Algebra Examples

$y \leq - \frac{5}{2} x - 2 y < - \frac{1}{2} x + 2$

Step 1

Solve $y \leq - \frac{5}{2} x - 2 y$ for $y$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $x$ and $\frac{5}{2}$ .

$y \leq - \frac{x \cdot 5}{2} - 2 y$

Step 1.1.2

Move $5$ to the left of $x$ .

$y \leq - \frac{5 x}{2} - 2 y$

Step 1.2

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

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Step 1.2.1

Add $2 y$ to both sides of the inequality.

$y + 2 y \leq - \frac{5 x}{2}$

Step 1.2.2

Add $y$ and $2 y$ .

$3 y \leq - \frac{5 x}{2}$

Step 1.3

Divide each term in $3 y \leq - \frac{5 x}{2}$ by $3$ and simplify.

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Step 1.3.1

Divide each term in $3 y \leq - \frac{5 x}{2}$ by $3$ .

$\frac{3 y}{3} \leq \frac{- \frac{5 x}{2}}{3}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $3$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{3 y}{3} \leq \frac{- \frac{5 x}{2}}{3}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y \leq \frac{- \frac{5 x}{2}}{3}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$y \leq - \frac{5 x}{2} \cdot \frac{1}{3}$

Step 1.3.3.2

Multiply $- \frac{5 x}{2} \cdot \frac{1}{3}$ .

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Step 1.3.3.2.1

Multiply $\frac{1}{3}$ by $\frac{5 x}{2}$ .

$y \leq - \frac{5 x}{3 \cdot 2}$

Step 1.3.3.2.2

Multiply $3$ by $2$ .

$y \leq - \frac{5 x}{6}$

Step 2

Solve $- \frac{5}{2} x - 2 y < - \frac{1}{2} x + 2$ for $y$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $x$ and $\frac{5}{2}$ .

$- \frac{x \cdot 5}{2} - 2 y < - \frac{1}{2} x + 2$

Step 2.1.2

Move $5$ to the left of $x$ .

$- \frac{5 x}{2} - 2 y < - \frac{1}{2} x + 2$

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

Add $\frac{5 x}{2}$ to both sides of the inequality.

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

Step 2.3.2

Combine the numerators over the common denominator.

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

Step 2.3.3

Add $- x$ and $5 x$ .

$- 2 y < 2 + \frac{4 x}{2}$

Step 2.3.4

Cancel the common factor of $4$ and $2$ .

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Step 2.3.4.1

Factor $2$ out of $4 x$ .

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

Step 2.3.4.2

Cancel the common factors.

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Step 2.3.4.2.1

Factor $2$ out of $2$ .

$- 2 y < 2 + \frac{2 (2 x)}{2 (1)}$

Step 2.3.4.2.2

Cancel the common factor.

$- 2 y < 2 + \frac{2 (2 x)}{2 \cdot 1}$

Step 2.3.4.2.3

Rewrite the expression.

$- 2 y < 2 + \frac{2 x}{1}$

Step 2.3.4.2.4

Divide $2 x$ by $1$ .

$- 2 y < 2 + 2 x$

Step 2.4

Divide each term in $- 2 y < 2 + 2 x$ by $- 2$ and simplify.

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Step 2.4.1

Divide each term in $- 2 y < 2 + 2 x$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $- 2$ .

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Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Divide $y$ by $1$ .

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Divide $2$ by $- 2$ .

$y > - 1 + \frac{2 x}{- 2}$

Step 2.4.3.1.2

Cancel the common factor of $2$ and $- 2$ .

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Step 2.4.3.1.2.1

Factor $2$ out of $2 x$ .

$y > - 1 + \frac{2 (x)}{- 2}$

Step 2.4.3.1.2.2

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$y > - 1 - 1 \cdot x$

Step 2.4.3.1.3

Rewrite $- 1 \cdot x$ as $- x$ .

$y > - 1 - x$

Step 3

Find the intersection of $y \leq - \frac{5 x}{6}$ and $y > - 1 - x$ .

$- 1 - x < y \leq - \frac{5 x}{6}$

Step 4