Solve the Inequality for x |x+1|<2

$| x + 1 | < 2$

Write $| x + 1 | < 2$ 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.

$x + 1 < 2$

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 $- 1$ .

$- (x + 1) < 2$

Write as a piecewise.

${\begin{cases} x + 1 < 2 & x \geq - 1 \\ - (x + 1) < 2 & x < - 1 \end{cases}$

Simplify $- (x + 1) < 2$ .

Apply the distributive property.

${\begin{cases} x + 1 < 2 & x \geq - 1 \\ - x - 1 \cdot 1 < 2 & x < - 1 \end{cases}$

Multiply $- 1$ by $1$ .

${\begin{cases} x + 1 < 2 & x \geq - 1 \\ - x - 1 < 2 & x < - 1 \end{cases}$

Solve $x + 1 < 2$ 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.

$x < 2 - 1$

Subtract $1$ from $2$ .

$x < 1$

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

$- 1 \leq x < 1$

Solve $- x - 1 < 2$ when $x < - 1$ .

Solve $- x - 1 < 2$ for $x$ .

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

Add $1$ to both sides of the inequality.

$- x < 2 + 1$

Add $2$ and $1$ .

$- x < 3$

Divide each term in $- x < 3$ by $- 1$ and simplify.

Divide each term in $- x < 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{3}{- 1}$

Divide $x$ by $1$ .

$x > \frac{3}{- 1}$

Simplify the right side.

Divide $3$ by $- 1$ .

$x > - 3$

Find the intersection of $x > - 3$ and $x < - 1$ .

$- 3 < x < - 1$

Find the union of the solutions.

$- 3 < x < 1$

The result can be shown in multiple forms.

Inequality Form:

$- 3 < x < 1$

Interval Notation:

$(- 3, 1)$

Solve the Inequality for x |x+1|<2

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