Solve the Inequality for x (|x+8|)/7>=1

$\frac{| x + 8 |}{7} \geq 1$

Step 1

Multiply both sides by $7$ .

$\frac{| x + 8 |}{7} \cdot 7 \geq 1 \cdot 7$

Step 2

Simplify.

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{| x + 8 |}{7} \cdot 7 \geq 1 \cdot 7$

Rewrite the expression.

$| x + 8 | \geq 1 \cdot 7$

Simplify the right side.

Multiply $7$ by $1$ .

$| x + 8 | \geq 7$

Step 3

Solve for $x$ .

Write $| x + 8 | \geq 7$ as a piecewise.

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

$x + 8 \geq 0$

Subtract $8$ from both sides of the inequality.

$x \geq - 8$

In the piece where $x + 8$ is non-negative, remove the absolute value.

$x + 8 \geq 7$

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

$x + 8 < 0$

Subtract $8$ from both sides of the inequality.

$x < - 8$

In the piece where $x + 8$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x + 8) \geq 7$

Write as a piecewise.

${\begin{cases} x + 8 \geq 7 & x \geq - 8 \\ - (x + 8) \geq 7 & x < - 8 \end{cases}$

Simplify $- (x + 8) \geq 7$ .

Apply the distributive property.

${\begin{cases} x + 8 \geq 7 & x \geq - 8 \\ - x - 1 \cdot 8 \geq 7 & x < - 8 \end{cases}$

Multiply $- 1$ by $8$ .

${\begin{cases} x + 8 \geq 7 & x \geq - 8 \\ - x - 8 \geq 7 & x < - 8 \end{cases}$

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

Subtract $8$ from both sides of the inequality.

$x \geq 7 - 8$

Subtract $8$ from $7$ .

$x \geq - 1$

Solve $- x - 8 \geq 7$ for $x$ .

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

Add $8$ to both sides of the inequality.

$- x \geq 7 + 8$

Add $7$ and $8$ .

$- x \geq 15$

Divide each term in $- x \geq 15$ by $- 1$ and simplify.

Divide each term in $- x \geq 15$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{15}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{15}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{15}{- 1}$

Simplify the right side.

Divide $15$ by $- 1$ .

$x \leq - 15$

Find the union of the solutions.

$x \leq - 15$ or $x \geq - 1$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 15 or x \geq - 1$

Interval Notation:

$(- \infty, - 15] \cup [- 1, \infty)$

Step 5

Solve the Inequality for x (|x+8|)/7>=1

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