Understanding Interval Notation and Compound Inequalities

Interval notation represents a set of numbers starting with the smallest value and ending with the largest value, separated by a comma.
The first slot in the notation is reserved for the lower bound of the set.
Closing operators, such as brackets $[ ]$ and parentheses $( )$ , indicate whether the boundary numbers are included in the set.

Parentheses $( )$ are used when the limit of the set is not attained. This applies to:
- Strict inequalities ( $<$ or $>$ ).
- Open circles on a number line.
- Infinity symbols ( $\infty$ or $-\infty$ ).
Brackets $[ ]$ are used when the boundary value is included in the set. This applies to:
- Non-strict inequalities ( $\le$ or $\ge$ ).
- Closed circles on a number line.
An interval can use different operators on each side depending on the specific conditions at each critical point.

For a simple inequality such as $x < 7$ , the notation is written as $(-\infty, 7)$ .
Solving a compound "and" inequality, such as $3 \le 5x - 2 < 7$ , yields a single interval:
- Add $2$ to all sides: $5 \le 5x < 9$ .
- Divide by $5$ : $1 \le x < \frac{9}{5}$ .
- The resulting interval notation is $[1, \frac{9}{5})$ , indicating that $1$ is included but the set stops just before attaining $\text{nine fifths}$ .

When an "or" inequality results in two arrows pointing in opposite directions on a number line, two separate intervals are required.
The Union symbol $\cup$ (a capital U without a tail) is used to join these intervals into one solution set.
Example calculation for an "or" case:
- Solving $-7x + 3 < -18$ involves dividing by a negative number and flipping the sign, resulting in $x > 3$ .
- Solving $3x \le 0$ results in $x \le 0$ .
- The combined notation for these two intervals is $(-\infty, 0] \cup (3, \infty)$ .