Solve for x |6x|+3<=0

$| 6 x | + 3 \leq 0$

Step 1

Write $| 6 x | + 3 \leq 0$ as a piecewise.

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

$6 x \geq 0$

Divide each term in $6 x \geq 0$ by $6$ and simplify.

Divide each term in $6 x \geq 0$ by $6$ .

$\frac{6 x}{6} \geq \frac{0}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} \geq \frac{0}{6}$

Divide $x$ by $1$ .

$x \geq \frac{0}{6}$

Simplify the right side.

Divide $0$ by $6$ .

$x \geq 0$

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

$6 x + 3 \leq 0$

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

$6 x < 0$

Divide each term in $6 x < 0$ by $6$ and simplify.

Divide each term in $6 x < 0$ by $6$ .

$\frac{6 x}{6} < \frac{0}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} < \frac{0}{6}$

Divide $x$ by $1$ .

$x < \frac{0}{6}$

Simplify the right side.

Divide $0$ by $6$ .

$x < 0$

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

$- 6 x + 3 \leq 0$

Write as a piecewise.

${\begin{cases} 6 x + 3 \leq 0 & x \geq 0 \\ - 6 x + 3 \leq 0 & x < 0 \end{cases}$

Step 2

Solve $6 x + 3 \leq 0$ when $x \geq 0$ .

Solve $6 x + 3 \leq 0$ for $x$ .

Subtract $3$ from both sides of the inequality.

$6 x \leq - 3$

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

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

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $- 3$ and $6$ .

Factor $3$ out of $- 3$ .

$x \leq \frac{3 (- 1)}{6}$

Cancel the common factors.

Factor $3$ out of $6$ .

$x \leq \frac{3 \cdot - 1}{3 \cdot 2}$

Cancel the common factor.

$x \leq \frac{3 \cdot - 1}{3 \cdot 2}$

Rewrite the expression.

$x \leq \frac{- 1}{2}$

Move the negative in front of the fraction.

$x \leq - \frac{1}{2}$

Find the intersection of $x \leq - \frac{1}{2}$ and $x \geq 0$ .

No solution

Step 3

Solve $- 6 x + 3 \leq 0$ when $x < 0$ .

Solve $- 6 x + 3 \leq 0$ for $x$ .

Subtract $3$ from both sides of the inequality.

$- 6 x \leq - 3$

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

Divide each term in $- 6 x \leq - 3$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $- 3$ and $- 6$ .

Factor $- 3$ out of $- 3$ .

$x \geq \frac{- 3 (1)}{- 6}$

Cancel the common factors.

Factor $- 3$ out of $- 6$ .

$x \geq \frac{- 3 \cdot 1}{- 3 \cdot 2}$

Cancel the common factor.

$x \geq \frac{- 3 \cdot 1}{- 3 \cdot 2}$

Rewrite the expression.

$x \geq \frac{1}{2}$

Find the intersection of $x \geq \frac{1}{2}$ and $x < 0$ .

No solution

Step 4

Find the union of the solutions.

No solution

Solve for x |6x|+3<=0

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