Pre-Algebra Examples

$| 3 z - 2 | + 5 > 9$

Step 1

Write $| 3 z - 2 | + 5 > 9$ 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.

$3 z - 2 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Add $2$ to both sides of the inequality.

$3 z \geq 2$

Step 1.2.2

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

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Step 1.2.2.1

Divide each term in $3 z \geq 2$ by $3$ .

$\frac{3 z}{3} \geq \frac{2}{3}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{3 z}{3} \geq \frac{2}{3}$

Step 1.2.2.2.1.2

Divide $z$ by $1$ .

$z \geq \frac{2}{3}$

Step 1.3

In the piece where $3 z - 2$ is non-negative, remove the absolute value.

$3 z - 2 + 5 > 9$

Step 1.4

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

$3 z - 2 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Add $2$ to both sides of the inequality.

$3 z < 2$

Step 1.5.2

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

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Step 1.5.2.1

Divide each term in $3 z < 2$ by $3$ .

$\frac{3 z}{3} < \frac{2}{3}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $3$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{3 z}{3} < \frac{2}{3}$

Step 1.5.2.2.1.2

Divide $z$ by $1$ .

$z < \frac{2}{3}$

Step 1.6

In the piece where $3 z - 2$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 z - 2) + 5 > 9$

Step 1.7

Write as a piecewise.

${\begin{cases} 3 z - 2 + 5 > 9 & z \geq \frac{2}{3} \\ - (3 z - 2) + 5 > 9 & z < \frac{2}{3} \end{cases}$

Step 1.8

Add $- 2$ and $5$ .

${\begin{cases} 3 z + 3 > 9 & z \geq \frac{2}{3} \\ - (3 z - 2) + 5 > 9 & z < \frac{2}{3} \end{cases}$

Step 1.9

Simplify $- (3 z - 2) + 5 > 9$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 3 z + 3 > 9 & z \geq \frac{2}{3} \\ - (3 z) - - 2 + 5 > 9 & z < \frac{2}{3} \end{cases}$

Step 1.9.1.2

Multiply $3$ by $- 1$ .

${\begin{cases} 3 z + 3 > 9 & z \geq \frac{2}{3} \\ - 3 z - - 2 + 5 > 9 & z < \frac{2}{3} \end{cases}$

Step 1.9.1.3

Multiply $- 1$ by $- 2$ .

${\begin{cases} 3 z + 3 > 9 & z \geq \frac{2}{3} \\ - 3 z + 2 + 5 > 9 & z < \frac{2}{3} \end{cases}$

Step 1.9.2

Add $2$ and $5$ .

${\begin{cases} 3 z + 3 > 9 & z \geq \frac{2}{3} \\ - 3 z + 7 > 9 & z < \frac{2}{3} \end{cases}$

Step 2

Solve $3 z + 3 > 9$ for $z$ .

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Step 2.1

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

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Step 2.1.1

Subtract $3$ from both sides of the inequality.

$3 z > 9 - 3$

Step 2.1.2

Subtract $3$ from $9$ .

$3 z > 6$

Step 2.2

Divide each term in $3 z > 6$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 z > 6$ by $3$ .

$\frac{3 z}{3} > \frac{6}{3}$

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.

$\frac{3 z}{3} > \frac{6}{3}$

Step 2.2.2.1.2

Divide $z$ by $1$ .

$z > \frac{6}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $6$ by $3$ .

$z > 2$

Step 3

Solve $- 3 z + 7 > 9$ for $z$ .

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Step 3.1

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

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Step 3.1.1

Subtract $7$ from both sides of the inequality.

$- 3 z > 9 - 7$

Step 3.1.2

Subtract $7$ from $9$ .

$- 3 z > 2$

Step 3.2

Divide each term in $- 3 z > 2$ by $- 3$ and simplify.

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Step 3.2.1

Divide each term in $- 3 z > 2$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $z$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

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

Step 4

Find the union of the solutions.

$z < - \frac{2}{3}$ or $z > 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$z < - \frac{2}{3} or z > 2$

Interval Notation:

$(- \infty, - \frac{2}{3}) \cup (2, \infty)$

Step 6