Question

Write the expression |x − 9| + |x − 6| without the absolute value symbols if x is in the given interval.

(−∞, 1)

Answer 1

Recall the definition of absolute value.

`|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}`

Now,

`x<1 \implies x-9 \le -8 < 0 \implies |x-9| = -(x-9) = 9 - x`

and

`x<1 \implies x-6 \le -6 < 0 \implies |x-6| = -(x-6) = 6 - x`

so that

`|x-9| + |x-6| = (9-x) + (6 - x) = \boxed{15 - 2x}`