Algebra Examples

$- 4 | 6 + 3 x | + 3 \leq - 21$

Step 1

Write $- 4 | 6 + 3 x | + 3 \leq - 21$ 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.

$6 + 3 x \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract $6$ from both sides of the inequality.

$3 x \geq - 6$

Step 1.2.2

Divide each term in $3 x \geq - 6$ by $3$ and simplify.

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Step 1.2.2.1

Divide each term in $3 x \geq - 6$ by $3$ .

$\frac{3 x}{3} \geq \frac{- 6}{3}$

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.

$\frac{3 x}{3} \geq \frac{- 6}{3}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 6}{3}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 6$ by $3$ .

$x \geq - 2$

Step 1.3

In the piece where $6 + 3 x$ is non-negative, remove the absolute value.

$- 4 (6 + 3 x) + 3 \leq - 21$

Step 1.4

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

$6 + 3 x < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Subtract $6$ from both sides of the inequality.

$3 x < - 6$

Step 1.5.2

Divide each term in $3 x < - 6$ by $3$ and simplify.

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Step 1.5.2.1

Divide each term in $3 x < - 6$ by $3$ .

$\frac{3 x}{3} < \frac{- 6}{3}$

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.

$\frac{3 x}{3} < \frac{- 6}{3}$

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 6}{3}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Divide $- 6$ by $3$ .

$x < - 2$

Step 1.6

In the piece where $6 + 3 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- 4 (- (6 + 3 x)) + 3 \leq - 21$

Step 1.7

Write as a piecewise.

${\begin{cases} - 4 (6 + 3 x) + 3 \leq - 21 & x \geq - 2 \\ - 4 (- (6 + 3 x)) + 3 \leq - 21 & x < - 2 \end{cases}$

Step 1.8

Simplify $- 4 (6 + 3 x) + 3 \leq - 21$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Apply the distributive property.

${\begin{cases} - 4 \cdot 6 - 4 (3 x) + 3 \leq - 21 & x \geq - 2 \\ - 4 (- (6 + 3 x)) + 3 \leq - 21 & x < - 2 \end{cases}$

Step 1.8.1.2

Multiply $- 4$ by $6$ .

${\begin{cases} - 24 - 4 (3 x) + 3 \leq - 21 & x \geq - 2 \\ - 4 (- (6 + 3 x)) + 3 \leq - 21 & x < - 2 \end{cases}$

Step 1.8.1.3

Multiply $3$ by $- 4$ .

${\begin{cases} - 24 - 12 x + 3 \leq - 21 & x \geq - 2 \\ - 4 (- (6 + 3 x)) + 3 \leq - 21 & x < - 2 \end{cases}$

Step 1.8.2

Add $- 24$ and $3$ .

${\begin{cases} - 12 x - 21 \leq - 21 & x \geq - 2 \\ - 4 (- (6 + 3 x)) + 3 \leq - 21 & x < - 2 \end{cases}$

Step 1.9

Simplify $- 4 (- (6 + 3 x)) + 3 \leq - 21$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} - 12 x - 21 \leq - 21 & x \geq - 2 \\ - 4 (- 1 \cdot 6 - (3 x)) + 3 \leq - 21 & x < - 2 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $6$ .

${\begin{cases} - 12 x - 21 \leq - 21 & x \geq - 2 \\ - 4 (- 6 - (3 x)) + 3 \leq - 21 & x < - 2 \end{cases}$

Step 1.9.1.3

Multiply $3$ by $- 1$ .

${\begin{cases} - 12 x - 21 \leq - 21 & x \geq - 2 \\ - 4 (- 6 - 3 x) + 3 \leq - 21 & x < - 2 \end{cases}$

Step 1.9.1.4

Apply the distributive property.

${\begin{cases} - 12 x - 21 \leq - 21 & x \geq - 2 \\ - 4 \cdot - 6 - 4 (- 3 x) + 3 \leq - 21 & x < - 2 \end{cases}$

Step 1.9.1.5

Multiply $- 4$ by $- 6$ .

${\begin{cases} - 12 x - 21 \leq - 21 & x \geq - 2 \\ 24 - 4 (- 3 x) + 3 \leq - 21 & x < - 2 \end{cases}$

Step 1.9.1.6

Multiply $- 3$ by $- 4$ .

${\begin{cases} - 12 x - 21 \leq - 21 & x \geq - 2 \\ 24 + 12 x + 3 \leq - 21 & x < - 2 \end{cases}$

Step 1.9.2

Add $24$ and $3$ .

${\begin{cases} - 12 x - 21 \leq - 21 & x \geq - 2 \\ 12 x + 27 \leq - 21 & x < - 2 \end{cases}$

Step 2

Solve $- 12 x - 21 \leq - 21$ for $x$ .

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Step 2.1

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

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Step 2.1.1

Add $21$ to both sides of the inequality.

$- 12 x \leq - 21 + 21$

Step 2.1.2

Add $- 21$ and $21$ .

$- 12 x \leq 0$

Step 2.2

Divide each term in $- 12 x \leq 0$ by $- 12$ and simplify.

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Step 2.2.1

Divide each term in $- 12 x \leq 0$ by $- 12$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 12 x}{- 12} \geq \frac{0}{- 12}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 12$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 12 x}{- 12} \geq \frac{0}{- 12}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{- 12}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $- 12$ .

$x \geq 0$

Step 3

Solve $12 x + 27 \leq - 21$ for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $27$ from both sides of the inequality.

$12 x \leq - 21 - 27$

Step 3.1.2

Subtract $27$ from $- 21$ .

$12 x \leq - 48$

Step 3.2

Divide each term in $12 x \leq - 48$ by $12$ and simplify.

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Step 3.2.1

Divide each term in $12 x \leq - 48$ by $12$ .

$\frac{12 x}{12} \leq \frac{- 48}{12}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $12$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} \leq \frac{- 48}{12}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 48}{12}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 48$ by $12$ .

$x \leq - 4$

Step 4

Find the union of the solutions.

$x \leq - 4$ or $x \geq 0$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 4 or x \geq 0$

Interval Notation:

$(- \infty, - 4] \cup [0, \infty)$

Step 6