Solve for x |4x-6|>14

$| 4 x - 6 | > 14$

Step 1

Write $| 4 x - 6 | > 14$ as a piecewise.

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

$4 x - 6 \geq 0$

Solve the inequality.

Add $6$ to both sides of the inequality.

$4 x \geq 6$

Divide each term in $4 x \geq 6$ by $4$ and simplify.

Divide each term in $4 x \geq 6$ by $4$ .

$\frac{4 x}{4} \geq \frac{6}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{6}{4}$

Divide $x$ by $1$ .

$x \geq \frac{6}{4}$

Simplify the right side.

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

$x \geq \frac{2 (3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x \geq \frac{2 \cdot 3}{2 \cdot 2}$

Cancel the common factor.

$x \geq \frac{2 \cdot 3}{2 \cdot 2}$

Rewrite the expression.

$x \geq \frac{3}{2}$

In the piece where $4 x - 6$ is non-negative, remove the absolute value.

$4 x - 6 > 14$

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

$4 x - 6 < 0$

Solve the inequality.

Add $6$ to both sides of the inequality.

$4 x < 6$

Divide each term in $4 x < 6$ by $4$ and simplify.

Divide each term in $4 x < 6$ by $4$ .

$\frac{4 x}{4} < \frac{6}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} < \frac{6}{4}$

Divide $x$ by $1$ .

$x < \frac{6}{4}$

Simplify the right side.

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

$x < \frac{2 (3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x < \frac{2 \cdot 3}{2 \cdot 2}$

Cancel the common factor.

$x < \frac{2 \cdot 3}{2 \cdot 2}$

Rewrite the expression.

$x < \frac{3}{2}$

In the piece where $4 x - 6$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 x - 6) > 14$

Write as a piecewise.

${\begin{cases} 4 x - 6 > 14 & x \geq \frac{3}{2} \\ - (4 x - 6) > 14 & x < \frac{3}{2} \end{cases}$

Simplify $- (4 x - 6) > 14$ .

Apply the distributive property.

${\begin{cases} 4 x - 6 > 14 & x \geq \frac{3}{2} \\ - (4 x) - - 6 > 14 & x < \frac{3}{2} \end{cases}$

Multiply $4$ by $- 1$ .

${\begin{cases} 4 x - 6 > 14 & x \geq \frac{3}{2} \\ - 4 x - - 6 > 14 & x < \frac{3}{2} \end{cases}$

Multiply $- 1$ by $- 6$ .

${\begin{cases} 4 x - 6 > 14 & x \geq \frac{3}{2} \\ - 4 x + 6 > 14 & x < \frac{3}{2} \end{cases}$

Step 2

Solve $4 x - 6 > 14$ for $x$ .

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

Add $6$ to both sides of the inequality.

$4 x > 14 + 6$

Add $14$ and $6$ .

$4 x > 20$

Divide each term in $4 x > 20$ by $4$ and simplify.

Divide each term in $4 x > 20$ by $4$ .

$\frac{4 x}{4} > \frac{20}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} > \frac{20}{4}$

Divide $x$ by $1$ .

$x > \frac{20}{4}$

Simplify the right side.

Divide $20$ by $4$ .

$x > 5$

Step 3

Solve $- 4 x + 6 > 14$ for $x$ .

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

Subtract $6$ from both sides of the inequality.

$- 4 x > 14 - 6$

Subtract $6$ from $14$ .

$- 4 x > 8$

Divide each term in $- 4 x > 8$ by $- 4$ and simplify.

Divide each term in $- 4 x > 8$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} < \frac{8}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} < \frac{8}{- 4}$

Divide $x$ by $1$ .

$x < \frac{8}{- 4}$

Simplify the right side.

Divide $8$ by $- 4$ .

$x < - 2$

Step 4

Find the union of the solutions.

$x < - 2$ or $x > 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - 2 or x > 5$

Interval Notation:

$(- \infty, - 2) \cup (5, \infty)$

Step 6

Solve for x |4x-6|>14

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