Question

How do you solve and graph `abs(-h+1.5)<3`?

Answer 1

$$1.5 < h < 4.5$$ .

Explanation:

When you have an absolute value in an inequality you have to treat the two cases.
The first case is when the argument of the absolute value is positive $$-h+1.5\ge 0$$
that is when $$-h \ge -1.5$$
$$h\le1.5$$
and in this case you can simply remove the absolute value and treat it as a normal inequality
$$-h+1.5<3$$
$$-h<1.5$$
$$h>-1.5$$ .
So when `h` is smaller or equal to `1.5` it has to be greater than `-1.5` in order to satisfy the equation, or in formula
$$-1.5 < h \le 1.5$$

The second case is when
$$-h+1.5< 0$$ or
$$h> 1.5$$
in this case the inequality is
$$-(-h+1.5)<3$$
$$h-1.5<3$$
$$h<4.5$$
that is
$$1.5 < h < 4.5$$ .

Combining the two solutions (because the intervals overlaps in `1.5` we have that $$-1.5 < h < 4.5$$ and the graph is the segment between `-1.5` and `4.5`.