Pre-Algebra Examples

$- 4 x - 5 y > 20$ , $x^{2} + y^{2} < 36$

Step 1

Solve $- 4 x - 5 y > 20$ for $y$ .

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Step 1.1

Add $4 x$ to both sides of the inequality.

$- 5 y > 20 + 4 x$

Step 1.2

Divide each term in $- 5 y > 20 + 4 x$ by $- 5$ and simplify.

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Step 1.2.1

Divide each term in $- 5 y > 20 + 4 x$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 y}{- 5} < \frac{20}{- 5} + \frac{4 x}{- 5}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 5 y}{- 5} < \frac{20}{- 5} + \frac{4 x}{- 5}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y < \frac{20}{- 5} + \frac{4 x}{- 5}$

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 $20$ by $- 5$ .

$y < - 4 + \frac{4 x}{- 5}$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$y < - 4 - \frac{4 x}{5}$

Step 2

Solve $x^{2} + y^{2} < 36$ for $y$ .

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Step 2.1

Subtract $x^{2}$ from both sides of the inequality.

$y^{2} < 36 - x^{2}$

Step 2.2

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{y^{2}} < \sqrt{36 - x^{2}}$

Step 2.3

Simplify the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Pull terms out from under the radical.

$| y | < \sqrt{36 - x^{2}}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{36 - x^{2}}$ .

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Step 2.3.2.1.1

Rewrite $36$ as $6^{2}$ .

$| y | < \sqrt{6^{2} - x^{2}}$

Step 2.3.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = x$ .

$| y | < \sqrt{(6 + x) (6 - x)}$

Step 2.4

Write $| y | < \sqrt{(6 + x) (6 - x)}$ as a piecewise.

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Step 2.4.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$y \geq 0$

Step 2.4.2

In the piece where $y$ is non-negative, remove the absolute value.

$y < \sqrt{(6 + x) (6 - x)}$

Step 2.4.3

Find the domain of $y < \sqrt{(6 + x) (6 - x)}$ and find the intersection with $y \geq 0$ .

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Step 2.4.3.1

Find the domain of $y < \sqrt{(6 + x) (6 - x)}$ .

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Step 2.4.3.1.1

Set the radicand in $\sqrt{(6 + x) (6 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(6 + x) (6 - x) \geq 0$

Step 2.4.3.1.2

Solve for $y$ .

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Step 2.4.3.1.2.1

Simplify $(6 + x) (6 - x)$ .

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Step 2.4.3.1.2.1.1

Expand $(6 + x) (6 - x)$ using the FOIL Method.

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Step 2.4.3.1.2.1.1.1

Apply the distributive property.

$6 (6 - x) + x (6 - x) \geq 0$

Step 2.4.3.1.2.1.1.2

Apply the distributive property.

$6 \cdot 6 + 6 (- x) + x (6 - x) \geq 0$

Step 2.4.3.1.2.1.1.3

Apply the distributive property.

$6 \cdot 6 + 6 (- x) + x \cdot 6 + x (- x) \geq 0$

Step 2.4.3.1.2.1.2

Simplify and combine like terms.

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Step 2.4.3.1.2.1.2.1

Simplify each term.

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Step 2.4.3.1.2.1.2.1.1

Multiply $6$ by $6$ .

$36 + 6 (- x) + x \cdot 6 + x (- x) \geq 0$

Step 2.4.3.1.2.1.2.1.2

Multiply $- 1$ by $6$ .

$36 - 6 x + x \cdot 6 + x (- x) \geq 0$

Step 2.4.3.1.2.1.2.1.3

Move $6$ to the left of $x$ .

$36 - 6 x + 6 \cdot x + x (- x) \geq 0$

Step 2.4.3.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$36 - 6 x + 6 x - x \cdot x \geq 0$

Step 2.4.3.1.2.1.2.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.4.3.1.2.1.2.1.5.1

Move $x$ .

$36 - 6 x + 6 x - (x \cdot x) \geq 0$

Step 2.4.3.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

$36 - 6 x + 6 x - x^{2} \geq 0$

Step 2.4.3.1.2.1.2.2

Add $- 6 x$ and $6 x$ .

$36 + 0 - x^{2} \geq 0$

Step 2.4.3.1.2.1.2.3

Add $36$ and $0$ .

$36 - x^{2} \geq 0$

Step 2.4.3.1.2.2

Subtract $36$ from both sides of the inequality.

$- x^{2} \geq - 36$

Step 2.4.3.1.2.3

Divide each term in $- x^{2} \geq - 36$ by $- 1$ and simplify.

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Step 2.4.3.1.2.3.1

Divide each term in $- x^{2} \geq - 36$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 36}{- 1}$

Step 2.4.3.1.2.3.2

Simplify the left side.

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Step 2.4.3.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 36}{- 1}$

Step 2.4.3.1.2.3.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \leq \frac{- 36}{- 1}$

Step 2.4.3.1.2.3.3

Simplify the right side.

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Step 2.4.3.1.2.3.3.1

Divide $- 36$ by $- 1$ .

$x^{2} \leq 36$

Step 2.4.3.1.2.4

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{36}$

Step 2.4.3.1.2.5

Simplify the equation.

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Step 2.4.3.1.2.5.1

Simplify the left side.

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Step 2.4.3.1.2.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{36}$

Step 2.4.3.1.2.5.2

Simplify the right side.

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Step 2.4.3.1.2.5.2.1

Simplify $\sqrt{36}$ .

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Step 2.4.3.1.2.5.2.1.1

Rewrite $36$ as $6^{2}$ .

$| x | \leq \sqrt{6^{2}}$

Step 2.4.3.1.2.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 6 |$

Step 2.4.3.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$| x | \leq 6$

Step 2.4.3.1.2.6

Write $| x | \leq 6$ as a piecewise.

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Step 2.4.3.1.2.6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 2.4.3.1.2.6.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 6$

Step 2.4.3.1.2.6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 2.4.3.1.2.6.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 6$

Step 2.4.3.1.2.6.5

Write as a piecewise.

${\begin{cases} x \leq 6 & x \geq 0 \\ - x \leq 6 & x < 0 \end{cases}$

Step 2.4.3.1.2.7

Find the intersection of $x \leq 6$ and $x \geq 0$ .

$0 \leq x \leq 6$

Step 2.4.3.1.2.8

Solve $- x \leq 6$ when $x < 0$ .

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Step 2.4.3.1.2.8.1

Divide each term in $- x \leq 6$ by $- 1$ and simplify.

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Step 2.4.3.1.2.8.1.1

Divide each term in $- x \leq 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{6}{- 1}$

Step 2.4.3.1.2.8.1.2

Simplify the left side.

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Step 2.4.3.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{6}{- 1}$

Step 2.4.3.1.2.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{6}{- 1}$

Step 2.4.3.1.2.8.1.3

Simplify the right side.

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Step 2.4.3.1.2.8.1.3.1

Divide $6$ by $- 1$ .

$x \geq - 6$

Step 2.4.3.1.2.8.2

Find the intersection of $x \geq - 6$ and $x < 0$ .

$- 6 \leq x < 0$

Step 2.4.3.1.2.9

Find the union of the solutions.

$- 6 \leq x \leq 6$

Step 2.4.3.1.3

The domain is all values of $y$ that make the expression defined.

$[- 6, 6]$

Step 2.4.3.2

Find the intersection of $y \geq 0$ and $[- 6, 6]$ .

$0 \leq y \leq 6$

Step 2.4.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$y < 0$

Step 2.4.5

In the piece where $y$ is negative, remove the absolute value and multiply by $- 1$ .

$- y < \sqrt{(6 + x) (6 - x)}$

Step 2.4.6

Find the domain of $- y < \sqrt{(6 + x) (6 - x)}$ and find the intersection with $y < 0$ .

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Step 2.4.6.1

Find the domain of $- y < \sqrt{(6 + x) (6 - x)}$ .

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Step 2.4.6.1.1

Set the radicand in $\sqrt{(6 + x) (6 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(6 + x) (6 - x) \geq 0$

Step 2.4.6.1.2

Solve for $y$ .

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Step 2.4.6.1.2.1

Simplify $(6 + x) (6 - x)$ .

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Step 2.4.6.1.2.1.1

Expand $(6 + x) (6 - x)$ using the FOIL Method.

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Step 2.4.6.1.2.1.1.1

Apply the distributive property.

$6 (6 - x) + x (6 - x) \geq 0$

Step 2.4.6.1.2.1.1.2

Apply the distributive property.

$6 \cdot 6 + 6 (- x) + x (6 - x) \geq 0$

Step 2.4.6.1.2.1.1.3

Apply the distributive property.

$6 \cdot 6 + 6 (- x) + x \cdot 6 + x (- x) \geq 0$

Step 2.4.6.1.2.1.2

Simplify and combine like terms.

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Step 2.4.6.1.2.1.2.1

Simplify each term.

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Step 2.4.6.1.2.1.2.1.1

Multiply $6$ by $6$ .

$36 + 6 (- x) + x \cdot 6 + x (- x) \geq 0$

Step 2.4.6.1.2.1.2.1.2

Multiply $- 1$ by $6$ .

$36 - 6 x + x \cdot 6 + x (- x) \geq 0$

Step 2.4.6.1.2.1.2.1.3

Move $6$ to the left of $x$ .

$36 - 6 x + 6 \cdot x + x (- x) \geq 0$

Step 2.4.6.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$36 - 6 x + 6 x - x \cdot x \geq 0$

Step 2.4.6.1.2.1.2.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.4.6.1.2.1.2.1.5.1

Move $x$ .

$36 - 6 x + 6 x - (x \cdot x) \geq 0$

Step 2.4.6.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

$36 - 6 x + 6 x - x^{2} \geq 0$

Step 2.4.6.1.2.1.2.2

Add $- 6 x$ and $6 x$ .

$36 + 0 - x^{2} \geq 0$

Step 2.4.6.1.2.1.2.3

Add $36$ and $0$ .

$36 - x^{2} \geq 0$

Step 2.4.6.1.2.2

Subtract $36$ from both sides of the inequality.

$- x^{2} \geq - 36$

Step 2.4.6.1.2.3

Divide each term in $- x^{2} \geq - 36$ by $- 1$ and simplify.

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Step 2.4.6.1.2.3.1

Divide each term in $- x^{2} \geq - 36$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 36}{- 1}$

Step 2.4.6.1.2.3.2

Simplify the left side.

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Step 2.4.6.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 36}{- 1}$

Step 2.4.6.1.2.3.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \leq \frac{- 36}{- 1}$

Step 2.4.6.1.2.3.3

Simplify the right side.

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Step 2.4.6.1.2.3.3.1

Divide $- 36$ by $- 1$ .

$x^{2} \leq 36$

Step 2.4.6.1.2.4

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{36}$

Step 2.4.6.1.2.5

Simplify the equation.

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Step 2.4.6.1.2.5.1

Simplify the left side.

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Step 2.4.6.1.2.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{36}$

Step 2.4.6.1.2.5.2

Simplify the right side.

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Step 2.4.6.1.2.5.2.1

Simplify $\sqrt{36}$ .

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Step 2.4.6.1.2.5.2.1.1

Rewrite $36$ as $6^{2}$ .

$| x | \leq \sqrt{6^{2}}$

Step 2.4.6.1.2.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 6 |$

Step 2.4.6.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$| x | \leq 6$

Step 2.4.6.1.2.6

Write $| x | \leq 6$ as a piecewise.

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Step 2.4.6.1.2.6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 2.4.6.1.2.6.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 6$

Step 2.4.6.1.2.6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 2.4.6.1.2.6.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 6$

Step 2.4.6.1.2.6.5

Write as a piecewise.

${\begin{cases} x \leq 6 & x \geq 0 \\ - x \leq 6 & x < 0 \end{cases}$

Step 2.4.6.1.2.7

Find the intersection of $x \leq 6$ and $x \geq 0$ .

$0 \leq x \leq 6$

Step 2.4.6.1.2.8

Solve $- x \leq 6$ when $x < 0$ .

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Step 2.4.6.1.2.8.1

Divide each term in $- x \leq 6$ by $- 1$ and simplify.

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Step 2.4.6.1.2.8.1.1

Divide each term in $- x \leq 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{6}{- 1}$

Step 2.4.6.1.2.8.1.2

Simplify the left side.

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Step 2.4.6.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{6}{- 1}$

Step 2.4.6.1.2.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{6}{- 1}$

Step 2.4.6.1.2.8.1.3

Simplify the right side.

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Step 2.4.6.1.2.8.1.3.1

Divide $6$ by $- 1$ .

$x \geq - 6$

Step 2.4.6.1.2.8.2

Find the intersection of $x \geq - 6$ and $x < 0$ .

$- 6 \leq x < 0$

Step 2.4.6.1.2.9

Find the union of the solutions.

$- 6 \leq x \leq 6$

Step 2.4.6.1.3

The domain is all values of $y$ that make the expression defined.

$[- 6, 6]$

Step 2.4.6.2

Find the intersection of $y < 0$ and $[- 6, 6]$ .

$- 6 \leq y < 0$

Step 2.4.7

Write as a piecewise.

${\begin{cases} y < \sqrt{(6 + x) (6 - x)} & 0 \leq y \leq 6 \\ - y < \sqrt{(6 + x) (6 - x)} & - 6 \leq y < 0 \end{cases}$

Step 2.5

Find the intersection of $y < \sqrt{(6 + x) (6 - x)}$ and $0 \leq y \leq 6$ .

$0 \leq y \leq N o (Min i m u m)$

Step 2.6

Solve $- y < \sqrt{(6 + x) (6 - x)}$ when $- 6 \leq y < 0$ .

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Step 2.6.1

Divide each term in $- y < \sqrt{(6 + x) (6 - x)}$ by $- 1$ and simplify.

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Step 2.6.1.1

Divide each term in $- y < \sqrt{(6 + x) (6 - x)}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} > \frac{\sqrt{(6 + x) (6 - x)}}{- 1}$

Step 2.6.1.2

Simplify the left side.

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Step 2.6.1.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} > \frac{\sqrt{(6 + x) (6 - x)}}{- 1}$

Step 2.6.1.2.2

Divide $y$ by $1$ .

$y > \frac{\sqrt{(6 + x) (6 - x)}}{- 1}$

Step 2.6.1.3

Simplify the right side.

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Step 2.6.1.3.1

Move the negative one from the denominator of $\frac{\sqrt{(6 + x) (6 - x)}}{- 1}$ .

$y > - 1 \cdot \sqrt{(6 + x) (6 - x)}$

Step 2.6.1.3.2

Rewrite $- 1 \cdot \sqrt{(6 + x) (6 - x)}$ as $- \sqrt{(6 + x) (6 - x)}$ .

$y > - \sqrt{(6 + x) (6 - x)}$

Step 2.6.2

Find the intersection of $y > - \sqrt{(6 + x) (6 - x)}$ and $- 6 \leq y < 0$ .

No solution

Step 2.7

Find the union of the solutions.

$0 \leq y \leq i N o Min m^{2} u$

Step 3

Find the intersection of $y < - 4 - \frac{4 x}{5}$ and $0 \leq y \leq i N o Min m^{2} u$ .

$0 \leq y \leq N o (Min i m u m)$

Step 4