Algebra Examples

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

Step 1

Solve $x \leq - 3 y$ for $y$ .

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Step 1.1

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

$- 3 y \geq x$

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 3$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

Divide each term in $- 3 y < \frac{5}{3} x + 2$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $y$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Combine $\frac{5}{3}$ and $x$ .

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

Step 2.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3.1.3

Move the negative in front of the fraction.

$y > \frac{5 x}{3} (- \frac{1}{3}) + \frac{2}{- 3}$

Step 2.3.1.4

Multiply $\frac{5 x}{3} (- \frac{1}{3})$ .

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Step 2.3.1.4.1

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

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

Step 2.3.1.4.2

Multiply $3$ by $3$ .

$y > - \frac{5 x}{9} + \frac{2}{- 3}$

Step 2.3.1.5

Move the negative in front of the fraction.

$y > - \frac{5 x}{9} - \frac{2}{3}$

Step 3

Find the intersection of $y \leq - \frac{x}{3}$ and $y > - \frac{5 x}{9} - \frac{2}{3}$ .

$- \frac{5 x}{9} - \frac{2}{3} < y \leq - \frac{x}{3}$

Step 4