Solve for t |2t-1|-5>2

$| 2 t - 1 | - 5 > 2$

Step 1

Write $| 2 t - 1 | - 5 > 2$ as a piecewise.

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

$2 t - 1 \geq 0$

Solve the inequality.

Add $1$ to both sides of the inequality.

$2 t \geq 1$

Divide each term in $2 t \geq 1$ by $2$ and simplify.

Divide each term in $2 t \geq 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $t$ by $1$ .

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

In the piece where $2 t - 1$ is non-negative, remove the absolute value.

$2 t - 1 - 5 > 2$

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

$2 t - 1 < 0$

Solve the inequality.

Add $1$ to both sides of the inequality.

$2 t < 1$

Divide each term in $2 t < 1$ by $2$ and simplify.

Divide each term in $2 t < 1$ by $2$ .

$\frac{2 t}{2} < \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 t}{2} < \frac{1}{2}$

Divide $t$ by $1$ .

$t < \frac{1}{2}$

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

$- (2 t - 1) - 5 > 2$

Write as a piecewise.

${\begin{cases} 2 t - 1 - 5 > 2 & t \geq \frac{1}{2} \\ - (2 t - 1) - 5 > 2 & t < \frac{1}{2} \end{cases}$

Subtract $5$ from $- 1$ .

${\begin{cases} 2 t - 6 > 2 & t \geq \frac{1}{2} \\ - (2 t - 1) - 5 > 2 & t < \frac{1}{2} \end{cases}$

Simplify $- (2 t - 1) - 5 > 2$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 2 t - 6 > 2 & t \geq \frac{1}{2} \\ - (2 t) - - 1 - 5 > 2 & t < \frac{1}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 t - 6 > 2 & t \geq \frac{1}{2} \\ - 2 t - - 1 - 5 > 2 & t < \frac{1}{2} \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} 2 t - 6 > 2 & t \geq \frac{1}{2} \\ - 2 t + 1 - 5 > 2 & t < \frac{1}{2} \end{cases}$

Subtract $5$ from $1$ .

${\begin{cases} 2 t - 6 > 2 & t \geq \frac{1}{2} \\ - 2 t - 4 > 2 & t < \frac{1}{2} \end{cases}$

Step 2

Solve $2 t - 6 > 2$ for $t$ .

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

Add $6$ to both sides of the inequality.

$2 t > 2 + 6$

Add $2$ and $6$ .

$2 t > 8$

Divide each term in $2 t > 8$ by $2$ and simplify.

Divide each term in $2 t > 8$ by $2$ .

$\frac{2 t}{2} > \frac{8}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 t}{2} > \frac{8}{2}$

Divide $t$ by $1$ .

$t > \frac{8}{2}$

Simplify the right side.

Divide $8$ by $2$ .

$t > 4$

Step 3

Solve $- 2 t - 4 > 2$ for $t$ .

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

Add $4$ to both sides of the inequality.

$- 2 t > 2 + 4$

Add $2$ and $4$ .

$- 2 t > 6$

Divide each term in $- 2 t > 6$ by $- 2$ and simplify.

Divide each term in $- 2 t > 6$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 t}{- 2} < \frac{6}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 t}{- 2} < \frac{6}{- 2}$

Divide $t$ by $1$ .

$t < \frac{6}{- 2}$

Simplify the right side.

Divide $6$ by $- 2$ .

$t < - 3$

Step 4

Find the union of the solutions.

$t < - 3$ or $t > 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$t < - 3 or t > 4$

Interval Notation:

$(- \infty, - 3) \cup (4, \infty)$

Step 6

Solve for t |2t-1|-5>2

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