Solve for j |3j-1|+6>0

$| 3 j - 1 | + 6 > 0$

Step 1

Write $| 3 j - 1 | + 6 > 0$ as a piecewise.

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

$3 j - 1 \geq 0$

Solve the inequality.

Add $1$ to both sides of the inequality.

$3 j \geq 1$

Divide each term in $3 j \geq 1$ by $3$ and simplify.

Divide each term in $3 j \geq 1$ by $3$ .

$\frac{3 j}{3} \geq \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 j}{3} \geq \frac{1}{3}$

Divide $j$ by $1$ .

$j \geq \frac{1}{3}$

In the piece where $3 j - 1$ is non-negative, remove the absolute value.

$3 j - 1 + 6 > 0$

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

$3 j - 1 < 0$

Solve the inequality.

Add $1$ to both sides of the inequality.

$3 j < 1$

Divide each term in $3 j < 1$ by $3$ and simplify.

Divide each term in $3 j < 1$ by $3$ .

$\frac{3 j}{3} < \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 j}{3} < \frac{1}{3}$

Divide $j$ by $1$ .

$j < \frac{1}{3}$

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

$- (3 j - 1) + 6 > 0$

Write as a piecewise.

${\begin{cases} 3 j - 1 + 6 > 0 & j \geq \frac{1}{3} \\ - (3 j - 1) + 6 > 0 & j < \frac{1}{3} \end{cases}$

Add $- 1$ and $6$ .

${\begin{cases} 3 j + 5 > 0 & j \geq \frac{1}{3} \\ - (3 j - 1) + 6 > 0 & j < \frac{1}{3} \end{cases}$

Simplify $- (3 j - 1) + 6 > 0$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 3 j + 5 > 0 & j \geq \frac{1}{3} \\ - (3 j) - - 1 + 6 > 0 & j < \frac{1}{3} \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} 3 j + 5 > 0 & j \geq \frac{1}{3} \\ - 3 j - - 1 + 6 > 0 & j < \frac{1}{3} \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} 3 j + 5 > 0 & j \geq \frac{1}{3} \\ - 3 j + 1 + 6 > 0 & j < \frac{1}{3} \end{cases}$

Add $1$ and $6$ .

${\begin{cases} 3 j + 5 > 0 & j \geq \frac{1}{3} \\ - 3 j + 7 > 0 & j < \frac{1}{3} \end{cases}$

Step 2

Solve $3 j + 5 > 0$ when $j \geq \frac{1}{3}$ .

Solve $3 j + 5 > 0$ for $j$ .

Subtract $5$ from both sides of the inequality.

$3 j > - 5$

Divide each term in $3 j > - 5$ by $3$ and simplify.

Divide each term in $3 j > - 5$ by $3$ .

$\frac{3 j}{3} > \frac{- 5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 j}{3} > \frac{- 5}{3}$

Divide $j$ by $1$ .

$j > \frac{- 5}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$j > - \frac{5}{3}$

Find the intersection of $j > - \frac{5}{3}$ and $j \geq \frac{1}{3}$ .

$j \geq \frac{1}{3}$

Step 3

Solve $- 3 j + 7 > 0$ when $j < \frac{1}{3}$ .

Solve $- 3 j + 7 > 0$ for $j$ .

Subtract $7$ from both sides of the inequality.

$- 3 j > - 7$

Divide each term in $- 3 j > - 7$ by $- 3$ and simplify.

Divide each term in $- 3 j > - 7$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 j}{- 3} < \frac{- 7}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 j}{- 3} < \frac{- 7}{- 3}$

Divide $j$ by $1$ .

$j < \frac{- 7}{- 3}$

Simplify the right side.

Dividing two negative values results in a positive value.

$j < \frac{7}{3}$

Find the intersection of $j < \frac{7}{3}$ and $j < \frac{1}{3}$ .

$j < \frac{1}{3}$

Step 4

Find the union of the solutions for any value of $j$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Step 6

Solve for j |3j-1|+6>0

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