Solve the Inequality for x 4|x-2|-4<16

$4 | x - 2 | - 4 < 16$

Step 1

Write $4 | x - 2 | - 4 < 16$ 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.

$4 (x - 2) - 4 < 16$

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 $- 1$ .

$4 (- (x - 2)) - 4 < 16$

Write as a piecewise.

${\begin{cases} 4 (x - 2) - 4 < 16 & x \geq 2 \\ 4 (- (x - 2)) - 4 < 16 & x < 2 \end{cases}$

Simplify $4 (x - 2) - 4 < 16$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 4 x + 4 \cdot - 2 - 4 < 16 & x \geq 2 \\ 4 (- (x - 2)) - 4 < 16 & x < 2 \end{cases}$

Multiply $4$ by $- 2$ .

${\begin{cases} 4 x - 8 - 4 < 16 & x \geq 2 \\ 4 (- (x - 2)) - 4 < 16 & x < 2 \end{cases}$

Subtract $4$ from $- 8$ .

${\begin{cases} 4 x - 12 < 16 & x \geq 2 \\ 4 (- (x - 2)) - 4 < 16 & x < 2 \end{cases}$

Simplify $4 (- (x - 2)) - 4 < 16$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 4 x - 12 < 16 & x \geq 2 \\ 4 (- x - - 2) - 4 < 16 & x < 2 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} 4 x - 12 < 16 & x \geq 2 \\ 4 (- x + 2) - 4 < 16 & x < 2 \end{cases}$

Apply the distributive property.

${\begin{cases} 4 x - 12 < 16 & x \geq 2 \\ 4 (- x) + 4 \cdot 2 - 4 < 16 & x < 2 \end{cases}$

Multiply $- 1$ by $4$ .

${\begin{cases} 4 x - 12 < 16 & x \geq 2 \\ - 4 x + 4 \cdot 2 - 4 < 16 & x < 2 \end{cases}$

Multiply $4$ by $2$ .

${\begin{cases} 4 x - 12 < 16 & x \geq 2 \\ - 4 x + 8 - 4 < 16 & x < 2 \end{cases}$

Subtract $4$ from $8$ .

${\begin{cases} 4 x - 12 < 16 & x \geq 2 \\ - 4 x + 4 < 16 & x < 2 \end{cases}$

Step 2

Solve $4 x - 12 < 16$ when $x \geq 2$ .

Solve $4 x - 12 < 16$ for $x$ .

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

Add $12$ to both sides of the inequality.

$4 x < 16 + 12$

Add $16$ and $12$ .

$4 x < 28$

Divide each term in $4 x < 28$ by $4$ and simplify.

Divide each term in $4 x < 28$ by $4$ .

$\frac{4 x}{4} < \frac{28}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} < \frac{28}{4}$

Divide $x$ by $1$ .

$x < \frac{28}{4}$

Simplify the right side.

Divide $28$ by $4$ .

$x < 7$

Find the intersection of $x < 7$ and $x \geq 2$ .

$2 \leq x < 7$

Step 3

Solve $- 4 x + 4 < 16$ when $x < 2$ .

Solve $- 4 x + 4 < 16$ for $x$ .

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

Subtract $4$ from both sides of the inequality.

$- 4 x < 16 - 4$

Subtract $4$ from $16$ .

$- 4 x < 12$

Divide each term in $- 4 x < 12$ by $- 4$ and simplify.

Divide each term in $- 4 x < 12$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} > \frac{12}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} > \frac{12}{- 4}$

Divide $x$ by $1$ .

$x > \frac{12}{- 4}$

Simplify the right side.

Divide $12$ by $- 4$ .

$x > - 3$

Find the intersection of $x > - 3$ and $x < 2$ .

$- 3 < x < 2$

Step 4

Find the union of the solutions.

$- 3 < x < 7$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 3 < x < 7$

Interval Notation:

$(- 3, 7)$

Step 6

Solve the Inequality for x 4|x-2|-4<16

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