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Hon Alg 2: Unit 1 Number Lines and Interval Notation Interval Notation: Rather than writing out inequalities to describe the range or domain of values we use brackets and parentheses. Parentheses “(“ and “)” are like less than “<” or greater than “>” statements. Brackets “[“ and “]” are like less than or equal to “<” or greater than or equal to “>” statements. Infinity “∞” To show that numbers will always get larger we use positive infinity with a parentheses “∞)” To show the numbers will always get smaller we use negative infinity with parentheses “( -∞” EXAMPLES: Each number line is not drawn to scale. 8 1. _______________________________ -4 2. _______________________________ -3 2 3. _______________________________ -7 5 4. _______________________________ 5. _______________________________ Union “U”: unifies different sections of a number line together as an overall answer -1 -6 6. ___________________________________ -3 10 4 7. ___________________________________ 8. ___________________________________ 7 5 9 12 -5 -8 9. ___________________________________ -9 10. ___________________________________ -2 3 7 GRAPH INTERPRETATION WITH INTERVAL NOTATION Intervals represent the x-values (domain) of a graph, while y-values Number you are often describing the y-values (range) of the points. Line x-values Undefined “Ø” and Zero “0” y-value are important and specifically listed on the number line. The remaining number line describe where the graph has positive or negative y-values. EXAMPLE #1: Ø pos 0 pos -6 neg -2 1. What x-values give zero y-values? 0 pos 5 Ø neg 11 2. What x-values give undefined y-values? 3. Write interval notation for x-values when the graph is … 3a. negative. 3b. positive. 3c. greater than or equal to zero. EXAMPLE #2: 3d. less than or equal to zero. 0 neg -9 1. What x-values give zero y-values? pos 0 neg -5 Ø neg 0 0 pos 7 2. What x-values give undefined y-values? 3. Write interval notation for x-values when the graph is … 3a. negative. 3b. positive. 3c. greater than or equal to zero. EXAMPLE #3: 1. Draw your own number line Interpretation of the graph. 3d. less than or equal to zero. (-6, 3) (-4, 0) (3, 0) (0, -5) 2. Write interval notation for x-values when the graph is … 2a. negative. 2b. positive. (10, 0) (5, -5) (8, -5) EXAMPLE #4: Draw your own number line interpretation of the graph. -8 -6 -4 2 -2 4 6 8 14 16 10 Write interval notation for x-values when the graph is … a. negative. b. positive. EXAMPLE #5: Draw your own number line interpretation of the graph. -4 -3 -2 -1 1 2 3 4 5 6 7 8 3. Write interval notation for x-values when the graph is … 3a. negative. 3b. positive. 3c. greater than or equal to zero. 3d. less than or equal to zero. How can you determine a number line statement from an equation without graphing? Step 1: Find all the zeros and undefined x-values for the equation. Step 2: Try any x-value between consecutive zeros and/or undefined values to determine if the yvalues are positive or negative in that region. Example: y = (x + 4) (x – 6) (x +1) Why do you think -4, -1, and 6 are given in the number line as zeros? 0 0 0 -4 -1 6 PRACTICE: Consider why you think the given values are zeros or undefined? 1) y = 3(x + 7) (x – 2) (x – 5) 0 0 0 -7 2) y 2 5 ( x 2)( x 3) x 0 Ø 0 -2 0 3 3) y = (2x – 1) (x – 5) (x + 4) (x – 3) 0 0 1 -4 4) y 5) y /2 0 0 3 5 ( x 4)( x 6) ( x 6)( x 1) ( x 3)( x 3)( x 9) ( x 2) 0 Ø 0 Ø -1 4 6 Ø 0 0 2 -9 -3 3 Additional Practice: For each equation, make a complete number line statement. 1) y = (x + 5) (x – 7) ( x 3)( x 1) 4) y ( x 3)( x 6) 2) y ( x 1)( x 5) ( x 2) 5) y 3) y = (x + 3) (x + 6) (x – 2) ( x 4)( x 2)( x 1) x ( x 4)