Algebra Examples

$2 | 3 - 3 x | - 2 < 22$

Step 1

Write $2 | 3 - 3 x | - 2 < 22$ 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.

$3 - 3 x \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract $3$ from both sides of the inequality.

$- 3 x \geq - 3$

Step 1.2.2

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

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Step 1.2.2.1

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

$\frac{- 3 x}{- 3} \leq \frac{- 3}{- 3}$

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.

$\frac{- 3 x}{- 3} \leq \frac{- 3}{- 3}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 3}{- 3}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 3$ by $- 3$ .

$x \leq 1$

Step 1.3

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

$2 (3 - 3 x) - 2 < 22$

Step 1.4

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

$3 - 3 x < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Subtract $3$ from both sides of the inequality.

$- 3 x < - 3$

Step 1.5.2

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

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Step 1.5.2.1

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

$\frac{- 3 x}{- 3} > \frac{- 3}{- 3}$

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.

$\frac{- 3 x}{- 3} > \frac{- 3}{- 3}$

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 3}{- 3}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Divide $- 3$ by $- 3$ .

$x > 1$

Step 1.6

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

$2 (- (3 - 3 x)) - 2 < 22$

Step 1.7

Write as a piecewise.

${\begin{cases} 2 (3 - 3 x) - 2 < 22 & x \leq 1 \\ 2 (- (3 - 3 x)) - 2 < 22 & x > 1 \end{cases}$

Step 1.8

Simplify $2 (3 - 3 x) - 2 < 22$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Apply the distributive property.

${\begin{cases} 2 \cdot 3 + 2 (- 3 x) - 2 < 22 & x \leq 1 \\ 2 (- (3 - 3 x)) - 2 < 22 & x > 1 \end{cases}$

Step 1.8.1.2

Multiply $2$ by $3$ .

${\begin{cases} 6 + 2 (- 3 x) - 2 < 22 & x \leq 1 \\ 2 (- (3 - 3 x)) - 2 < 22 & x > 1 \end{cases}$

Step 1.8.1.3

Multiply $- 3$ by $2$ .

${\begin{cases} 6 - 6 x - 2 < 22 & x \leq 1 \\ 2 (- (3 - 3 x)) - 2 < 22 & x > 1 \end{cases}$

Step 1.8.2

Subtract $2$ from $6$ .

${\begin{cases} - 6 x + 4 < 22 & x \leq 1 \\ 2 (- (3 - 3 x)) - 2 < 22 & x > 1 \end{cases}$

Step 1.9

Simplify $2 (- (3 - 3 x)) - 2 < 22$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} - 6 x + 4 < 22 & x \leq 1 \\ 2 (- 1 \cdot 3 - (- 3 x)) - 2 < 22 & x > 1 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $3$ .

${\begin{cases} - 6 x + 4 < 22 & x \leq 1 \\ 2 (- 3 - (- 3 x)) - 2 < 22 & x > 1 \end{cases}$

Step 1.9.1.3

Multiply $- 3$ by $- 1$ .

${\begin{cases} - 6 x + 4 < 22 & x \leq 1 \\ 2 (- 3 + 3 x) - 2 < 22 & x > 1 \end{cases}$

Step 1.9.1.4

Apply the distributive property.

${\begin{cases} - 6 x + 4 < 22 & x \leq 1 \\ 2 \cdot - 3 + 2 (3 x) - 2 < 22 & x > 1 \end{cases}$

Step 1.9.1.5

Multiply $2$ by $- 3$ .

${\begin{cases} - 6 x + 4 < 22 & x \leq 1 \\ - 6 + 2 (3 x) - 2 < 22 & x > 1 \end{cases}$

Step 1.9.1.6

Multiply $3$ by $2$ .

${\begin{cases} - 6 x + 4 < 22 & x \leq 1 \\ - 6 + 6 x - 2 < 22 & x > 1 \end{cases}$

Step 1.9.2

Subtract $2$ from $- 6$ .

${\begin{cases} - 6 x + 4 < 22 & x \leq 1 \\ 6 x - 8 < 22 & x > 1 \end{cases}$

Step 2

Solve $- 6 x + 4 < 22$ when $x \leq 1$ .

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Step 2.1

Solve $- 6 x + 4 < 22$ for $x$ .

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Step 2.1.1

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

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Step 2.1.1.1

Subtract $4$ from both sides of the inequality.

$- 6 x < 22 - 4$

Step 2.1.1.2

Subtract $4$ from $22$ .

$- 6 x < 18$

Step 2.1.2

Divide each term in $- 6 x < 18$ by $- 6$ and simplify.

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Step 2.1.2.1

Divide each term in $- 6 x < 18$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 x}{- 6} > \frac{18}{- 6}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $- 6$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} > \frac{18}{- 6}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{18}{- 6}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Divide $18$ by $- 6$ .

$x > - 3$

Step 2.2

Find the intersection of $x > - 3$ and $x \leq 1$ .

$- 3 < x \leq 1$

Step 3

Solve $6 x - 8 < 22$ when $x > 1$ .

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Step 3.1

Solve $6 x - 8 < 22$ for $x$ .

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Step 3.1.1

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

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Step 3.1.1.1

Add $8$ to both sides of the inequality.

$6 x < 22 + 8$

Step 3.1.1.2

Add $22$ and $8$ .

$6 x < 30$

Step 3.1.2

Divide each term in $6 x < 30$ by $6$ and simplify.

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Step 3.1.2.1

Divide each term in $6 x < 30$ by $6$ .

$\frac{6 x}{6} < \frac{30}{6}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $6$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} < \frac{30}{6}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{30}{6}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $30$ by $6$ .

$x < 5$

Step 3.2

Find the intersection of $x < 5$ and $x > 1$ .

$1 < x < 5$

Step 4

Find the union of the solutions.

$- 3 < x < 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 3 < x < 5$

Interval Notation:

$(- 3, 5)$

Step 6