Solve the Inequality for x |19+x|<2

$| 19 + x | < 2$

Step 1

Write $| 19 + x | < 2$ as a piecewise.

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

$19 + x \geq 0$

Subtract $19$ from both sides of the inequality.

$x \geq - 19$

In the piece where $19 + x$ is non-negative, remove the absolute value.

$19 + x < 2$

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

$19 + x < 0$

Subtract $19$ from both sides of the inequality.

$x < - 19$

In the piece where $19 + x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (19 + x) < 2$

Write as a piecewise.

${\begin{cases} 19 + x < 2 & x \geq - 19 \\ - (19 + x) < 2 & x < - 19 \end{cases}$

Simplify $- (19 + x) < 2$ .

Apply the distributive property.

${\begin{cases} 19 + x < 2 & x \geq - 19 \\ - 1 \cdot 19 - x < 2 & x < - 19 \end{cases}$

Multiply $- 1$ by $19$ .

${\begin{cases} 19 + x < 2 & x \geq - 19 \\ - 19 - x < 2 & x < - 19 \end{cases}$

Step 2

Solve $19 + x < 2$ when $x \geq - 19$ .

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

Subtract $19$ from both sides of the inequality.

$x < 2 - 19$

Subtract $19$ from $2$ .

$x < - 17$

Find the intersection of $x < - 17$ and $x \geq - 19$ .

$- 19 \leq x < - 17$

Step 3

Solve $- 19 - x < 2$ when $x < - 19$ .

Solve $- 19 - x < 2$ for $x$ .

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

Add $19$ to both sides of the inequality.

$- x < 2 + 19$

Add $2$ and $19$ .

$- x < 21$

Divide each term in $- x < 21$ by $- 1$ and simplify.

Divide each term in $- x < 21$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{21}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{21}{- 1}$

Divide $x$ by $1$ .

$x > \frac{21}{- 1}$

Simplify the right side.

Divide $21$ by $- 1$ .

$x > - 21$

Find the intersection of $x > - 21$ and $x < - 19$ .

$- 21 < x < - 19$

Step 4

Find the union of the solutions.

$- 21 < x < - 17$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 21 < x < - 17$

Interval Notation:

$(- 21, - 17)$

Step 6

Solve the Inequality for x |19+x|<2

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