Solve the Inequality for x |2x-4|+2<6

$| 2 x - 4 | + 2 < 6$

Write $| 2 x - 4 | + 2 < 6$ as a piecewise.

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

$2 x - 4 \geq 0$

Solve the inequality.

Add $4$ to both sides of the inequality.

$2 x \geq 4$

Divide each term in $2 x \geq 4$ by $2$ and simplify.

Divide each term in $2 x \geq 4$ by $2$ .

$\frac{2 x}{2} \geq \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{4}{2}$

Divide $x$ by $1$ .

$x \geq \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$x \geq 2$

In the piece where $2 x - 4$ is non-negative, remove the absolute value.

$2 x - 4 + 2 < 6$

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

$2 x - 4 < 0$

Solve the inequality.

Add $4$ to both sides of the inequality.

$2 x < 4$

Divide each term in $2 x < 4$ by $2$ and simplify.

Divide each term in $2 x < 4$ by $2$ .

$\frac{2 x}{2} < \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{4}{2}$

Divide $x$ by $1$ .

$x < \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$x < 2$

In the piece where $2 x - 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 x - 4) + 2 < 6$

Write as a piecewise.

${\begin{cases} 2 x - 4 + 2 < 6 & x \geq 2 \\ - (2 x - 4) + 2 < 6 & x < 2 \end{cases}$

Add $- 4$ and $2$ .

${\begin{cases} 2 x - 2 < 6 & x \geq 2 \\ - (2 x - 4) + 2 < 6 & x < 2 \end{cases}$

Simplify $- (2 x - 4) + 2 < 6$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 2 x - 2 < 6 & x \geq 2 \\ - (2 x) - - 4 + 2 < 6 & x < 2 \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x - 2 < 6 & x \geq 2 \\ - 2 x - - 4 + 2 < 6 & x < 2 \end{cases}$

Multiply $- 1$ by $- 4$ .

${\begin{cases} 2 x - 2 < 6 & x \geq 2 \\ - 2 x + 4 + 2 < 6 & x < 2 \end{cases}$

Add $4$ and $2$ .

${\begin{cases} 2 x - 2 < 6 & x \geq 2 \\ - 2 x + 6 < 6 & x < 2 \end{cases}$

Solve $2 x - 2 < 6$ when $x \geq 2$ .

Solve $2 x - 2 < 6$ for $x$ .

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

Add $2$ to both sides of the inequality.

$2 x < 6 + 2$

Add $6$ and $2$ .

$2 x < 8$

Divide each term in $2 x < 8$ by $2$ and simplify.

Divide each term in $2 x < 8$ by $2$ .

$\frac{2 x}{2} < \frac{8}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{8}{2}$

Divide $x$ by $1$ .

$x < \frac{8}{2}$

Simplify the right side.

Divide $8$ by $2$ .

$x < 4$

Find the intersection of $x < 4$ and $x \geq 2$ .

$2 \leq x < 4$

Solve $- 2 x + 6 < 6$ when $x < 2$ .

Solve $- 2 x + 6 < 6$ for $x$ .

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

Subtract $6$ from both sides of the inequality.

$- 2 x < 6 - 6$

Subtract $6$ from $6$ .

$- 2 x < 0$

Divide each term in $- 2 x < 0$ by $- 2$ and simplify.

Divide each term in $- 2 x < 0$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{0}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} > \frac{0}{- 2}$

Divide $x$ by $1$ .

$x > \frac{0}{- 2}$

Simplify the right side.

Divide $0$ by $- 2$ .

$x > 0$

Find the intersection of $x > 0$ and $x < 2$ .

$0 < x < 2$

Find the union of the solutions.

$0 < x < 4$

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 4$

Interval Notation:

$(0, 4)$

Solve the Inequality for x |2x-4|+2<6

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