Finite Math Examples

$y - \frac{2}{3} > \frac{4}{3} y - 2$

Step 1

Combine $\frac{4}{3}$ and $y$ .

$y - \frac{2}{3} > \frac{4 y}{3} - 2$

Step 2

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

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Step 2.1

Subtract $\frac{4 y}{3}$ from both sides of the inequality.

$y - \frac{2}{3} - \frac{4 y}{3} > - 2$

Step 2.2

To write $y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y \cdot \frac{3}{3} - \frac{4 y}{3} - \frac{2}{3} > - 2$

Step 2.3

Combine $y$ and $\frac{3}{3}$ .

$\frac{y \cdot 3}{3} - \frac{4 y}{3} - \frac{2}{3} > - 2$

Step 2.4

Combine the numerators over the common denominator.

$\frac{y \cdot 3 - 4 y}{3} - \frac{2}{3} > - 2$

Step 2.5

Combine the numerators over the common denominator.

$\frac{y \cdot 3 - 4 y - 2}{3} > - 2$

Step 2.6

Move $3$ to the left of $y$ .

$\frac{3 y - 4 y - 2}{3} > - 2$

Step 2.7

Subtract $4 y$ from $3 y$ .

$\frac{- y - 2}{3} > - 2$

Step 2.8

Factor $- 1$ out of $- y$ .

$\frac{- (y) - 2}{3} > - 2$

Step 2.9

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 2.10

Factor $- 1$ out of $- (y) - 1 (2)$ .

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

Step 2.11

Rewrite $- (y + 2)$ as $- 1 (y + 2)$ .

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

Step 2.12

Move the negative in front of the fraction.

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

Step 3

Divide each term in $- \frac{y + 2}{3} > - 2$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- \frac{y + 2}{3} > - 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2

Divide $\frac{y + 2}{3}$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $- 2$ by $- 1$ .

$\frac{y + 2}{3} < 2$

Step 4

Multiply both sides by $3$ .

$\frac{y + 2}{3} \cdot 3 < 2 \cdot 3$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $3$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{y + 2}{3} \cdot 3 < 2 \cdot 3$

Step 5.1.1.2

Rewrite the expression.

$y + 2 < 2 \cdot 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $2$ by $3$ .

$y + 2 < 6$

Step 6

Move all terms not containing $y$ to the right side of the inequality.

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Step 6.1

Subtract $2$ from both sides of the inequality.

$y < 6 - 2$

Step 6.2

Subtract $2$ from $6$ .

$y < 4$

Step 7

Convert the inequality to interval notation.

$(- \infty, 4)$

Step 8