Pre-Algebra Examples

$| - 3 a + 2 | > 7$

Step 1

Write $| - 3 a + 2 | > 7$ as a piecewise.

Tap for more steps...

Step 1.1

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

$- 3 a + 2 \geq 0$

Step 1.2

Solve the inequality.

Tap for more steps...

Step 1.2.1

Subtract $2$ from both sides of the inequality.

$- 3 a \geq - 2$

Step 1.2.2

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

Tap for more steps...

Step 1.2.2.1

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

$\frac{- 3 a}{- 3} \leq \frac{- 2}{- 3}$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 a}{- 3} \leq \frac{- 2}{- 3}$

Step 1.2.2.2.1.2

Divide $a$ by $1$ .

$a \leq \frac{- 2}{- 3}$

Step 1.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.3.1

Dividing two negative values results in a positive value.

$a \leq \frac{2}{3}$

Step 1.3

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

$- 3 a + 2 > 7$

Step 1.4

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

$- 3 a + 2 < 0$

Step 1.5

Solve the inequality.

Tap for more steps...

Step 1.5.1

Subtract $2$ from both sides of the inequality.

$- 3 a < - 2$

Step 1.5.2

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

Tap for more steps...

Step 1.5.2.1

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

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

Step 1.5.2.2

Simplify the left side.

Tap for more steps...

Step 1.5.2.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 1.5.2.2.1.1

Cancel the common factor.

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

Step 1.5.2.2.1.2

Divide $a$ by $1$ .

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

Step 1.5.2.3

Simplify the right side.

Tap for more steps...

Step 1.5.2.3.1

Dividing two negative values results in a positive value.

$a > \frac{2}{3}$

Step 1.6

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

$- (- 3 a + 2) > 7$

Step 1.7

Write as a piecewise.

${\begin{cases} - 3 a + 2 > 7 & a \leq \frac{2}{3} \\ - (- 3 a + 2) > 7 & a > \frac{2}{3} \end{cases}$

Step 1.8

Simplify $- (- 3 a + 2) > 7$ .

Tap for more steps...

Step 1.8.1

Apply the distributive property.

${\begin{cases} - 3 a + 2 > 7 & a \leq \frac{2}{3} \\ - (- 3 a) - 1 \cdot 2 > 7 & a > \frac{2}{3} \end{cases}$

Step 1.8.2

Multiply $- 3$ by $- 1$ .

${\begin{cases} - 3 a + 2 > 7 & a \leq \frac{2}{3} \\ 3 a - 1 \cdot 2 > 7 & a > \frac{2}{3} \end{cases}$

Step 1.8.3

Multiply $- 1$ by $2$ .

${\begin{cases} - 3 a + 2 > 7 & a \leq \frac{2}{3} \\ 3 a - 2 > 7 & a > \frac{2}{3} \end{cases}$

Step 2

Solve $- 3 a + 2 > 7$ for $a$ .

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

Subtract $2$ from both sides of the inequality.

$- 3 a > 7 - 2$

Step 2.1.2

Subtract $2$ from $7$ .

$- 3 a > 5$

Step 2.2

Divide each term in $- 3 a > 5$ by $- 3$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $- 3 a > 5$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 a}{- 3} < \frac{5}{- 3}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 3 a}{- 3} < \frac{5}{- 3}$

Step 2.2.2.1.2

Divide $a$ by $1$ .

$a < \frac{5}{- 3}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Move the negative in front of the fraction.

$a < - \frac{5}{3}$

Step 3

Solve $3 a - 2 > 7$ for $a$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Add $2$ to both sides of the inequality.

$3 a > 7 + 2$

Step 3.1.2

Add $7$ and $2$ .

$3 a > 9$

Step 3.2

Divide each term in $3 a > 9$ by $3$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $3 a > 9$ by $3$ .

$\frac{3 a}{3} > \frac{9}{3}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 a}{3} > \frac{9}{3}$

Step 3.2.2.1.2

Divide $a$ by $1$ .

$a > \frac{9}{3}$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Divide $9$ by $3$ .

$a > 3$

Step 4

Find the union of the solutions.

$a < - \frac{5}{3}$ or $a > 3$

Step 5