Simplify |x+1|+5>=9

$| x + 1 | + 5 \geq 9$

Step 1

Write

$| x + 1 | + 5 \geq 9$ 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.

$x + 1 + 5 \geq 9$

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

$- 1$ .

$- (x + 1) + 5 \geq 9$

Write as a piecewise.

${\begin{cases} x + 1 + 5 \geq 9 & x \geq - 1 \\ - (x + 1) + 5 \geq 9 & x < - 1 \end{cases}$

Add

$1$ and

$5$ .

${\begin{cases} x + 6 \geq 9 & x \geq - 1 \\ - (x + 1) + 5 \geq 9 & x < - 1 \end{cases}$

Simplify

$- (x + 1) + 5 \geq 9$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} x + 6 \geq 9 & x \geq - 1 \\ - x - 1 \cdot 1 + 5 \geq 9 & x < - 1 \end{cases}$

Multiply

$- 1$ by

$1$ .

${\begin{cases} x + 6 \geq 9 & x \geq - 1 \\ - x - 1 + 5 \geq 9 & x < - 1 \end{cases}$

Add

$- 1$ and

$5$ .

${\begin{cases} x + 6 \geq 9 & x \geq - 1 \\ - x + 4 \geq 9 & x < - 1 \end{cases}$

Step 2

Move all terms not containing

$x$ to the right side of the inequality.

Subtract

$6$ from both sides of the inequality.

$x \geq 9 - 6$

Subtract

$6$ from

$9$ .

$x \geq 3$

Step 3

Solve

$- x + 4 \geq 9$ for

$x$ .

Move all terms not containing

$x$ to the right side of the inequality.

Subtract

$4$ from both sides of the inequality.

$- x \geq 9 - 4$

Subtract

$4$ from

$9$ .

$- x \geq 5$

Divide each term in

$- x \geq 5$ by

$- 1$ and simplify.

Divide each term in

$- x \geq 5$ by

$- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide

$x$ by

$1$ .

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

Simplify the right side.

Divide

$5$ by

$- 1$ .

$x \leq - 5$

Step 4

Find the union of the solutions.

$x \leq - 5$ or

$x \geq 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 5 or x \geq 3$

Interval Notation:

$(- \infty, - 5] \cup [3, \infty)$

Step 6

Simplify |x+1|+5>=9

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