Algebra Examples

$1 < | x + 2 | < 3$

Step 1

Find the $x$ values that make $1 < | x + 2 |$ true.

Tap for more steps...

Step 1.1

Rewrite so $x$ is on the left side of the inequality.

$| x + 2 | > 1$

Step 1.2

Write $| x + 2 | > 1$ as a piecewise.

Tap for more steps...

Step 1.2.1

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

$x + 2 \geq 0$

Step 1.2.2

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 1.2.3

In the piece where $x + 2$ is non-negative, remove the absolute value.

$x + 2 > 1$

Step 1.2.4

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

$x + 2 < 0$

Step 1.2.5

Subtract $2$ from both sides of the inequality.

$x < - 2$

Step 1.2.6

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

$- (x + 2) > 1$

Step 1.2.7

Write as a piecewise.

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

Step 1.2.8

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

Tap for more steps...

Step 1.2.8.1

Apply the distributive property.

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

Step 1.2.8.2

Multiply $- 1$ by $2$ .

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

Step 1.3

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

Tap for more steps...

Step 1.3.1

Subtract $2$ from both sides of the inequality.

$x > 1 - 2$

Step 1.3.2

Subtract $2$ from $1$ .

$x > - 1$

Step 1.4

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

Tap for more steps...

Step 1.4.1

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

Tap for more steps...

Step 1.4.1.1

Add $2$ to both sides of the inequality.

$- x > 1 + 2$

Step 1.4.1.2

Add $1$ and $2$ .

$- x > 3$

Step 1.4.2

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

Tap for more steps...

Step 1.4.2.1

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.

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

Step 1.4.2.2

Simplify the left side.

Tap for more steps...

Step 1.4.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.4.2.2.2

Divide $x$ by $1$ .

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

Step 1.4.2.3

Simplify the right side.

Tap for more steps...

Step 1.4.2.3.1

Divide $3$ by $- 1$ .

$x < - 3$

Step 1.5

Find the union of the solutions.

$x < - 3$ or $x > - 1$

Step 2

Find the $x$ values that make $| x + 2 | < 3$ true.

Tap for more steps...

Step 2.1

Write $| x + 2 | < 3$ as a piecewise.

Tap for more steps...

Step 2.1.1

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

$x + 2 \geq 0$

Step 2.1.2

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 2.1.3

In the piece where $x + 2$ is non-negative, remove the absolute value.

$x + 2 < 3$

Step 2.1.4

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

$x + 2 < 0$

Step 2.1.5

Subtract $2$ from both sides of the inequality.

$x < - 2$

Step 2.1.6

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

$- (x + 2) < 3$

Step 2.1.7

Write as a piecewise.

${\begin{cases} x + 2 < 3 & x \geq - 2 \\ - (x + 2) < 3 & x < - 2 \end{cases}$

Step 2.1.8

Simplify $- (x + 2) < 3$ .

Tap for more steps...

Step 2.1.8.1

Apply the distributive property.

${\begin{cases} x + 2 < 3 & x \geq - 2 \\ - x - 1 \cdot 2 < 3 & x < - 2 \end{cases}$

Step 2.1.8.2

Multiply $- 1$ by $2$ .

${\begin{cases} x + 2 < 3 & x \geq - 2 \\ - x - 2 < 3 & x < - 2 \end{cases}$

Step 2.2

Solve $x + 2 < 3$ when $x \geq - 2$ .

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

Subtract $2$ from both sides of the inequality.

$x < 3 - 2$

Step 2.2.1.2

Subtract $2$ from $3$ .

$x < 1$

Step 2.2.2

Find the intersection of $x < 1$ and $x \geq - 2$ .

$- 2 \leq x < 1$

Step 2.3

Solve $- x - 2 < 3$ when $x < - 2$ .

Tap for more steps...

Step 2.3.1

Solve $- x - 2 < 3$ for $x$ .

Tap for more steps...

Step 2.3.1.1

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

Tap for more steps...

Step 2.3.1.1.1

Add $2$ to both sides of the inequality.

$- x < 3 + 2$

Step 2.3.1.1.2

Add $3$ and $2$ .

$- x < 5$

Step 2.3.1.2

Divide each term in $- x < 5$ by $- 1$ and simplify.

Tap for more steps...

Step 2.3.1.2.1

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

$\frac{- x}{- 1} > \frac{5}{- 1}$

Step 2.3.1.2.2

Simplify the left side.

Tap for more steps...

Step 2.3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{5}{- 1}$

Step 2.3.1.2.2.2

Divide $x$ by $1$ .

$x > \frac{5}{- 1}$

Step 2.3.1.2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1.2.3.1

Divide $5$ by $- 1$ .

$x > - 5$

Step 2.3.2

Find the intersection of $x > - 5$ and $x < - 2$ .

$- 5 < x < - 2$

Step 2.4

Find the union of the solutions.

$- 5 < x < 1$

Step 3

The solution is the intersection of the intervals.

$x < - 3$ or $x > - 1$

$- 5 < x < 1$

Step 4

Find the intersection.

$- 5 < x < - 3$ or $- 1 < x < 1$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 5 < x < - 3 or - 1 < x < 1$

Interval Notation:

$(- 5, - 3) \cup (- 1, 1)$

Step 6