Solve the Inequality for x |18+x/2|>=10

$| 18 + \frac{x}{2} | \geq 10$

Step 1

Write $| 18 + \frac{x}{2} | \geq 10$ as a piecewise.

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

$18 + \frac{x}{2} \geq 0$

Solve the inequality.

Subtract $18$ from both sides of the inequality.

$\frac{x}{2} \geq - 18$

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 \geq - 18 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x}{2} \cdot 2 \geq - 18 \cdot 2$

Rewrite the expression.

$x \geq - 18 \cdot 2$

Simplify the right side.

Multiply $- 18$ by $2$ .

$x \geq - 36$

In the piece where $18 + \frac{x}{2}$ is non-negative, remove the absolute value.

$18 + \frac{x}{2} \geq 10$

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

$18 + \frac{x}{2} < 0$

Solve the inequality.

Subtract $18$ from both sides of the inequality.

$\frac{x}{2} < - 18$

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 < - 18 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x}{2} \cdot 2 < - 18 \cdot 2$

Rewrite the expression.

$x < - 18 \cdot 2$

Simplify the right side.

Multiply $- 18$ by $2$ .

$x < - 36$

In the piece where $18 + \frac{x}{2}$ is negative, remove the absolute value and multiply by $- 1$ .

$- (18 + \frac{x}{2}) \geq 10$

Write as a piecewise.

${\begin{cases} 18 + \frac{x}{2} \geq 10 & x \geq - 36 \\ - (18 + \frac{x}{2}) \geq 10 & x < - 36 \end{cases}$

Simplify $- (18 + \frac{x}{2}) \geq 10$ .

Apply the distributive property.

${\begin{cases} 18 + \frac{x}{2} \geq 10 & x \geq - 36 \\ - 1 \cdot 18 - \frac{x}{2} \geq 10 & x < - 36 \end{cases}$

Multiply $- 1$ by $18$ .

${\begin{cases} 18 + \frac{x}{2} \geq 10 & x \geq - 36 \\ - 18 - \frac{x}{2} \geq 10 & x < - 36 \end{cases}$

Step 2

Solve $18 + \frac{x}{2} \geq 10$ for $x$ .

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

Subtract $18$ from both sides of the inequality.

$\frac{x}{2} \geq 10 - 18$

Subtract $18$ from $10$ .

$\frac{x}{2} \geq - 8$

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 \geq - 8 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x}{2} \cdot 2 \geq - 8 \cdot 2$

Rewrite the expression.

$x \geq - 8 \cdot 2$

Simplify the right side.

Multiply $- 8$ by $2$ .

$x \geq - 16$

Step 3

Solve $- 18 - \frac{x}{2} \geq 10$ for $x$ .

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

Add $18$ to both sides of the inequality.

$- \frac{x}{2} \geq 10 + 18$

Add $10$ and $18$ .

$- \frac{x}{2} \geq 28$

Divide each term in $- \frac{x}{2} \geq 28$ by $- 1$ and simplify.

Divide each term in $- \frac{x}{2} \geq 28$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{2}}{- 1} \leq \frac{28}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{x}{2}}{1} \leq \frac{28}{- 1}$

Divide $\frac{x}{2}$ by $1$ .

$\frac{x}{2} \leq \frac{28}{- 1}$

Simplify the right side.

Divide $28$ by $- 1$ .

$\frac{x}{2} \leq - 28$

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 \leq - 28 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x}{2} \cdot 2 \leq - 28 \cdot 2$

Rewrite the expression.

$x \leq - 28 \cdot 2$

Simplify the right side.

Multiply $- 28$ by $2$ .

$x \leq - 56$

Step 4

Find the union of the solutions.

$x \leq - 56$ or $x \geq - 16$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 56 or x \geq - 16$

Interval Notation:

$(- \infty, - 56] \cup [- 16, \infty)$

Step 6

Solve the Inequality for x |18+x/2|>=10

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