Algebra Examples

$y \geq 3 - x$ $y < 5 x - 2$

Step 1

Simplify the first inequality.

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Step 1.1

Rewrite so $x$ is on the left side of the inequality.

$3 - x \leq y$ and $y < 5 x - 2$

Step 1.2

Subtract $3$ from both sides of the inequality.

$- x \leq y - 3$ and $y < 5 x - 2$

Step 1.3

Divide each term in $- x \leq y - 3$ by $- 1$ and simplify.

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Step 1.3.1

Divide each term in $- x \leq y - 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{y}{- 1} + \frac{- 3}{- 1}$ and $y < 5 x - 2$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{y}{- 1} + \frac{- 3}{- 1}$ and $y < 5 x - 2$

Step 1.3.2.2

Divide $x$ by $1$ .

$x \geq \frac{y}{- 1} + \frac{- 3}{- 1}$ and $y < 5 x - 2$

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

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

$x \geq - 1 \cdot y + \frac{- 3}{- 1}$ and $y < 5 x - 2$

Step 1.3.3.1.2

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

$x \geq - y + \frac{- 3}{- 1}$ and $y < 5 x - 2$

Step 1.3.3.1.3

Divide $- 3$ by $- 1$ .

$x \geq - y + 3$ and $y < 5 x - 2$

Step 2

Simplify the second inequality.

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Step 2.1

Rewrite so $x$ is on the left side of the inequality.

$x \geq - y + 3$ and $5 x - 2 > y$

Step 2.2

Add $2$ to both sides of the inequality.

$x \geq - y + 3$ and $5 x > y + 2$

Step 2.3

Divide each term in $5 x > y + 2$ by $5$ and simplify.

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Step 2.3.1

Divide each term in $5 x > y + 2$ by $5$ .

$x \geq - y + 3$ and $\frac{5 x}{5} > \frac{y}{5} + \frac{2}{5}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $5$ .

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Step 2.3.2.1.1

Cancel the common factor.

$x \geq - y + 3$ and $\frac{5 x}{5} > \frac{y}{5} + \frac{2}{5}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x \geq - y + 3$ and $x > \frac{y}{5} + \frac{2}{5}$

Step 3