Algebra Examples

$5 x - 9 y \geq 8$ $7 x - 4 y < 10$

Step 1

Solve $5 x - 9 y \geq 8$ for $y$ .

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Step 1.1

Subtract $5 x$ from both sides of the inequality.

$- 9 y \geq 8 - 5 x$

Step 1.2

Divide each term in $- 9 y \geq 8 - 5 x$ by $- 9$ and simplify.

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Step 1.2.1

Divide each term in $- 9 y \geq 8 - 5 x$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 y}{- 9} \leq \frac{8}{- 9} + \frac{- 5 x}{- 9}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 9 y}{- 9} \leq \frac{8}{- 9} + \frac{- 5 x}{- 9}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y \leq \frac{8}{- 9} + \frac{- 5 x}{- 9}$

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

Move the negative in front of the fraction.

$y \leq - \frac{8}{9} + \frac{- 5 x}{- 9}$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$y \leq - \frac{8}{9} + \frac{5 x}{9}$

Step 2

Solve $7 x - 4 y < 10$ for $y$ .

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Step 2.1

Subtract $7 x$ from both sides of the inequality.

$- 4 y < 10 - 7 x$

Step 2.2

Divide each term in $- 4 y < 10 - 7 x$ by $- 4$ and simplify.

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Step 2.2.1

Divide each term in $- 4 y < 10 - 7 x$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 y}{- 4} > \frac{10}{- 4} + \frac{- 7 x}{- 4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} > \frac{10}{- 4} + \frac{- 7 x}{- 4}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y > \frac{10}{- 4} + \frac{- 7 x}{- 4}$

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

Cancel the common factor of $10$ and $- 4$ .

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Step 2.2.3.1.1.1

Factor $2$ out of $10$ .

$y > \frac{2 (5)}{- 4} + \frac{- 7 x}{- 4}$

Step 2.2.3.1.1.2

Cancel the common factors.

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Step 2.2.3.1.1.2.1

Factor $2$ out of $- 4$ .

$y > \frac{2 \cdot 5}{2 \cdot - 2} + \frac{- 7 x}{- 4}$

Step 2.2.3.1.1.2.2

Cancel the common factor.

$y > \frac{2 \cdot 5}{2 \cdot - 2} + \frac{- 7 x}{- 4}$

Step 2.2.3.1.1.2.3

Rewrite the expression.

$y > \frac{5}{- 2} + \frac{- 7 x}{- 4}$

Step 2.2.3.1.2

Move the negative in front of the fraction.

$y > - \frac{5}{2} + \frac{- 7 x}{- 4}$

Step 2.2.3.1.3

Dividing two negative values results in a positive value.

$y > - \frac{5}{2} + \frac{7 x}{4}$

Step 3

Find the intersection of $y \leq - \frac{8}{9} + \frac{5 x}{9}$ and $y > - \frac{5}{2} + \frac{7 x}{4}$ .

$- \frac{5}{2} + \frac{7 x}{4} < y \leq - \frac{8}{9} + \frac{5 x}{9}$

Step 4