Solve for x |2x-6|<4

$| 2 x - 6 | < 4$

Step 1

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

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

$2 x - 6 \geq 0$

Solve the inequality.

Add $6$ to both sides of the inequality.

$2 x \geq 6$

Divide each term in $2 x \geq 6$ by $2$ and simplify.

Divide each term in $2 x \geq 6$ by $2$ .

$\frac{2 x}{2} \geq \frac{6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{6}{2}$

Divide $x$ by $1$ .

$x \geq \frac{6}{2}$

Simplify the right side.

Divide $6$ by $2$ .

$x \geq 3$

In the piece where $2 x - 6$ is non-negative, remove the absolute value.

$2 x - 6 < 4$

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

$2 x - 6 < 0$

Solve the inequality.

Add $6$ to both sides of the inequality.

$2 x < 6$

Divide each term in $2 x < 6$ by $2$ and simplify.

Divide each term in $2 x < 6$ by $2$ .

$\frac{2 x}{2} < \frac{6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{6}{2}$

Divide $x$ by $1$ .

$x < \frac{6}{2}$

Simplify the right side.

Divide $6$ by $2$ .

$x < 3$

In the piece where $2 x - 6$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 x - 6) < 4$

Write as a piecewise.

${\begin{cases} 2 x - 6 < 4 & x \geq 3 \\ - (2 x - 6) < 4 & x < 3 \end{cases}$

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

Apply the distributive property.

${\begin{cases} 2 x - 6 < 4 & x \geq 3 \\ - (2 x) - - 6 < 4 & x < 3 \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x - 6 < 4 & x \geq 3 \\ - 2 x - - 6 < 4 & x < 3 \end{cases}$

Multiply $- 1$ by $- 6$ .

${\begin{cases} 2 x - 6 < 4 & x \geq 3 \\ - 2 x + 6 < 4 & x < 3 \end{cases}$

Step 2

Solve $2 x - 6 < 4$ when $x \geq 3$ .

Solve $2 x - 6 < 4$ for $x$ .

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

Add $6$ to both sides of the inequality.

$2 x < 4 + 6$

Add $4$ and $6$ .

$2 x < 10$

Divide each term in $2 x < 10$ by $2$ and simplify.

Divide each term in $2 x < 10$ by $2$ .

$\frac{2 x}{2} < \frac{10}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{10}{2}$

Divide $x$ by $1$ .

$x < \frac{10}{2}$

Simplify the right side.

Divide $10$ by $2$ .

$x < 5$

Find the intersection of $x < 5$ and $x \geq 3$ .

$3 \leq x < 5$

Step 3

Solve $- 2 x + 6 < 4$ when $x < 3$ .

Solve $- 2 x + 6 < 4$ for $x$ .

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

Subtract $6$ from both sides of the inequality.

$- 2 x < 4 - 6$

Subtract $6$ from $4$ .

$- 2 x < - 2$

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

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 2$ by $- 2$ .

$x > 1$

Find the intersection of $x > 1$ and $x < 3$ .

$1 < x < 3$

Step 4

Find the union of the solutions.

$1 < x < 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$1 < x < 5$

Interval Notation:

$(1, 5)$

Step 6

Solve for x |2x-6|<4

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