Verify Equation

[nycmath]

Precalculus
David Cohen
Edition 3
Chapter 1, Section 1.2

See attachment.

[Math Tutor]

Before plugging in numbers, it helps to see WHY the formula works. Split the fraction:

(a + b + |a - b|)/2 = (a + b)/2 + |a - b|/2

The first piece is the midpoint (average) of a and b. The second piece is half the distance between them. Start at the middle and walk half the distance toward the bigger side - you land exactly on the larger number. That is the whole idea.

Now verify the three cases.

(a) a = 1, b = 2
|a - b| = |1 - 2| = |-1| = 1
(1 + 2 + 1)/2 = 4/2 = 2
And max(1, 2) = 2. It checks. [OK]

(b) a = -1, b = -5
Careful with the signs here. Since -1 is to the RIGHT of -5 on the number line, the larger number is -1, not -5.
a - b = -1 - (-5) = -1 + 5 = 4, so |a - b| = 4
a + b = -1 + (-5) = -6
(-6 + 4)/2 = -2/2 = -1
And max(-1, -5) = -1. It checks. [OK]

(c) a = b = 10
|a - b| = |10 - 10| = 0
(10 + 10 + 0)/2 = 20/2 = 10
And max(10, 10) = 10, the common value. It checks. [OK]

Notice what case (c) shows: when a = b the absolute-value term vanishes and the formula collapses to the average, which is that common value. That is exactly why the definition of max was extended to cover a = b.

[nycmath]

Before plugging in numbers, it helps to see WHY the formula works. Split the fraction:

(a + b + |a - b|)/2 = (a + b)/2 + |a - b|/2

The first piece is the midpoint (average) of a and b. The second piece is half the distance between them. Start at the middle and walk half the distance toward the bigger side - you land exactly on the larger number. That is the whole idea.

Now verify the three cases.

(a) a = 1, b = 2
|a - b| = |1 - 2| = |-1| = 1
(1 + 2 + 1)/2 = 4/2 = 2
And max(1, 2) = 2. It checks. [OK]

(b) a = -1, b = -5
Careful with the signs here. Since -1 is to the RIGHT of -5 on the number line, the larger number is -1, not -5.
a - b = -1 - (-5) = -1 + 5 = 4, so |a - b| = 4
a + b = -1 + (-5) = -6
(-6 + 4)/2 = -2/2 = -1
And max(-1, -5) = -1. It checks. [OK]

(c) a = b = 10
|a - b| = |10 - 10| = 0
(10 + 10 + 0)/2 = 20/2 = 10
And max(10, 10) = 10, the common value. It checks. [OK]

Notice what case (c) shows: when a = b the absolute-value term vanishes and the formula collapses to the average, which is that common value. That is exactly why the definition of max was extended to cover a = b.

Another detailed reply from Math Tutor. Very cool.