Interval Notation

Interval Notation: Detailed Notes

Purpose:

Interval notation is a concise way to express the solution set for an inequality or a range of numbers [1].

General Approach:

The primary recommended step is to graph the solution on a number line first [1].
- Why graph? Graphing provides a visual representation that simplifies understanding the boundaries and direction of the solution set, making it much easier to accurately translate to interval notation [1].
When translating the graph to interval notation, you should always write the solution from left to right [1]. This ensures consistency and correctness in the order of the endpoints.

Key Symbols and Their Meanings:

Parentheses $( )$ [1]:
- Meaning: Indicate that a number is not included in the solution set [1].
- Number Line Representation: Associated with an open circle on a number line, signifying that the exact point is excluded [1].
- Usage with Infinity: Always used with infinity symbols (positive $\infty$ or negative $-\infty$ ) because infinity represents an unbounded concept, not a specific included number [1].
- Inequalities: Used for strict inequalities: "greater than" ( > ) or "less than" ( < ) [1].
Brackets $[ ]$ [1]:
- Meaning: Indicate that a number is included in the solution set [1].
- Number Line Representation: Associated with a closed circle on a number line, signifying that the exact point is part of the solution [1].
- Inequalities: Used for inclusive inequalities: "greater than or equal to" ( $\ge$ ) or "less than or equal to" ( $\le$ ) [1].
Infinity Symbols ( $\infty$ , $-\infty$ ) [1]:
- Positive Infinity ( $\infty$ ): Always located to the right on a number line, representing numbers that increase without bound [1].
- Negative Infinity ( $-\infty$ ): Always located to the left on a number line, representing numbers that decrease without bound [1].
- Notation Rule: Always associated with parentheses because infinity is not a specific number that can be included in a set [1].
- Appearance: The symbol itself resembles a sideways figure eight.
Union Symbol $(U)$ [1]:
- Purpose: Used to connect two or more separated parts of a solution set [1].
- Application: Applies when the solution involves an "or" condition, meaning the variable x can satisfy one range or another, but not necessarily fall within a single continuous interval.

Steps for Expressing Solutions:

Plot the Solution on a Number Line:
1. Draw the Number Line: Create a horizontal line, typically placing $0$ in the middle for reference. Label the ends with $-\infty$ on the far left and $+\infty$ on the far right [1].
2. Identify Relevant Numbers: Mark the critical numbers from your inequality on the number line. Ensure they are placed in their correct relative positions (e.g., $-1$ to the left of $2$ ).
3. Determine Circle Type:
  - If the inequality uses > or < , place an open circle at the corresponding number. This signifies the number is not included [1].
  - If the inequality uses $\ge$ or $\le$ , place a closed circle at the corresponding number. This signifies the number is included [1].
4. Shade the Correct Direction/Region:
  - For "greater than" ( > ) or "greater than or equal to" ( $\ge$ ), shade to the right of the circle [1].
  - For "less than" ( < ) or "less than or equal to" ( $\le$ ), shade to the left of the circle [1].
  - For compound inequalities like a < x < b (an "AND" type), the solution is the region between the two numbers. Shade the segment of the number line connecting the two circles [1].
  - For compound inequalities like x < a \text{ or } x > b (an "OR" type), shade two separate regions: one to the left of $a$ and one to the right of $b$ [1]. There will be a gap between the shaded regions.
Translate to Interval Notation:
1. Read the Graph from Left to Right: Always start from the leftmost point of your shaded region (which could be $-\infty$ ) and move towards the rightmost point (which could be $+\infty$ ) [1].
2. Choose Appropriate Symbols for Endpoints:
  - If an endpoint is marked with an open circle on the number line, use a parenthesis $($ or $)$ for that number in interval notation [1].
  - If an endpoint is marked with a closed circle on the number line, use a bracket $[$ or $]$ for that number in interval notation [1].
3. Handle Infinity Symbols: Always use parentheses $($ or $)$ when infinity ( $\infty$ ) or negative infinity ( $-\infty$ ) is an endpoint [1].
4. Connect Separated Parts with a Union Symbol: If your number line graph shows two or more separated shaded regions, write each region in its own interval notation and then connect them using the union symbol $U$ [1].

Examples:

Inequality: x > 4 [1]
- Step 1: Plot on Number Line:
  - Place an open circle at $4$ (because of > , $4$ is not included).
  - Shade to the right of $4$ towards $+\infty$ (because of > , meaning values greater than $4$ ).
- Step 2: Write in Interval Notation:
  - Reading from left to right, the shaded region starts just after $4$ and goes indefinitely to the right.
  - Since $4$ is not included, use a parenthesis: $(4$ .
  - Since the shading goes to positive infinity, use a parenthesis for infinity: $\infty )$ .
  - Interval Notation: $(4, \infty)$ [1]
Inequality: $x \ge 2$ [1]
- Step 1: Plot on Number Line:
  - Place a closed circle at $2$ (because of $\ge$ , $2$ is included).
  - Shade to the right of $2$ towards $+\infty$ (because of $\ge$ ).
- Step 2: Write in Interval Notation:
  - Reading from left to right, the shaded region starts at $2$ and goes indefinitely to the right.
  - Since $2$ is included, use a bracket: $[2$ .
  - Since the shading goes to positive infinity, use a parenthesis for infinity: $\infty )$ .
  - Interval Notation: $[2, \infty)$ [1]
Inequality: x < 3 [1]
- Step 1: Plot on Number Line:
  - Place an open circle at $3$ (because of < , $3$ is not included).
  - Shade to the left of $3$ towards $-\infty$ (because of < ).
- Step 2: Write in Interval Notation:
  - Reading from left to right, the shaded region starts from negative infinity and goes up to just before $3$ .
  - Negative infinity is always represented with a parenthesis: $(-\infty$ .
  - Since $3$ is not included, use a parenthesis: $3)$ .
  - Interval Notation: $(-\infty, 3)$ [1]
Inequality: $x \le -1$ [1]
- Step 1: Plot on Number Line:
  - Place a closed circle at $-1$ (because of $\le$ , $-1$ is included).
  - Shade to the left of $-1$ towards $-\infty$ (because of $\le$ ).
- Step 2: Write in Interval Notation:
  - Reading from left to right, the shaded region starts from negative infinity and goes up to $-1$ .
  - Negative infinity is always represented with a parenthesis: $(-\infty$ .
  - Since $-1$ is included, use a bracket: $-1]$ .
  - Interval Notation: $(-\infty, -1]$ [1]
Compound Inequality (AND type): 2 < x \le 6 [1]
- Step 1: Plot on Number Line:
  - Place an open circle at $2$ (because of < ).
  - Place a closed circle at $6$ (because of $\le$ ).
  - Shade the region between $2$ and $6$ (because $x$ is greater than $2$ AND less than or equal to $6$ ).
- Step 2: Write in Interval Notation:
  - Reading from left to right, the shaded region is contained between $2$ and $6$ .
  - Since $2$ is not included, use a parenthesis: $(2$ .
  - Since $6$ is included, use a bracket: $6]$ .
  - Interval Notation: $(2, 6]$ [1]
Compound Inequality (AND type): -3 \le x < 4 [1]
- Step 1: Plot on Number Line:
  - Place a closed circle at $-3$ (because of $\le$ ).
  - Place an open circle at $4$ (because of < ).
  - Shade the region between $-3$ and $4$ .
- Step 2: Write in Interval Notation:
  - Reading from left to right, the shaded region is contained between $-3$ and $4$ .
  - Since $-3$ is included, use a bracket: $[-3$ .
  - Since $4$ is not included, use a parenthesis: $4)$ .
  - Interval Notation: $[-3, 4)$ [1]
Compound Inequality (OR type): x < -2 \text{ or } x \ge 5 [1]
- Step 1: Plot on Number Line:
  - For x < -2 : Place an open circle at $-2$ and shade to the left towards $-\infty$ .
  - For $x \ge 5$ : Place a closed circle at $5$ and shade to the right towards $+\infty$ .
  - Observe the gap between $-2$ and $5$ . This indicates two separate intervals.
- Step 2: Write in Interval Notation:
  - Left part: The shaded region goes from $-\infty$ to just before $-2$ . This is written as $(-\infty, -2)$ . Both ends use parentheses because $-\infty$ is always parenthesis and $-2$ is not included (open circle) [1].
  - Right part: The shaded region goes from $5$ to $+\infty$ . This is written as $[5, \infty)$ . $5$ is included (closed circle), so it gets a bracket; $+\infty$ always gets a parenthesis [1].
  - Connect with Union: Since there are two separate parts, use the union symbol $U$ .
  - Interval Notation: $(-\infty, -2) \text{ U } [5, \infty)$ [1]
Compound Inequality (OR type): x \le 1 \text{ or } x > 2 [1]
- Step 1: Plot on Number Line:
  - For $x \le 1$ : Place a closed circle at $1$ and shade to the left towards $-\infty$ .
  - For x > 2 : Place an open circle at $2$ and shade to the right towards $+\infty$ .
  - Observe the gap between $1$ and $2$ . This indicates two separate intervals.
- Step 2: Write in Interval Notation:
  *