Solve for t |7t-2|<=12

$| 7 t - 2 | \leq 12$

Step 1

Write $| 7 t - 2 | \leq 12$ as a piecewise.

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

$7 t - 2 \geq 0$

Solve the inequality.

Add $2$ to both sides of the inequality.

$7 t \geq 2$

Divide each term in $7 t \geq 2$ by $7$ and simplify.

Divide each term in $7 t \geq 2$ by $7$ .

$\frac{7 t}{7} \geq \frac{2}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 t}{7} \geq \frac{2}{7}$

Divide $t$ by $1$ .

$t \geq \frac{2}{7}$

In the piece where $7 t - 2$ is non-negative, remove the absolute value.

$7 t - 2 \leq 12$

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

$7 t - 2 < 0$

Solve the inequality.

Add $2$ to both sides of the inequality.

$7 t < 2$

Divide each term in $7 t < 2$ by $7$ and simplify.

Divide each term in $7 t < 2$ by $7$ .

$\frac{7 t}{7} < \frac{2}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 t}{7} < \frac{2}{7}$

Divide $t$ by $1$ .

$t < \frac{2}{7}$

In the piece where $7 t - 2$ is negative, remove the absolute value and multiply by $- 1$ .

$- (7 t - 2) \leq 12$

Write as a piecewise.

${\begin{cases} 7 t - 2 \leq 12 & t \geq \frac{2}{7} \\ - (7 t - 2) \leq 12 & t < \frac{2}{7} \end{cases}$

Simplify $- (7 t - 2) \leq 12$ .

Apply the distributive property.

${\begin{cases} 7 t - 2 \leq 12 & t \geq \frac{2}{7} \\ - (7 t) - - 2 \leq 12 & t < \frac{2}{7} \end{cases}$

Multiply $7$ by $- 1$ .

${\begin{cases} 7 t - 2 \leq 12 & t \geq \frac{2}{7} \\ - 7 t - - 2 \leq 12 & t < \frac{2}{7} \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} 7 t - 2 \leq 12 & t \geq \frac{2}{7} \\ - 7 t + 2 \leq 12 & t < \frac{2}{7} \end{cases}$

Step 2

Solve $7 t - 2 \leq 12$ when $t \geq \frac{2}{7}$ .

Solve $7 t - 2 \leq 12$ for $t$ .

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

Add $2$ to both sides of the inequality.

$7 t \leq 12 + 2$

Add $12$ and $2$ .

$7 t \leq 14$

Divide each term in $7 t \leq 14$ by $7$ and simplify.

Divide each term in $7 t \leq 14$ by $7$ .

$\frac{7 t}{7} \leq \frac{14}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 t}{7} \leq \frac{14}{7}$

Divide $t$ by $1$ .

$t \leq \frac{14}{7}$

Simplify the right side.

Divide $14$ by $7$ .

$t \leq 2$

Find the intersection of $t \leq 2$ and $t \geq \frac{2}{7}$ .

$\frac{2}{7} \leq t \leq 2$

Step 3

Solve $- 7 t + 2 \leq 12$ when $t < \frac{2}{7}$ .

Solve $- 7 t + 2 \leq 12$ for $t$ .

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

Subtract $2$ from both sides of the inequality.

$- 7 t \leq 12 - 2$

Subtract $2$ from $12$ .

$- 7 t \leq 10$

Divide each term in $- 7 t \leq 10$ by $- 7$ and simplify.

Divide each term in $- 7 t \leq 10$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 t}{- 7} \geq \frac{10}{- 7}$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 t}{- 7} \geq \frac{10}{- 7}$

Divide $t$ by $1$ .

$t \geq \frac{10}{- 7}$

Simplify the right side.

Move the negative in front of the fraction.

$t \geq - \frac{10}{7}$

Find the intersection of $t \geq - \frac{10}{7}$ and $t < \frac{2}{7}$ .

$- \frac{10}{7} \leq t < \frac{2}{7}$

Step 4

Find the union of the solutions.

$- \frac{10}{7} \leq t \leq 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- \frac{10}{7} \leq t \leq 2$

Interval Notation:

$[- \frac{10}{7}, 2]$

Step 6

Solve for t |7t-2|<=12

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