Solve the Inequality for x 8|x-2|+3>=19

$8 | x - 2 | + 3 \geq 19$

Step 1

Write $8 | x - 2 | + 3 \geq 19$ as a piecewise.

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

$x - 2 \geq 0$

Add $2$ to both sides of the inequality.

$x \geq 2$

In the piece where $x - 2$ is non-negative, remove the absolute value.

$8 (x - 2) + 3 \geq 19$

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

$x - 2 < 0$

Add $2$ to both sides of the inequality.

$x < 2$

In the piece where $x - 2$ is negative, remove the absolute value and multiply by $- 1$ .

$8 (- (x - 2)) + 3 \geq 19$

Write as a piecewise.

${\begin{cases} 8 (x - 2) + 3 \geq 19 & x \geq 2 \\ 8 (- (x - 2)) + 3 \geq 19 & x < 2 \end{cases}$

Simplify $8 (x - 2) + 3 \geq 19$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 8 x + 8 \cdot - 2 + 3 \geq 19 & x \geq 2 \\ 8 (- (x - 2)) + 3 \geq 19 & x < 2 \end{cases}$

Multiply $8$ by $- 2$ .

${\begin{cases} 8 x - 16 + 3 \geq 19 & x \geq 2 \\ 8 (- (x - 2)) + 3 \geq 19 & x < 2 \end{cases}$

Add $- 16$ and $3$ .

${\begin{cases} 8 x - 13 \geq 19 & x \geq 2 \\ 8 (- (x - 2)) + 3 \geq 19 & x < 2 \end{cases}$

Simplify $8 (- (x - 2)) + 3 \geq 19$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 8 x - 13 \geq 19 & x \geq 2 \\ 8 (- x - - 2) + 3 \geq 19 & x < 2 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} 8 x - 13 \geq 19 & x \geq 2 \\ 8 (- x + 2) + 3 \geq 19 & x < 2 \end{cases}$

Apply the distributive property.

${\begin{cases} 8 x - 13 \geq 19 & x \geq 2 \\ 8 (- x) + 8 \cdot 2 + 3 \geq 19 & x < 2 \end{cases}$

Multiply $- 1$ by $8$ .

${\begin{cases} 8 x - 13 \geq 19 & x \geq 2 \\ - 8 x + 8 \cdot 2 + 3 \geq 19 & x < 2 \end{cases}$

Multiply $8$ by $2$ .

${\begin{cases} 8 x - 13 \geq 19 & x \geq 2 \\ - 8 x + 16 + 3 \geq 19 & x < 2 \end{cases}$

Add $16$ and $3$ .

${\begin{cases} 8 x - 13 \geq 19 & x \geq 2 \\ - 8 x + 19 \geq 19 & x < 2 \end{cases}$

Step 2

Solve $8 x - 13 \geq 19$ for $x$ .

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

Add $13$ to both sides of the inequality.

$8 x \geq 19 + 13$

Add $19$ and $13$ .

$8 x \geq 32$

Divide each term in $8 x \geq 32$ by $8$ and simplify.

Divide each term in $8 x \geq 32$ by $8$ .

$\frac{8 x}{8} \geq \frac{32}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} \geq \frac{32}{8}$

Divide $x$ by $1$ .

$x \geq \frac{32}{8}$

Simplify the right side.

Divide $32$ by $8$ .

$x \geq 4$

Step 3

Solve $- 8 x + 19 \geq 19$ for $x$ .

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

Subtract $19$ from both sides of the inequality.

$- 8 x \geq 19 - 19$

Subtract $19$ from $19$ .

$- 8 x \geq 0$

Divide each term in $- 8 x \geq 0$ by $- 8$ and simplify.

Divide each term in $- 8 x \geq 0$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} \leq \frac{0}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 x}{- 8} \leq \frac{0}{- 8}$

Divide $x$ by $1$ .

$x \leq \frac{0}{- 8}$

Simplify the right side.

Divide $0$ by $- 8$ .

$x \leq 0$

Step 4

Find the union of the solutions.

$x \leq 0$ or $x \geq 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq 0 or x \geq 4$

Interval Notation:

$(- \infty, 0] \cup [4, \infty)$

Step 6

Solve the Inequality for x 8|x-2|+3>=19

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