Solve for t |6t-7|-8>=3

$| 6 t - 7 | - 8 \geq 3$

Step 1

Write $| 6 t - 7 | - 8 \geq 3$ as a piecewise.

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

$6 t - 7 \geq 0$

Solve the inequality.

Add $7$ to both sides of the inequality.

$6 t \geq 7$

Divide each term in $6 t \geq 7$ by $6$ and simplify.

Divide each term in $6 t \geq 7$ by $6$ .

$\frac{6 t}{6} \geq \frac{7}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 t}{6} \geq \frac{7}{6}$

Divide $t$ by $1$ .

$t \geq \frac{7}{6}$

In the piece where $6 t - 7$ is non-negative, remove the absolute value.

$6 t - 7 - 8 \geq 3$

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

$6 t - 7 < 0$

Solve the inequality.

Add $7$ to both sides of the inequality.

$6 t < 7$

Divide each term in $6 t < 7$ by $6$ and simplify.

Divide each term in $6 t < 7$ by $6$ .

$\frac{6 t}{6} < \frac{7}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 t}{6} < \frac{7}{6}$

Divide $t$ by $1$ .

$t < \frac{7}{6}$

In the piece where $6 t - 7$ is negative, remove the absolute value and multiply by $- 1$ .

$- (6 t - 7) - 8 \geq 3$

Write as a piecewise.

${\begin{cases} 6 t - 7 - 8 \geq 3 & t \geq \frac{7}{6} \\ - (6 t - 7) - 8 \geq 3 & t < \frac{7}{6} \end{cases}$

Subtract $8$ from $- 7$ .

${\begin{cases} 6 t - 15 \geq 3 & t \geq \frac{7}{6} \\ - (6 t - 7) - 8 \geq 3 & t < \frac{7}{6} \end{cases}$

Simplify $- (6 t - 7) - 8 \geq 3$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 6 t - 15 \geq 3 & t \geq \frac{7}{6} \\ - (6 t) - - 7 - 8 \geq 3 & t < \frac{7}{6} \end{cases}$

Multiply $6$ by $- 1$ .

${\begin{cases} 6 t - 15 \geq 3 & t \geq \frac{7}{6} \\ - 6 t - - 7 - 8 \geq 3 & t < \frac{7}{6} \end{cases}$

Multiply $- 1$ by $- 7$ .

${\begin{cases} 6 t - 15 \geq 3 & t \geq \frac{7}{6} \\ - 6 t + 7 - 8 \geq 3 & t < \frac{7}{6} \end{cases}$

Subtract $8$ from $7$ .

${\begin{cases} 6 t - 15 \geq 3 & t \geq \frac{7}{6} \\ - 6 t - 1 \geq 3 & t < \frac{7}{6} \end{cases}$

Step 2

Solve $6 t - 15 \geq 3$ for $t$ .

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

Add $15$ to both sides of the inequality.

$6 t \geq 3 + 15$

Add $3$ and $15$ .

$6 t \geq 18$

Divide each term in $6 t \geq 18$ by $6$ and simplify.

Divide each term in $6 t \geq 18$ by $6$ .

$\frac{6 t}{6} \geq \frac{18}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 t}{6} \geq \frac{18}{6}$

Divide $t$ by $1$ .

$t \geq \frac{18}{6}$

Simplify the right side.

Divide $18$ by $6$ .

$t \geq 3$

Step 3

Solve $- 6 t - 1 \geq 3$ for $t$ .

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

Add $1$ to both sides of the inequality.

$- 6 t \geq 3 + 1$

Add $3$ and $1$ .

$- 6 t \geq 4$

Divide each term in $- 6 t \geq 4$ by $- 6$ and simplify.

Divide each term in $- 6 t \geq 4$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 t}{- 6} \leq \frac{4}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 t}{- 6} \leq \frac{4}{- 6}$

Divide $t$ by $1$ .

$t \leq \frac{4}{- 6}$

Simplify the right side.

Cancel the common factor of $4$ and $- 6$ .

Factor $2$ out of $4$ .

$t \leq \frac{2 (2)}{- 6}$

Cancel the common factors.

Factor $2$ out of $- 6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 4

Find the union of the solutions.

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

Step 5

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(- \infty, - \frac{2}{3}] \cup [3, \infty)$

Step 6

Solve for t |6t-7|-8>=3

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