Algebra Examples

$y < - \frac{1}{3} x + 1 y \leq 2 x - 3$

Step 1

Solve $y < - \frac{1}{3} x + 1 y$ for $y$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $x$ and $\frac{1}{3}$ .

$y < - \frac{x}{3} + 1 y$

Step 1.1.2

Multiply $y$ by $1$ .

$y < - \frac{x}{3} + y$

Step 1.2

Move all terms containing $y$ to the left side of the inequality.

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Step 1.2.1

Subtract $y$ from both sides of the inequality.

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

Step 1.2.2

Subtract $y$ from $y$ .

$0 < - \frac{x}{3}$

Step 1.3

Multiply both sides by $3$ .

$0 \cdot 3 < - \frac{x}{3} \cdot 3$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the left side.

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Step 1.4.1.1

Multiply $0$ by $3$ .

$0 < - \frac{x}{3} \cdot 3$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Cancel the common factor of $3$ .

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Step 1.4.2.1.1

Move the leading negative in $- \frac{x}{3}$ into the numerator.

$0 < \frac{- x}{3} \cdot 3$

Step 1.4.2.1.2

Cancel the common factor.

$0 < \frac{- x}{3} \cdot 3$

Step 1.4.2.1.3

Rewrite the expression.

$0 < - x$

Step 1.5

Solve for $x$ .

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Step 1.5.1

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

$- x > 0$

Step 1.5.2

Divide each term in $- x > 0$ by $- 1$ and simplify.

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Step 1.5.2.1

Divide each term in $- x > 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{0}{- 1}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{0}{- 1}$

Step 1.5.2.2.2

Divide $x$ by $1$ .

$x < \frac{0}{- 1}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Divide $0$ by $- 1$ .

$x < 0$

Step 1.6

The variable $y$ got canceled.

All real numbers

Step 2

Solve $- \frac{1}{3} x + 1 y \leq 2 x - 3$ for $y$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $x$ and $\frac{1}{3}$ .

$- \frac{x}{3} + 1 y \leq 2 x - 3$

Step 2.1.2

Multiply $y$ by $1$ .

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

Step 2.2

Add $\frac{x}{3}$ to both sides of the inequality.

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

Step 3

Find the intersection.

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

Step 4