 4}{8}$

Step 2.3.3.3.3.1.2

Multiply $- 3$ by $2$ .

$y \leq \frac{\frac{x (- 1 - 6)}{2} + 4}{8}$

Step 2.3.3.3.3.1.3

Subtract $6$ from $- 1$ .

$y \leq \frac{\frac{x \cdot - 7}{2} + 4}{8}$

Step 2.3.3.3.3.2

Move $- 7$ to the left of $x$ .

$y \leq \frac{\frac{- 7 \cdot x}{2} + 4}{8}$

Step 2.3.3.3.3.3

Move the negative in front of the fraction.

$y \leq \frac{- \frac{7 x}{2} + 4}{8}$

Step 2.3.3.4

Simplify the numerator.

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Step 2.3.3.4.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y \leq \frac{- \frac{7 x}{2} + 4 \cdot \frac{2}{2}}{8}$

Step 2.3.3.4.2

Combine $4$ and $\frac{2}{2}$ .

$y \leq \frac{- \frac{7 x}{2} + \frac{4 \cdot 2}{2}}{8}$

Step 2.3.3.4.3

Combine the numerators over the common denominator.

$y \leq \frac{\frac{- 7 x + 4 \cdot 2}{2}}{8}$

Step 2.3.3.4.4

Multiply $4$ by $2$ .

$y \leq \frac{\frac{- 7 x + 8}{2}}{8}$

Step 2.3.3.5

Multiply the numerator by the reciprocal of the denominator.

$y \leq \frac{- 7 x + 8}{2} \cdot \frac{1}{8}$

Step 2.3.3.6

Multiply $\frac{- 7 x + 8}{2} \cdot \frac{1}{8}$ .

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Step 2.3.3.6.1

Multiply $\frac{- 7 x + 8}{2}$ by $\frac{1}{8}$ .

$y \leq \frac{- 7 x + 8}{2 \cdot 8}$

Step 2.3.3.6.2

Multiply $2$ by $8$ .

$y \leq \frac{- 7 x + 8}{16}$

Step 2.3.3.7

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

$y \leq \frac{- (7 x) + 8}{16}$

Step 2.3.3.8

Rewrite $8$ as $- 1 (- 8)$ .

$y \leq \frac{- (7 x) - 1 (- 8)}{16}$

Step 2.3.3.9

Factor $- 1$ out of $- (7 x) - 1 (- 8)$ .

$y \leq \frac{- (7 x - 8)}{16}$

Step 2.3.3.10

Simplify the expression.

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Step 2.3.3.10.1

Rewrite $- (7 x - 8)$ as $- 1 (7 x - 8)$ .

$y \leq \frac{- 1 (7 x - 8)}{16}$

Step 2.3.3.10.2

Move the negative in front of the fraction.

$y \leq - \frac{7 x - 8}{16}$

Step 3

Find the intersection of $y < - \frac{3 x}{7}$ and $y \leq - \frac{7 x - 8}{16}$ .

$y \leq N o (Min i m u m)$

Step 4