Notes 2.3 & 2.2: Inequalities and Sets

Flashcards based on lecture notes covering set operations (union and intersection), graphing inequalities, compound inequalities, and inequality sign reversal rules.

Inequality Symbol Reversal Rule

When multiplying or dividing both sides of an inequality by a negative number, the inequality symbol must be reversed.

Union of Sets ($\cup$)

The set containing all elements belonging to set $A$, set $B$, or both. For example, if $A = \{1, 2, 3\}$ and $B = \{2, 3, 4, 5\}$, then $A \cup B = \{1, 2, 3, 4, 5\}$.

Intersection of Sets ($\cap$)

The set containing only elements that belong to both set $A$ and set $B$. For example, if $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, then $A \cap B = \{3\}$.

Compound Inequalities

Two inequalities joined by "and" or "or".

Graph of $x > -2$

A number line representation with an open circle at $-2$ and an arrow pointing to the right.

Graph of $x \le 1$

A number line representation with a closed circle at $1$ and an arrow pointing to the left.

Union Example: $x < 2 \cup x \ge 5$

A compound inequality joined by union, expressed in interval notation as $x \in (-\infty, 2) \cup [5, \infty)$.

Intersection Example: $x \ge 2 \text{ and } x \le 5$

A compound inequality joined by intersection, expressed in interval notation as $x \in [2, 5]$.

Compound Inequality Example: $-2 \le 3x + 1 \le 7$

A compound "and" inequality solved as $-1 \le x \le 2$, which is written in interval notation as $x \in [-1, 2]$.

Compound Inequality with Negative Coefficient: $-2 \le -3x + 1 \le 7$

A compound inequality where dividing by $-3$ reverses the inequality signs, giving $-2 \le x \le 1$, or $x \in [-2, 1]$ in interval notation.