Graph 6|x+1|>=18

$6 | x + 1 | \geq 18$

Step 1

Write $6 | x + 1 | \geq 18$ as a piecewise.

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

$x + 1 \geq 0$

Subtract $1$ from both sides of the inequality.

$x \geq - 1$

In the piece where $x + 1$ is non-negative, remove the absolute value.

$6 (x + 1) \geq 18$

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

$x + 1 < 0$

Subtract $1$ from both sides of the inequality.

$x < - 1$

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

$6 (- (x + 1)) \geq 18$

Write as a piecewise.

${\begin{cases} 6 (x + 1) \geq 18 & x \geq - 1 \\ 6 (- (x + 1)) \geq 18 & x < - 1 \end{cases}$

Simplify $6 (x + 1) \geq 18$ .

Apply the distributive property.

${\begin{cases} 6 x + 6 \cdot 1 \geq 18 & x \geq - 1 \\ 6 (- (x + 1)) \geq 18 & x < - 1 \end{cases}$

Multiply $6$ by $1$ .

${\begin{cases} 6 x + 6 \geq 18 & x \geq - 1 \\ 6 (- (x + 1)) \geq 18 & x < - 1 \end{cases}$

Simplify $6 (- (x + 1)) \geq 18$ .

Apply the distributive property.

${\begin{cases} 6 x + 6 \geq 18 & x \geq - 1 \\ 6 (- x - 1 \cdot 1) \geq 18 & x < - 1 \end{cases}$

Multiply $- 1$ by $1$ .

${\begin{cases} 6 x + 6 \geq 18 & x \geq - 1 \\ 6 (- x - 1) \geq 18 & x < - 1 \end{cases}$

Apply the distributive property.

${\begin{cases} 6 x + 6 \geq 18 & x \geq - 1 \\ 6 (- x) + 6 \cdot - 1 \geq 18 & x < - 1 \end{cases}$

Multiply $- 1$ by $6$ .

${\begin{cases} 6 x + 6 \geq 18 & x \geq - 1 \\ - 6 x + 6 \cdot - 1 \geq 18 & x < - 1 \end{cases}$

Multiply $6$ by $- 1$ .

${\begin{cases} 6 x + 6 \geq 18 & x \geq - 1 \\ - 6 x - 6 \geq 18 & x < - 1 \end{cases}$

Step 2

Solve $6 x + 6 \geq 18$ for $x$ .

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

Subtract $6$ from both sides of the inequality.

$6 x \geq 18 - 6$

Subtract $6$ from $18$ .

$6 x \geq 12$

Divide each term in $6 x \geq 12$ by $6$ and simplify.

Divide each term in $6 x \geq 12$ by $6$ .

$\frac{6 x}{6} \geq \frac{12}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} \geq \frac{12}{6}$

Divide $x$ by $1$ .

$x \geq \frac{12}{6}$

Simplify the right side.

Divide $12$ by $6$ .

$x \geq 2$

Step 3

Solve $- 6 x - 6 \geq 18$ for $x$ .

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

Add $6$ to both sides of the inequality.

$- 6 x \geq 18 + 6$

Add $18$ and $6$ .

$- 6 x \geq 24$

Divide each term in $- 6 x \geq 24$ by $- 6$ and simplify.

Divide each term in $- 6 x \geq 24$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 x}{- 6} \leq \frac{24}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} \leq \frac{24}{- 6}$

Divide $x$ by $1$ .

$x \leq \frac{24}{- 6}$

Simplify the right side.

Divide $24$ by $- 6$ .

$x \leq - 4$

Step 4

Find the union of the solutions.

$x \leq - 4$ or $x \geq 2$

Step 5

Graph 6|x+1|>=18

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