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# solHMK2 - Section 2.2#3 If x y z then |x y | |y z | = y x z...

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#3) If x y z , then | x - y | + | y - z | = y - x + z - y = z - x = | x - z | . If y < x z , then | y - z | = z - y > z - x = | x - z | ; therefore | x - y | + | y - z | > | z - x | since | x - y | > 0. Finally, if x z < y , then | y - x | = y - x > z - x = | x - z | ; therefore | x - y | + | y - z | > | z - x | since | x - y | > 0. Geometrically, the triangle inequality is actually equality if and only if y is on the line between x and z . #7) If x 2, then | x + 1 | + | x - 2 | = x + 1 + x - 2 = 2 x - 1. 2 x - 1 = 7 has solution x = 4 which indeed lies in the region x 2. If - 1 x < 2, then | x + 1 | + | x - 2 | = x + 1 + 2 - x = 3. 3 6 = 7, so there is no solution in this range. Finally, if x < - 1, then | x + 1 | + | x - 2 | = - x - 1 + 2 - x = 1 - 2 x . 1 - 2 x = 7 has solution x = - 3 < - 1. Thus, the equation has two solutions: x = 4 and x = - 3. 1 -1 1 y = |x| - |x-1| #9) #12) (a) { ( t, t ) and ( t, - t ): t R } . |y| = |x|

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solHMK2 - Section 2.2#3 If x y z then |x y | |y z | = y x z...

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