Algebra Examples

$2 x - y \geq - 2$ $8 x - y < 5$

Step 1

Solve $2 x - y \geq - 2$ for $y$ .

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Step 1.1

Subtract $2 x$ from both sides of the inequality.

$- y \geq - 2 - 2 x$

Step 1.2

Divide each term in $- y \geq - 2 - 2 x$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- y \geq - 2 - 2 x$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.2.2

Divide $y$ by $1$ .

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

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 $- 2$ by $- 1$ .

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

Step 1.2.3.1.2

Move the negative one from the denominator of $\frac{- 2 x}{- 1}$ .

$y \leq 2 - 1 \cdot (- 2 x)$

Step 1.2.3.1.3

Rewrite $- 1 \cdot (- 2 x)$ as $- (- 2 x)$ .

$y \leq 2 - (- 2 x)$

Step 1.2.3.1.4

Multiply $- 2$ by $- 1$ .

$y \leq 2 + 2 x$

Step 2

Solve $8 x - y < 5$ for $y$ .

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Step 2.1

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

$- y < 5 - 8 x$

Step 2.2

Divide each term in $- y < 5 - 8 x$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- y < 5 - 8 x$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} > \frac{5}{- 1} + \frac{- 8 x}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} > \frac{5}{- 1} + \frac{- 8 x}{- 1}$

Step 2.2.2.2

Divide $y$ by $1$ .

$y > \frac{5}{- 1} + \frac{- 8 x}{- 1}$

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

Divide $5$ by $- 1$ .

$y > - 5 + \frac{- 8 x}{- 1}$

Step 2.2.3.1.2

Move the negative one from the denominator of $\frac{- 8 x}{- 1}$ .

$y > - 5 - 1 \cdot (- 8 x)$

Step 2.2.3.1.3

Rewrite $- 1 \cdot (- 8 x)$ as $- (- 8 x)$ .

$y > - 5 - (- 8 x)$

Step 2.2.3.1.4

Multiply $- 8$ by $- 1$ .

$y > - 5 + 8 x$

Step 3

Find the intersection of $y \leq 2 + 2 x$ and $y > - 5 + 8 x$ .

$- 5 + 8 x < y \leq 2 + 2 x$

Step 4