Finite Math Examples

$3 x - 2 y > - 13 - x + 2 y > 5$

Step 1

Solve $3 x - 2 y > - 13 - x + 2 y$ 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 $2 y$ from both sides of the inequality.

$3 x - 2 y - 2 y > - 13 - x$

Step 1.1.2

Subtract $2 y$ from $- 2 y$ .

$3 x - 4 y > - 13 - x$

Step 1.2

Move all terms not containing $y$ to the right side of the inequality.

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Step 1.2.1

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

$- 4 y > - 13 - x - 3 x$

Step 1.2.2

Subtract $3 x$ from $- x$ .

$- 4 y > - 13 - 4 x$

Step 1.3

Divide each term in $- 4 y > - 13 - 4 x$ by $- 4$ and simplify.

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Step 1.3.1

Divide each term in $- 4 y > - 13 - 4 x$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 y}{- 4} < \frac{- 13}{- 4} + \frac{- 4 x}{- 4}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 4$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} < \frac{- 13}{- 4} + \frac{- 4 x}{- 4}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y < \frac{- 13}{- 4} + \frac{- 4 x}{- 4}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Dividing two negative values results in a positive value.

$y < \frac{13}{4} + \frac{- 4 x}{- 4}$

Step 1.3.3.1.2

Cancel the common factor of $- 4$ .

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Step 1.3.3.1.2.1

Cancel the common factor.

$y < \frac{13}{4} + \frac{- 4 x}{- 4}$

Step 1.3.3.1.2.2

Divide $x$ by $1$ .

$y < \frac{13}{4} + x$

Step 2

Solve $- 13 - x + 2 y > 5$ for $y$ .

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Step 2.1

Move all terms not containing $y$ to the right side of the inequality.

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Step 2.1.1

Add $13$ to both sides of the inequality.

$- x + 2 y > 5 + 13$

Step 2.1.2

Add $x$ to both sides of the inequality.

$2 y > 5 + 13 + x$

Step 2.1.3

Add $5$ and $13$ .

$2 y > 18 + x$

Step 2.2

Divide each term in $2 y > 18 + x$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 y > 18 + x$ by $2$ .

$\frac{2 y}{2} > \frac{18}{2} + \frac{x}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} > \frac{18}{2} + \frac{x}{2}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y > \frac{18}{2} + \frac{x}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $18$ by $2$ .

$y > 9 + \frac{x}{2}$

Step 3

Find the intersection of $y < \frac{13}{4} + x$ and $y > 9 + \frac{x}{2}$ .

$9 + \frac{x}{2} < y < \frac{13}{4} + x$

Step 4