using absolute function to translate the inequality

How to use the absolute value function to translate each of the following statements into a single inequality.

(a) $$\ x ∈ (-4,10)$$

(b) $$\ x ∈ (-\infty,2] \cup[9,\infty)$$

I think in the first one the absolute value of $$\ x$$ should be greater than 4 and less than 10. is that correct? because the distance from $$\ x$$ to $$\ 0$$ should be between $$\ 4$$ and$$\ 10$$ in order for $$\ x$$ to belong in this interval.

a) Since $$3$$ is in the midlle of $$(-4,10)$$ we see that $$x$$ is at most $$7$$ (but not equal) distance from $$3$$, so $$|x-3|<7$$

b) Since $$x$$ is at least $${7\over 2}$$ from $${11\over 2}$$ we have $$|x-{11\over 2}|\geq {7\over 2}$$

• can u please explain what's is case if the interval has a value instead of negative infinity and infinity in b – John C. Dec 15 '19 at 22:34

$$x \in (-4,10)\implies$$

$$-4 < x < 10\implies$$

$$-4 + k < x + k < 10 + k$$

If we have $$M=|10+k| = |-4+k|$$ (and presumably $$-M= -4k < 0 < 10+k=M$$) then we would have $$-M < x + k < M$$ so $$|x + k| .

So if $$10+k = -(-4+k)=4-k$$ then $$2k = -6$$ and $$k -3$$ and $$M=10 -3=7$$ and

$$-4 -3 < x - 3< 10-3$$

$$-7 < x-3 < 7$$

$$|x-3| < 7$$.

b) $$x ∈ (-\infty,2] \cup[9,\infty)$$ means

$$x \le 2$$ or $$x \ge 9$$.

Again $$x + k \le 2+k$$ or $$x \ge 9+k$$. If we can get $$M=9+k = -(2+k)$$ we would have $$|x+k | \ge M$$.

$$9+k = -(2+k) \implies k=-5.5$$ so $$x-5.5 \le -3.5$$ and $$x-5.5 \ge 3.5$$ so $$|x-5.5| \ge 3.5$$.