Algebra Examples

$6 x - 2 y < 106 x + 3 y \geq 15$

Step 1

Solve $6 x - 2 y < 106 x + 3 y$ 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 $3 y$ from both sides of the inequality.

$6 x - 2 y - 3 y < 106 x$

Step 1.1.2

Subtract $3 y$ from $- 2 y$ .

$6 x - 5 y < 106 x$

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 $6 x$ from both sides of the inequality.

$- 5 y < 106 x - 6 x$

Step 1.2.2

Subtract $6 x$ from $106 x$ .

$- 5 y < 100 x$

Step 1.3

Divide each term in $- 5 y < 100 x$ by $- 5$ and simplify.

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Step 1.3.1

Divide each term in $- 5 y < 100 x$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 y}{- 5} > \frac{100 x}{- 5}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 5$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{- 5 y}{- 5} > \frac{100 x}{- 5}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y > \frac{100 x}{- 5}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Cancel the common factor of $100$ and $- 5$ .

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Step 1.3.3.1.1

Factor $5$ out of $100 x$ .

$y > \frac{5 (20 x)}{- 5}$

Step 1.3.3.1.2

Move the negative one from the denominator of $\frac{20 x}{- 1}$ .

$y > - 1 \cdot (20 x)$

Step 1.3.3.2

Simplify the expression.

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Step 1.3.3.2.1

Rewrite $- 1 \cdot (20 x)$ as $- (20 x)$ .

$y > - (20 x)$

Step 1.3.3.2.2

Multiply $20$ by $- 1$ .

$y > - 20 x$

Step 2

Solve $106 x + 3 y \geq 15$ for $y$ .

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Step 2.1

Subtract $106 x$ from both sides of the inequality.

$3 y \geq 15 - 106 x$

Step 2.2

Divide each term in $3 y \geq 15 - 106 x$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 y \geq 15 - 106 x$ by $3$ .

$\frac{3 y}{3} \geq \frac{15}{3} + \frac{- 106 x}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} \geq \frac{15}{3} + \frac{- 106 x}{3}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y \geq \frac{15}{3} + \frac{- 106 x}{3}$

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 $15$ by $3$ .

$y \geq 5 + \frac{- 106 x}{3}$

Step 2.2.3.1.2

Move the negative in front of the fraction.

$y \geq 5 - \frac{106 x}{3}$

Step 3

Find the intersection of $y > - 20 x$ and $y \geq 5 - \frac{106 x}{3}$ .

$y > - 20 x$ and $y \geq 5 - \frac{106 x}{3}$

Step 4