Solve for x |x+1|<8

$| x + 1 | < 8$

Step 1

Write $| x + 1 | < 8$ as a piecewise.

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

$x + 1 \geq 0$

Subtract $1$ from both sides of the inequality.

$x \geq - 1$

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

$x + 1 < 8$

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

$x + 1 < 0$

Subtract $1$ from both sides of the inequality.

$x < - 1$

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

$- (x + 1) < 8$

Write as a piecewise.

${\begin{cases} x + 1 < 8 & x \geq - 1 \\ - (x + 1) < 8 & x < - 1 \end{cases}$

Simplify $- (x + 1) < 8$ .

Apply the distributive property.

${\begin{cases} x + 1 < 8 & x \geq - 1 \\ - x - 1 \cdot 1 < 8 & x < - 1 \end{cases}$

Multiply $- 1$ by $1$ .

${\begin{cases} x + 1 < 8 & x \geq - 1 \\ - x - 1 < 8 & x < - 1 \end{cases}$

Step 2

Solve $x + 1 < 8$ when $x \geq - 1$ .

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

Subtract $1$ from both sides of the inequality.

$x < 8 - 1$

Subtract $1$ from $8$ .

$x < 7$

Find the intersection of $x < 7$ and $x \geq - 1$ .

$- 1 \leq x < 7$

Step 3

Solve $- x - 1 < 8$ when $x < - 1$ .

Solve $- x - 1 < 8$ for $x$ .

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

Add $1$ to both sides of the inequality.

$- x < 8 + 1$

Add $8$ and $1$ .

$- x < 9$

Divide each term in $- x < 9$ by $- 1$ and simplify.

Divide each term in $- x < 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{9}{- 1}$

Divide $x$ by $1$ .

$x > \frac{9}{- 1}$

Simplify the right side.

Divide $9$ by $- 1$ .

$x > - 9$

Find the intersection of $x > - 9$ and $x < - 1$ .

$- 9 < x < - 1$

Step 4

Find the union of the solutions.

$- 9 < x < 7$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 9 < x < 7$

Interval Notation:

$(- 9, 7)$

Step 6

Solve for x |x+1|<8

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