How to Use Union in Inequalities?

Union in inequalities is used to combine sets of numbers that satisfy either one inequality or another.

When you encounter multiple inequalities connected by the word "or," you are dealing with the union of their solution sets. This means the solution includes any value of the variable that satisfies at least one of the inequalities.

According to the provided reference, "When we see a statement like 'x < 7 or x ≥ 11', written in set notation as {x : x < 7 or x ≥ 11}, the word or denotes the union of the two sets of numbers which satisfy each inequality. Thus, {x : x < 7 or x ≥ 11} = {x : x < 7}∪{x : x ≥ 11}. This is the set of values which satisfy either x < 7 or x ≥ 11."

Essentially, the union symbol ($\cup$) represents the combination of two or more sets. In the context of inequalities, these sets are the collections of numbers that make each individual inequality true.

Understanding the "Or" Connective

• "Or" means Union: If you have inequality A or inequality B, the solution set is the union of the solution set for A and the solution set for B.
• Satisfying "Or": A number satisfies an "or" statement if it is in the solution set of the first inequality, or in the solution set of the second inequality, or in both (though with simple inequalities like those often connected by "or", the solution sets may not overlap).

Representing Solutions Using Union

Solutions involving the union of inequalities are typically expressed in one of the following ways:

1. Inequality Notation: Keep the original "or" statement, e.g., $x < 7$ or $x \geq 11$.
2. Set-Builder Notation: Use the union symbol, e.g., ${x \mid x < 7} \cup {x \mid x \geq 11}$.
3. Interval Notation: Write the solution sets for each inequality as intervals and connect them with the union symbol.
   • For $x < 7$, the interval is $(-\infty, 7)$.
   • For $x \geq 11$, the interval is $[11, \infty)$.
   • The union is $(-\infty, 7) \cup [11, \infty)$.

Practical Example

Let's solve and represent the solution for the statement: $x + 2 \leq 3$ or $2x > 10$.

Here's how to approach it using the concept of union:

1. Solve Each Inequality Separately:

   • Inequality 1: $x + 2 \leq 3$
     • Subtract 2 from both sides: $x \leq 3 - 2$
     • $x \leq 1$
     • Solution set 1: ${x \mid x \leq 1}$ or $(-\infty, 1]$
   • Inequality 2: $2x > 10$
     • Divide both sides by 2: $x > 10 / 2$
     • $x > 5$
     • Solution set 2: ${x \mid x > 5}$ or $(5, \infty)$

2. Combine Using Union: Since the original statement used "or," the solution is the union of the individual solution sets.

   • In set-builder notation: ${x \mid x \leq 1} \cup {x \mid x > 5}$
   • In interval notation: $(-\infty, 1] \cup (5, \infty)$
   • In inequality notation: $x \leq 1$ or $x > 5$

This means any number less than or equal to 1, or any number greater than 5, will satisfy the original statement. Numbers between 1 (exclusive) and 5 (inclusive) do not satisfy the statement.

Using union in inequalities allows you to clearly define the set of all possible values that fulfill compound conditions linked by "or".