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Let's first assume that both a and b are positive. In this case, |a| = a and |b| = b. So, the conditions become: 1. a + b = 2 2. a + b = 8 Since both conditions cannot be true simultaneously, this case doesn't provide a valid example.

Now let's assume that a is positive and b is negative. In this case, |a| = a and |b| = -b. So, the conditions become: 1. a - b = 2 2. a - b = 8 Since both conditions cannot be true simultaneously, this case doesn't provide a valid example.

In this case, a is negative and b is positive. So, |a| = -a and |b| = b, and the conditions become: 1. -a + b = 2 2. -a + b = 8 One valid example is when a = -3 and b = 5 because both conditions are satisfied: 1. -(-3) + 5 = 3 + 5 = 8 2. -(-3) + 5 = 3 + 5 = 8

Finally, let's assume that both a and b are negative. In this case, |a| = -a and |b| = -b. So, the conditions become: 1. -(a + b) = 2 2. -a - b = 8 Let's solve the equations: From equation 1, we get a + b = -2. Now we can substitute a + b = -2 in equation 2: -(-2) = 8 2 = 8 Since the above equation is not true, this case doesn't provide a valid example. From our analysis, we found one case where the conditions hold, which leads to the first example: Example 1: a = -3, b = 5 To find more examples, we can slightly modify the values of a and b while maintaining the validity: Example 2: a = -4, b = 6 Example 3: a = -2.5, b = 4.5 Example 4: a = -3.5, b = 5.5 Thus, we have found four valid examples of pairs of real numbers a and b that satisfy both conditions |a+b| = 2 and |a|+|b| = 8.