How would the equation be exactly transformed given that either a or b or both are negative?
I'd like to stress that this is a hypothetical situation, because in the question we show that b > a > 0.
So let's try the proposed conditions. |a + 3| + |-b| + |b – a| + |ab| = ?
1) b < 0 In this case we can simplify |-b| = -b, because -b > 0. But we can NOT simplify any other absolute value, because we know nothing about a.
2) a < 0 In this case we can NOT simplify any absolute value, because
|a + 3| can be a + 3 (if a is from -3 to 0) OR |a + 3| can be -a – 3 (if a is less than -3)
Any other absolute value contains b and we know nothing about it.
3) a < 0 and b < 0 In this case we can simplify: |-b| = -b, because -b > 0; |ab| = ab, because ab > 0.
We can NOT simplify: |a + 3| for the same reasons as in 2) |b – a| can be b – a (if b > a) |b – a| can be -b + a (if b < a).
You may try to plug some numbers in the absolute values we could NOT simplify: 3) a < 0 and b < 0 a = -1 < 0 , b = -4 < 0 |a + 3| = |-1 + 3| = -1 + 3 = 2 in this case we used |a + 3| = a + 3, because (a = -1 is between -3 and 0) |b – a| = |-4 – (-1)| = -(-4) + (-1) = 3 in this case we used |b – a| = -b + a, because (b = -4 < -1 = a)
a = -4 < 0 , b = -1 < 0 |a + 3| = |-4 + 3| = -(-4) – 3 = 1 in this case we used |a + 3| = -a – 3, because (a = -1 is less than -3) |b – a| = |-1 – (-4)| = -1 – (-4) = 3 in this case we used |b – a| = b – a, because (b = -1 > -4 = a)
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