Solve the Inequality for x |3x|>=x+20

$| 3 x | \geq x + 20$

Write $| 3 x | \geq x + 20$ as a piecewise.

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

$3 x \geq 0$

Divide each term in $3 x \geq 0$ by $3$ and simplify.

Divide each term in $3 x \geq 0$ by $3$ .

$\frac{3 x}{3} \geq \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{0}{3}$

Divide $x$ by $1$ .

$x \geq \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x \geq 0$

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

$3 x \geq x + 20$

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

$3 x < 0$

Divide each term in $3 x < 0$ by $3$ and simplify.

Divide each term in $3 x < 0$ by $3$ .

$\frac{3 x}{3} < \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{0}{3}$

Divide $x$ by $1$ .

$x < \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x < 0$

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

$- 3 x \geq x + 20$

Write as a piecewise.

${\begin{cases} 3 x \geq x + 20 & x \geq 0 \\ - 3 x \geq x + 20 & x < 0 \end{cases}$

Solve $3 x \geq x + 20$ for $x$ .

Move all terms containing $x$ to the left side of the inequality.

Subtract $x$ from both sides of the inequality.

$3 x - x \geq 20$

Subtract $x$ from $3 x$ .

$2 x \geq 20$

Divide each term in $2 x \geq 20$ by $2$ and simplify.

Divide each term in $2 x \geq 20$ by $2$ .

$\frac{2 x}{2} \geq \frac{20}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{20}{2}$

Divide $x$ by $1$ .

$x \geq \frac{20}{2}$

Simplify the right side.

Divide $20$ by $2$ .

$x \geq 10$

Solve $- 3 x \geq x + 20$ for $x$ .

Move all terms containing $x$ to the left side of the inequality.

Subtract $x$ from both sides of the inequality.

$- 3 x - x \geq 20$

Subtract $x$ from $- 3 x$ .

$- 4 x \geq 20$

Divide each term in $- 4 x \geq 20$ by $- 4$ and simplify.

Divide each term in $- 4 x \geq 20$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} \leq \frac{20}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} \leq \frac{20}{- 4}$

Divide $x$ by $1$ .

$x \leq \frac{20}{- 4}$

Simplify the right side.

Divide $20$ by $- 4$ .

$x \leq - 5$

Find the union of the solutions.

$x \leq - 5$ or $x \geq 10$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 5 or x \geq 10$

Interval Notation:

$(- \infty, - 5] \cup [10, \infty)$

Solve the Inequality for x |3x|>=x+20

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