Solve for y |y-1|<=6

$| y - 1 | \leq 6$

Write $| y - 1 | \leq 6$ as a piecewise.

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

$y - 1 \geq 0$

Add $1$ to both sides of the inequality.

$y \geq 1$

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

$y - 1 \leq 6$

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

$y - 1 < 0$

Add $1$ to both sides of the inequality.

$y < 1$

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

$- (y - 1) \leq 6$

Write as a piecewise.

${\begin{cases} y - 1 \leq 6 & y \geq 1 \\ - (y - 1) \leq 6 & y < 1 \end{cases}$

Simplify $- (y - 1) \leq 6$ .

Apply the distributive property.

${\begin{cases} y - 1 \leq 6 & y \geq 1 \\ - y - - 1 \leq 6 & y < 1 \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} y - 1 \leq 6 & y \geq 1 \\ - y + 1 \leq 6 & y < 1 \end{cases}$

Solve $y - 1 \leq 6$ when $y \geq 1$ .

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

Add $1$ to both sides of the inequality.

$y \leq 6 + 1$

Add $6$ and $1$ .

$y \leq 7$

Find the intersection of $y \leq 7$ and $y \geq 1$ .

$1 \leq y \leq 7$

Solve $- y + 1 \leq 6$ when $y < 1$ .

Solve $- y + 1 \leq 6$ for $y$ .

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

Subtract $1$ from both sides of the inequality.

$- y \leq 6 - 1$

Subtract $1$ from $6$ .

$- y \leq 5$

Divide each term in $- y \leq 5$ by $- 1$ and simplify.

Divide each term in $- y \leq 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \geq \frac{5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} \geq \frac{5}{- 1}$

Divide $y$ by $1$ .

$y \geq \frac{5}{- 1}$

Simplify the right side.

Divide $5$ by $- 1$ .

$y \geq - 5$

Find the intersection of $y \geq - 5$ and $y < 1$ .

$- 5 \leq y < 1$

Find the union of the solutions.

$- 5 \leq y \leq 7$

The result can be shown in multiple forms.

Inequality Form:

$- 5 \leq y \leq 7$

Interval Notation:

$[- 5, 7]$

Solve for y |y-1|<=6

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