Finite Math Examples

$| \frac{6 y + 4}{2} | + 8 > 2$

Step 1

Write $| \frac{6 y + 4}{2} | + 8 > 2$ as a piecewise.

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Step 1.1

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

$\frac{6 y + 4}{2} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Cancel the common factor of $6 y + 4$ and $2$ .

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Step 1.2.1.1

Factor $2$ out of $6 y$ .

$\frac{2 (3 y) + 4}{2} \geq 0$

Step 1.2.1.2

Factor $2$ out of $4$ .

$\frac{2 (3 y) + 2 \cdot 2}{2} \geq 0$

Step 1.2.1.3

Factor $2$ out of $2 (3 y) + 2 (2)$ .

$\frac{2 (3 y + 2)}{2} \geq 0$

Step 1.2.1.4

Cancel the common factors.

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Step 1.2.1.4.1

Factor $2$ out of $2$ .

$\frac{2 (3 y + 2)}{2 (1)} \geq 0$

Step 1.2.1.4.2

Cancel the common factor.

$\frac{2 (3 y + 2)}{2 \cdot 1} \geq 0$

Step 1.2.1.4.3

Rewrite the expression.

$\frac{3 y + 2}{1} \geq 0$

Step 1.2.1.4.4

Divide $3 y + 2$ by $1$ .

$3 y + 2 \geq 0$

Step 1.2.2

Solve for $y$ .

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Step 1.2.2.1

Subtract $2$ from both sides of the inequality.

$3 y \geq - 2$

Step 1.2.2.2

Divide each term in $3 y \geq - 2$ by $3$ and simplify.

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Step 1.2.2.2.1

Divide each term in $3 y \geq - 2$ by $3$ .

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

Step 1.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.3.1

Move the negative in front of the fraction.

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

Step 1.3

In the piece where $\frac{6 y + 4}{2}$ is non-negative, remove the absolute value.

$\frac{6 y + 4}{2} + 8 > 2$

Step 1.4

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

$\frac{6 y + 4}{2} < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Cancel the common factor of $6 y + 4$ and $2$ .

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Step 1.5.1.1

Factor $2$ out of $6 y$ .

$\frac{2 (3 y) + 4}{2} < 0$

Step 1.5.1.2

Factor $2$ out of $4$ .

$\frac{2 (3 y) + 2 \cdot 2}{2} < 0$

Step 1.5.1.3

Factor $2$ out of $2 (3 y) + 2 (2)$ .

$\frac{2 (3 y + 2)}{2} < 0$

Step 1.5.1.4

Cancel the common factors.

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Step 1.5.1.4.1

Factor $2$ out of $2$ .

$\frac{2 (3 y + 2)}{2 (1)} < 0$

Step 1.5.1.4.2

Cancel the common factor.

$\frac{2 (3 y + 2)}{2 \cdot 1} < 0$

Step 1.5.1.4.3

Rewrite the expression.

$\frac{3 y + 2}{1} < 0$

Step 1.5.1.4.4

Divide $3 y + 2$ by $1$ .

$3 y + 2 < 0$

Step 1.5.2

Solve for $y$ .

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Step 1.5.2.1

Subtract $2$ from both sides of the inequality.

$3 y < - 2$

Step 1.5.2.2

Divide each term in $3 y < - 2$ by $3$ and simplify.

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Step 1.5.2.2.1

Divide each term in $3 y < - 2$ by $3$ .

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

Step 1.5.2.2.2

Simplify the left side.

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Step 1.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.5.2.2.2.1.1

Cancel the common factor.

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

Step 1.5.2.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.5.2.2.3

Simplify the right side.

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Step 1.5.2.2.3.1

Move the negative in front of the fraction.

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

Step 1.6

In the piece where $\frac{6 y + 4}{2}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{6 y + 4}{2} + 8 > 2$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{6 y + 4}{2} + 8 > 2 & y \geq - \frac{2}{3} \\ - \frac{6 y + 4}{2} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.8

Simplify $\frac{6 y + 4}{2} + 8 > 2$ .

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Step 1.8.1

Cancel the common factor of $6 y + 4$ and $2$ .

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Step 1.8.1.1

Factor $2$ out of $6 y$ .

${\begin{cases} \frac{2 (3 y) + 4}{2} + 8 > 2 & y \geq - \frac{2}{3} \\ - \frac{6 y + 4}{2} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.8.1.2

Factor $2$ out of $4$ .

${\begin{cases} \frac{2 (3 y) + 2 \cdot 2}{2} + 8 > 2 & y \geq - \frac{2}{3} \\ - \frac{6 y + 4}{2} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.8.1.3

Factor $2$ out of $2 (3 y) + 2 (2)$ .

${\begin{cases} \frac{2 (3 y + 2)}{2} + 8 > 2 & y \geq - \frac{2}{3} \\ - \frac{6 y + 4}{2} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.8.1.4

Cancel the common factors.

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Step 1.8.1.4.1

Factor $2$ out of $2$ .

${\begin{cases} \frac{2 (3 y + 2)}{2 (1)} + 8 > 2 & y \geq - \frac{2}{3} \\ - \frac{6 y + 4}{2} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.8.1.4.2

Cancel the common factor.

${\begin{cases} \frac{2 (3 y + 2)}{2 \cdot 1} + 8 > 2 & y \geq - \frac{2}{3} \\ - \frac{6 y + 4}{2} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.8.1.4.3

Rewrite the expression.

${\begin{cases} \frac{3 y + 2}{1} + 8 > 2 & y \geq - \frac{2}{3} \\ - \frac{6 y + 4}{2} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.8.1.4.4

Divide $3 y + 2$ by $1$ .

${\begin{cases} 3 y + 2 + 8 > 2 & y \geq - \frac{2}{3} \\ - \frac{6 y + 4}{2} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.8.2

Add $2$ and $8$ .

${\begin{cases} 3 y + 10 > 2 & y \geq - \frac{2}{3} \\ - \frac{6 y + 4}{2} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.9

Simplify $- \frac{6 y + 4}{2} + 8 > 2$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Cancel the common factor of $6 y + 4$ and $2$ .

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Step 1.9.1.1.1

Factor $2$ out of $6 y$ .

${\begin{cases} 3 y + 10 > 2 & y \geq - \frac{2}{3} \\ - \frac{2 (3 y) + 4}{2} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.9.1.1.2

Factor $2$ out of $4$ .

${\begin{cases} 3 y + 10 > 2 & y \geq - \frac{2}{3} \\ - \frac{2 (3 y) + 2 \cdot 2}{2} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.9.1.1.3

Factor $2$ out of $2 (3 y) + 2 (2)$ .

${\begin{cases} 3 y + 10 > 2 & y \geq - \frac{2}{3} \\ - \frac{2 (3 y + 2)}{2} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.9.1.1.4

Cancel the common factors.

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Step 1.9.1.1.4.1

Factor $2$ out of $2$ .

${\begin{cases} 3 y + 10 > 2 & y \geq - \frac{2}{3} \\ - \frac{2 (3 y + 2)}{2 (1)} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.9.1.1.4.2

Cancel the common factor.

${\begin{cases} 3 y + 10 > 2 & y \geq - \frac{2}{3} \\ - \frac{2 (3 y + 2)}{2 \cdot 1} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.9.1.1.4.3

Rewrite the expression.

${\begin{cases} 3 y + 10 > 2 & y \geq - \frac{2}{3} \\ - \frac{3 y + 2}{1} + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.9.1.1.4.4

Divide $3 y + 2$ by $1$ .

${\begin{cases} 3 y + 10 > 2 & y \geq - \frac{2}{3} \\ - (3 y + 2) + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.9.1.2

Apply the distributive property.

${\begin{cases} 3 y + 10 > 2 & y \geq - \frac{2}{3} \\ - (3 y) - 1 \cdot 2 + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.9.1.3

Multiply $3$ by $- 1$ .

${\begin{cases} 3 y + 10 > 2 & y \geq - \frac{2}{3} \\ - 3 y - 1 \cdot 2 + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.9.1.4

Multiply $- 1$ by $2$ .

${\begin{cases} 3 y + 10 > 2 & y \geq - \frac{2}{3} \\ - 3 y - 2 + 8 > 2 & y < - \frac{2}{3} \end{cases}$

Step 1.9.2

Add $- 2$ and $8$ .

${\begin{cases} 3 y + 10 > 2 & y \geq - \frac{2}{3} \\ - 3 y + 6 > 2 & y < - \frac{2}{3} \end{cases}$

Step 2

Solve $3 y + 10 > 2$ when $y \geq - \frac{2}{3}$ .

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Step 2.1

Solve $3 y + 10 > 2$ for $y$ .

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Step 2.1.1

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

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Step 2.1.1.1

Subtract $10$ from both sides of the inequality.

$3 y > 2 - 10$

Step 2.1.1.2

Subtract $10$ from $2$ .

$3 y > - 8$

Step 2.1.2

Divide each term in $3 y > - 8$ by $3$ and simplify.

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Step 2.1.2.1

Divide each term in $3 y > - 8$ by $3$ .

$\frac{3 y}{3} > \frac{- 8}{3}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $3$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} > \frac{- 8}{3}$

Step 2.1.2.2.1.2

Divide $y$ by $1$ .

$y > \frac{- 8}{3}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Move the negative in front of the fraction.

$y > - \frac{8}{3}$

Step 2.2

Find the intersection of $y > - \frac{8}{3}$ and $y \geq - \frac{2}{3}$ .

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

Step 3

Solve $- 3 y + 6 > 2$ when $y < - \frac{2}{3}$ .

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Step 3.1

Solve $- 3 y + 6 > 2$ for $y$ .

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Step 3.1.1

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

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Step 3.1.1.1

Subtract $6$ from both sides of the inequality.

$- 3 y > 2 - 6$

Step 3.1.1.2

Subtract $6$ from $2$ .

$- 3 y > - 4$

Step 3.1.2

Divide each term in $- 3 y > - 4$ by $- 3$ and simplify.

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Step 3.1.2.1

Divide each term in $- 3 y > - 4$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 y}{- 3} < \frac{- 4}{- 3}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} < \frac{- 4}{- 3}$

Step 3.1.2.2.1.2

Divide $y$ by $1$ .

$y < \frac{- 4}{- 3}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Dividing two negative values results in a positive value.

$y < \frac{4}{3}$

Step 3.2

Find the intersection of $y < \frac{4}{3}$ and $y < - \frac{2}{3}$ .

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

Step 4

Find the union of the solutions.

All real numbers

Step 5

Convert the inequality to interval notation.

$(- \infty, \infty)$

Step 6