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58 Chapter 1/Prerequisites for Calculus
The absolute value of a sum of two numbers is never larger than the sum of their absolute values. When we put this in symbols. we get the imponant triangle inequality.
T
he Triangle Inequality|a+ b\≤ |a| +| b| for all numbers a and b
EXAMPLE 6 The number a + b is less than |a| +|b| if a and b have different signs. In all other cases, |a + b| equals |a | +| b|
|-3 + 5| = |2| = 2<|-3| + |5| = 8
|3 + 5| = |8| = 8 = |3| +|5|
|-З +0| = |-3| = 3 = |-3|+ |0|
|-3-5| = |-8| = 8= -3 +-|5|.
Notice that absolute value bars in expressions such as | - 3 + 5| also work like parentheses: We do the arithmetic inside before we take the absolute value.
Absolute Values and Distance
The numbers a - b and \b - a |are always equal because
\a - b\ = |(-1)(6 - a)| =|1| |b - a | = |b - a|. (4
They give the distance between the points a and b on the number line (Fig. 1.82). The
Number-line Distance
|a - b| = |b - a |for all numbers a and b. This number is the distance between a and b on the number line.
Absolute Values and Intervals
The connection between absolute values and distance gives us a new way write formulas for intervals.
T
he inequality |a| < 5 says that the distance from a to the origin is than 5. This is the same as saying that a lies between - 5 and 5 on the number line. In symbols.|a|<5 -5
The set of numbers a with |a|<5 is the open interval from -5 to 3 (Fig. 1.83