Write the Absolute Value as Piecewise |5-12t|

$| 5 - 12 t |$

Step 1

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

$5 - 12 t \geq 0$

Step 2

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$- 12 t \geq - 5$

Divide each term in $- 12 t \geq - 5$ by $- 12$ and simplify.

Divide each term in $- 12 t \geq - 5$ by $- 12$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 12 t}{- 12} \leq \frac{- 5}{- 12}$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 t}{- 12} \leq \frac{- 5}{- 12}$

Divide $t$ by $1$ .

$t \leq \frac{- 5}{- 12}$

Simplify the right side.

Dividing two negative values results in a positive value.

$t \leq \frac{5}{12}$

Step 3

In the piece where $5 - 12 t$ is non-negative, remove the absolute value.

$5 - 12 t$

Step 4

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

$5 - 12 t < 0$

Step 5

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$- 12 t < - 5$

Divide each term in $- 12 t < - 5$ by $- 12$ and simplify.

Divide each term in $- 12 t < - 5$ by $- 12$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 12 t}{- 12} > \frac{- 5}{- 12}$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 t}{- 12} > \frac{- 5}{- 12}$

Divide $t$ by $1$ .

$t > \frac{- 5}{- 12}$

Simplify the right side.

Dividing two negative values results in a positive value.

$t > \frac{5}{12}$

Step 6

In the piece where $5 - 12 t$ is negative, remove the absolute value and multiply by $- 1$ .

$- (5 - 12 t)$

Step 7

Write as a piecewise.

${\begin{cases} 5 - 12 t & t \leq \frac{5}{12} \\ - (5 - 12 t) & t > \frac{5}{12} \end{cases}$

Step 8

Simplify $- (5 - 12 t)$ .

Apply the distributive property.

${\begin{cases} 5 - 12 t & t \leq \frac{5}{12} \\ - 1 \cdot 5 - (- 12 t) & t > \frac{5}{12} \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 5 - 12 t & t \leq \frac{5}{12} \\ - 5 - (- 12 t) & t > \frac{5}{12} \end{cases}$

Multiply $- 12$ by $- 1$ .

${\begin{cases} 5 - 12 t & t \leq \frac{5}{12} \\ - 5 + 12 t & t > \frac{5}{12} \end{cases}$

Write the Absolute Value as Piecewise |5-12t|

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