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
xcan 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:
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].
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 ).
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].
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:
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].
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].
Handle Infinity Symbols: Always use parentheses ( or ) when infinity ( \infty ) or negative infinity ( -\infty ) is an endpoint [1].
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:
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