Solve the Inequality for x |x+14|+3>17

$| x + 14 | + 3 > 17$

Step 1

Write $| x + 14 | + 3 > 17$ as a piecewise.

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

$x + 14 \geq 0$

Subtract $14$ from both sides of the inequality.

$x \geq - 14$

In the piece where $x + 14$ is non-negative, remove the absolute value.

$x + 14 + 3 > 17$

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

$x + 14 < 0$

Subtract $14$ from both sides of the inequality.

$x < - 14$

In the piece where $x + 14$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x + 14) + 3 > 17$

Write as a piecewise.

${\begin{cases} x + 14 + 3 > 17 & x \geq - 14 \\ - (x + 14) + 3 > 17 & x < - 14 \end{cases}$

Add $14$ and $3$ .

${\begin{cases} x + 17 > 17 & x \geq - 14 \\ - (x + 14) + 3 > 17 & x < - 14 \end{cases}$

Simplify $- (x + 14) + 3 > 17$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} x + 17 > 17 & x \geq - 14 \\ - x - 1 \cdot 14 + 3 > 17 & x < - 14 \end{cases}$

Multiply $- 1$ by $14$ .

${\begin{cases} x + 17 > 17 & x \geq - 14 \\ - x - 14 + 3 > 17 & x < - 14 \end{cases}$

Add $- 14$ and $3$ .

${\begin{cases} x + 17 > 17 & x \geq - 14 \\ - x - 11 > 17 & x < - 14 \end{cases}$

Step 2

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

Subtract $17$ from both sides of the inequality.

$x > 17 - 17$

Subtract $17$ from $17$ .

$x > 0$

Step 3

Solve $- x - 11 > 17$ for $x$ .

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

Add $11$ to both sides of the inequality.

$- x > 17 + 11$

Add $17$ and $11$ .

$- x > 28$

Divide each term in $- x > 28$ by $- 1$ and simplify.

Divide each term in $- x > 28$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{28}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{28}{- 1}$

Divide $x$ by $1$ .

$x < \frac{28}{- 1}$

Simplify the right side.

Divide $28$ by $- 1$ .

$x < - 28$

Step 4

Find the union of the solutions.

$x < - 28$ or $x > 0$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - 28 or x > 0$

Interval Notation:

$(- \infty, - 28) \cup (0, \infty)$

Step 6

Solve the Inequality for x |x+14|+3>17

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