Solve for x |x+5|>=7

$| x + 5 | \geq 7$

Write $| x + 5 | \geq 7$ as a piecewise.

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

$x + 5 \geq 0$

Subtract $5$ from both sides of the inequality.

$x \geq - 5$

In the piece where $x + 5$ is non-negative, remove the absolute value.

$x + 5 \geq 7$

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

$x + 5 < 0$

Subtract $5$ from both sides of the inequality.

$x < - 5$

In the piece where $x + 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x + 5) \geq 7$

Write as a piecewise.

${\begin{cases} x + 5 \geq 7 & x \geq - 5 \\ - (x + 5) \geq 7 & x < - 5 \end{cases}$

Simplify $- (x + 5) \geq 7$ .

Apply the distributive property.

${\begin{cases} x + 5 \geq 7 & x \geq - 5 \\ - x - 1 \cdot 5 \geq 7 & x < - 5 \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} x + 5 \geq 7 & x \geq - 5 \\ - x - 5 \geq 7 & x < - 5 \end{cases}$

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

Subtract $5$ from both sides of the inequality.

$x \geq 7 - 5$

Subtract $5$ from $7$ .

$x \geq 2$

Solve $- x - 5 \geq 7$ for $x$ .

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

Add $5$ to both sides of the inequality.

$- x \geq 7 + 5$

Add $7$ and $5$ .

$- x \geq 12$

Divide each term in $- x \geq 12$ by $- 1$ and simplify.

Divide each term in $- x \geq 12$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{12}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{12}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{12}{- 1}$

Simplify the right side.

Divide $12$ by $- 1$ .

$x \leq - 12$

Find the union of the solutions.

$x \leq - 12$ or $x \geq 2$

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 12 or x \geq 2$

Interval Notation:

$(- \infty, - 12] \cup [2, \infty)$

Solve for x |x+5|>=7

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