Problem 50 - College Algebra

Question

Write the interval in set-builder notation. $$ [-4,1] \cup[9, \infty) $$

Step-by-Step Solution

Answer

\( \{ x \mid -4 \leq x \leq 1 \text{ or } x \geq 9 \} \)

Step 1: Understand Interval Notation

The given interval in interval notation is \([-4, 1] \cup [9, \, \infty)\). This means the union of two intervals. The first is \([-4, 1]\), which includes all numbers from \(-4\) to \(1\) inclusive. The second is \([9, \, \infty)\), which includes all numbers starting from \(9\) to infinity.

Step 2: Translate Each Interval to Set-Builder Notation

Set-builder notation expresses the set of elements meeting a certain condition. For the interval \([-4, 1]\), the condition is: all numbers \(x\) such that \(-4 \leq x \leq 1\). For \([9, \, \infty)\), the condition is: all numbers \(x\) such that \(x \geq 9\).

Step 3: Write the Union of the Sets in Set-Builder Notation

Write the set-builder notation of the union of the two intervals. We combine the two sets of conditions using the logical 'or', as the union represents all elements that satisfy either condition. Thus, the set-builder notation is: \[ \{ x \mid -4 \leq x \leq 1 \text{ or } x \geq 9 \} \]

Key Concepts

Set-Builder Notation

Set-builder notation is a concise way to describe a set by specifying a property that its members must satisfy. It involves a combination of elements and a condition.
For example, if you want to express a set of integers between -4 and 1, you can use set-builder notation to say: "all numbers \(x\) that satisfy the condition \(-4 \leq x \leq 1\)."

• The general formula is: \( \{ x \mid \text{condition on } x \} \).
• "\(x\mid\)" means "\(x\) such that" in this context.

Let's apply this to find the set-builder notation for intervals in mathematics. Notice how it allows complex sets to be described in simple terms using conditions.

Union of Intervals

A union of intervals is a combination of two or more intervals to include all numbers that belong to either interval. The union is denoted by the symbol \(\cup\).
It's like including items from two different collections into one without repetition. For instance, consider two intervals: \([-4, 1]\) and \([9, \infty)\). The union of these intervals, \([-4, 1] \cup [9, \infty)\), encompasses all numbers from both intervals without omitting any values in between.
When translating a union of intervals into set-builder notation, we use the logical "or" to join conditions from the different intervals. This expresses that elements must satisfy at least one of these conditions, thereby covering both sets effectively.

Conditions in Sets

Conditions in sets define the rules that an element must follow to be part of the set. These are often described in set-builder notation using logical symbols, such as "and" (\(\land\)) and "or" (\(\lor\)).

• The "and" condition means elements must satisfy all given requirements.
• The "or" condition allows elements to satisfy at least one condition.

For the union of intervals like \([-4, 1] \cup [9, \infty)\), the set-builder notation uses "or" to highlight that elements can either be between -4 and 1 or greater than or equal to 9. So, the corresponding set is \(\{x \mid -4 \leq x \leq 1 \text{ or } x \geq 9\}\). This flexibility in defining conditions allows for a robust method of representing complex sets through simple mathematical expressions.