Solve for t |3t-2|<=4

$| 3 t - 2 | \leq 4$

Step 1

Write $| 3 t - 2 | \leq 4$ as a piecewise.

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

$3 t - 2 \geq 0$

Solve the inequality.

Add $2$ to both sides of the inequality.

$3 t \geq 2$

Divide each term in $3 t \geq 2$ by $3$ and simplify.

Divide each term in $3 t \geq 2$ by $3$ .

$\frac{3 t}{3} \geq \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 t}{3} \geq \frac{2}{3}$

Divide $t$ by $1$ .

$t \geq \frac{2}{3}$

In the piece where $3 t - 2$ is non-negative, remove the absolute value.

$3 t - 2 \leq 4$

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

$3 t - 2 < 0$

Solve the inequality.

Add $2$ to both sides of the inequality.

$3 t < 2$

Divide each term in $3 t < 2$ by $3$ and simplify.

Divide each term in $3 t < 2$ by $3$ .

$\frac{3 t}{3} < \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 t}{3} < \frac{2}{3}$

Divide $t$ by $1$ .

$t < \frac{2}{3}$

In the piece where $3 t - 2$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 t - 2) \leq 4$

Write as a piecewise.

${\begin{cases} 3 t - 2 \leq 4 & t \geq \frac{2}{3} \\ - (3 t - 2) \leq 4 & t < \frac{2}{3} \end{cases}$

Simplify $- (3 t - 2) \leq 4$ .

Apply the distributive property.

${\begin{cases} 3 t - 2 \leq 4 & t \geq \frac{2}{3} \\ - (3 t) - - 2 \leq 4 & t < \frac{2}{3} \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} 3 t - 2 \leq 4 & t \geq \frac{2}{3} \\ - 3 t - - 2 \leq 4 & t < \frac{2}{3} \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} 3 t - 2 \leq 4 & t \geq \frac{2}{3} \\ - 3 t + 2 \leq 4 & t < \frac{2}{3} \end{cases}$

Step 2

Solve $3 t - 2 \leq 4$ when $t \geq \frac{2}{3}$ .

Solve $3 t - 2 \leq 4$ for $t$ .

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

Add $2$ to both sides of the inequality.

$3 t \leq 4 + 2$

Add $4$ and $2$ .

$3 t \leq 6$

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

Divide each term in $3 t \leq 6$ by $3$ .

$\frac{3 t}{3} \leq \frac{6}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 t}{3} \leq \frac{6}{3}$

Divide $t$ by $1$ .

$t \leq \frac{6}{3}$

Simplify the right side.

Divide $6$ by $3$ .

$t \leq 2$

Find the intersection of $t \leq 2$ and $t \geq \frac{2}{3}$ .

$\frac{2}{3} \leq t \leq 2$

Step 3

Solve $- 3 t + 2 \leq 4$ when $t < \frac{2}{3}$ .

Solve $- 3 t + 2 \leq 4$ for $t$ .

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

Subtract $2$ from both sides of the inequality.

$- 3 t \leq 4 - 2$

Subtract $2$ from $4$ .

$- 3 t \leq 2$

Divide each term in $- 3 t \leq 2$ by $- 3$ and simplify.

Divide each term in $- 3 t \leq 2$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 t}{- 3} \geq \frac{2}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 t}{- 3} \geq \frac{2}{- 3}$

Divide $t$ by $1$ .

$t \geq \frac{2}{- 3}$

Simplify the right side.

Move the negative in front of the fraction.

$t \geq - \frac{2}{3}$

Find the intersection of $t \geq - \frac{2}{3}$ and $t < \frac{2}{3}$ .

$- \frac{2}{3} \leq t < \frac{2}{3}$

Step 4

Find the union of the solutions.

$- \frac{2}{3} \leq t \leq 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- \frac{2}{3} \leq t \leq 2$

Interval Notation:

$[- \frac{2}{3}, 2]$

Step 6

Solve for t |3t-2|<=4

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