Solve the Inequality for x 5|x-2|-2>58

$5 | x - 2 | - 2 > 58$

Write $5 | x - 2 | - 2 > 58$ 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.

$5 (x - 2) - 2 > 58$

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 $- 1$ .

$5 (- (x - 2)) - 2 > 58$

Write as a piecewise.

${\begin{cases} 5 (x - 2) - 2 > 58 & x \geq 2 \\ 5 (- (x - 2)) - 2 > 58 & x < 2 \end{cases}$

Simplify $5 (x - 2) - 2 > 58$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 5 x + 5 \cdot - 2 - 2 > 58 & x \geq 2 \\ 5 (- (x - 2)) - 2 > 58 & x < 2 \end{cases}$

Multiply $5$ by $- 2$ .

${\begin{cases} 5 x - 10 - 2 > 58 & x \geq 2 \\ 5 (- (x - 2)) - 2 > 58 & x < 2 \end{cases}$

Subtract $2$ from $- 10$ .

${\begin{cases} 5 x - 12 > 58 & x \geq 2 \\ 5 (- (x - 2)) - 2 > 58 & x < 2 \end{cases}$

Simplify $5 (- (x - 2)) - 2 > 58$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 5 x - 12 > 58 & x \geq 2 \\ 5 (- x - - 2) - 2 > 58 & x < 2 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} 5 x - 12 > 58 & x \geq 2 \\ 5 (- x + 2) - 2 > 58 & x < 2 \end{cases}$

Apply the distributive property.

${\begin{cases} 5 x - 12 > 58 & x \geq 2 \\ 5 (- x) + 5 \cdot 2 - 2 > 58 & x < 2 \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 5 x - 12 > 58 & x \geq 2 \\ - 5 x + 5 \cdot 2 - 2 > 58 & x < 2 \end{cases}$

Multiply $5$ by $2$ .

${\begin{cases} 5 x - 12 > 58 & x \geq 2 \\ - 5 x + 10 - 2 > 58 & x < 2 \end{cases}$

Subtract $2$ from $10$ .

${\begin{cases} 5 x - 12 > 58 & x \geq 2 \\ - 5 x + 8 > 58 & x < 2 \end{cases}$

Solve $5 x - 12 > 58$ for $x$ .

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

Add $12$ to both sides of the inequality.

$5 x > 58 + 12$

Add $58$ and $12$ .

$5 x > 70$

Divide each term in $5 x > 70$ by $5$ and simplify.

Divide each term in $5 x > 70$ by $5$ .

$\frac{5 x}{5} > \frac{70}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} > \frac{70}{5}$

Divide $x$ by $1$ .

$x > \frac{70}{5}$

Simplify the right side.

Divide $70$ by $5$ .

$x > 14$

Solve $- 5 x + 8 > 58$ for $x$ .

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

Subtract $8$ from both sides of the inequality.

$- 5 x > 58 - 8$

Subtract $8$ from $58$ .

$- 5 x > 50$

Divide each term in $- 5 x > 50$ by $- 5$ and simplify.

Divide each term in $- 5 x > 50$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} < \frac{50}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} < \frac{50}{- 5}$

Divide $x$ by $1$ .

$x < \frac{50}{- 5}$

Simplify the right side.

Divide $50$ by $- 5$ .

$x < - 10$

Find the union of the solutions.

$x < - 10$ or $x > 14$

The result can be shown in multiple forms.

Inequality Form:

$x < - 10 or x > 14$

Interval Notation:

$(- \infty, - 10) \cup (14, \infty)$

Solve the Inequality for x 5|x-2|-2>58

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