Solve for x |x-12|<15

$| x - 12 | < 15$

Step 1

Write $| x - 12 | < 15$ as a piecewise.

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

$x - 12 \geq 0$

Add $12$ to both sides of the inequality.

$x \geq 12$

In the piece where $x - 12$ is non-negative, remove the absolute value.

$x - 12 < 15$

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

$x - 12 < 0$

Add $12$ to both sides of the inequality.

$x < 12$

In the piece where $x - 12$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 12) < 15$

Write as a piecewise.

${\begin{cases} x - 12 < 15 & x \geq 12 \\ - (x - 12) < 15 & x < 12 \end{cases}$

Simplify $- (x - 12) < 15$ .

Apply the distributive property.

${\begin{cases} x - 12 < 15 & x \geq 12 \\ - x - - 12 < 15 & x < 12 \end{cases}$

Multiply $- 1$ by $- 12$ .

${\begin{cases} x - 12 < 15 & x \geq 12 \\ - x + 12 < 15 & x < 12 \end{cases}$

Step 2

Solve $x - 12 < 15$ when $x \geq 12$ .

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

Add $12$ to both sides of the inequality.

$x < 15 + 12$

Add $15$ and $12$ .

$x < 27$

Find the intersection of $x < 27$ and $x \geq 12$ .

$12 \leq x < 27$

Step 3

Solve $- x + 12 < 15$ when $x < 12$ .

Solve $- x + 12 < 15$ for $x$ .

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

Subtract $12$ from both sides of the inequality.

$- x < 15 - 12$

Subtract $12$ from $15$ .

$- x < 3$

Divide each term in $- x < 3$ by $- 1$ and simplify.

Divide each term in $- x < 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{3}{- 1}$

Divide $x$ by $1$ .

$x > \frac{3}{- 1}$

Simplify the right side.

Divide $3$ by $- 1$ .

$x > - 3$

Find the intersection of $x > - 3$ and $x < 12$ .

$- 3 < x < 12$

Step 4

Find the union of the solutions.

$- 3 < x < 27$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 3 < x < 27$

Interval Notation:

$(- 3, 27)$

Step 6

Solve for x |x-12|<15

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