1.1 Absolute Value and Comparing Real Numbers

Integers and Comparisons

• Understanding integers: whole numbers that can be positive, negative, or zero.

• Comparing Numbers:

  • Use symbols: < (less than), > (greater than).

    • Example Comparisons:

      • 17 > 6

      • 29 < 143

      • -8 > -52 (thinking about the number line helps)

      • -75 < 0

Number Line Concept

• The number line helps visualize comparisons:

  • Positive numbers are to the right.

  • Negative numbers are to the left, with zero in the middle.

  • Example: 8 > -52 because 8 is positioned further right (closer to zero) than -52.

Ordering Numbers

• Writing numbers from smallest to largest:

  • List:

    • Positive numbers: 5, 2 (order: 2, 5)

    • Negative numbers: -1, -6, -9 (order: -9, -6, -1)

  • Final ordered list: -9, -6, -1, 0, 2, 5

Opposites of Numbers

• The opposite of a number flips its sign:

  • Opposite of 19 is -19

  • Opposite of -45 is 45

Absolute Values

• Definition: Absolute value is the distance from zero; always non-negative.

• Key Examples:

  • ABS(76) = 76

  • ABS(-34) = 34

  • ABS(0) = 0

Using a Calculator for Absolute Values

• Finding absolute values:

  • On a calculator, access through the "Math" menu, under "Number" section.

  • Demonstration with:

    • ABS(76) and ABS(-34).

Negative Absolute Values

• Adding a negative in front of the absolute value changes the result:

  • Example: -ABS(29) = -29

  • Example: -ABS(-15) = -15

Evaluating Negative Absolute Values

• Example: Evaluating -ABS(x) when x = -2:

  • Calculation: -ABS(-2) = -2 (input in the same way as in the calculator).

Evaluating Expressions with Absolute Values

• Expression: Negative absolute value of x + y when x = 5, y = 20:

  • Calculation: -ABS(5 + 20) = -25

Conclusion

• Recap on integers, comparisons, and understanding absolute values.