# How do you solve the inequality 2<=abs(4-x)<7 and write your answer in interval notation?

Jun 2, 2017

$- 3 < x \le 2$ OR $6 \le x < 11$

#### Explanation:

First, recall the rules of absolute values in inequality expressions.

Absolute value is less than something rule
If $\left\mid A \right\mid < B$, that's the same as saying $- B < A < B$

Absolute value is greater than something rule
If $B \le \left\mid A \right\mid$, that's the same as saying either $B \le A$ or $A \le - B$

The inequality we are given can be broken into two parts.

First part: Less than rule
$\left\mid 4 - x \right\mid < 7$

Applying the "less than rule" gives
$- 7 < 4 - x < 7$

Subtract $4$ from all three sides
$- 11 < - x < 3$

Divide by $- 1$ and flip the inequality signs
$- 3 < x < 11$

Second part: Greater than rule
$2 \le \left\mid 4 - x \right\mid$

Applying the "greater than rule" gives
Either $2 \le 4 - x$ or $4 - x \le - 2$

Solving both for $x$ gives

Either $x \le 2$ or $x \ge 6$

Finally, combine all solutions into one

The solution: $- 3 < x < 11$ gives the most-negative and most-positive outside boundaries.

The solution: Either $x \le 2$ or $x \ge 6$ gives the inner boundaries.

ANSWER: $- 3 < x \le 2$ OR $6 \le x < 11$

Here is what the graph would look like: