How do you solve the inequality -6abs(2-x)<-12?

May 11, 2018

The inequality is fulfilled when
$x < 0$ or $x > 4$, i.e. $x \notin \left[0 , 4\right]$

Explanation:

Let's draw the graph of this inequality:

As we see on the graph x must be outside the area [0, 4] to solve this equation, i.e. $x < 0$ or $x > 4$

To solve this matematically we need to remember that $| 2 - x |$ means the absolut value of $\left(2 - x\right)$, i.e.
$2 - x$ when $x \le 2$
$x - 2$ when $x \ge 2$

We, therefore must consider two inequalities, one when $x \le 2$, the other when $x \ge 2$

$x \le 2$:
$- 6 \left(2 - x\right) = - 12 + 6 x < - 12$ Add 12 on both sides:
$- 12 + 6 x + 12 < - 12 + 12$
or $6 x < 0$. Therefore $x < 0$

$x \ge 2$:
$- 6 \left(x - 2\right) = - 6 x + 12 < - 12$ Add $6 x + 12$ on both sides:
$- 6 x + 12 + 6 x + 12 < - 12 + 6 x + 12$
$24 < 6 x$ If we divide on 6 on both sides, we end up with:
$x > 4$

The inequality is fulfilled when
$x < 0$ or $x > 4$, in other words $x \notin \left[0 , 4\right]$