The equation |a + 5| = −5 − a is not true for any value of a, while the equation |a + 5| = a + 5 is true for values of a greater than or equal to -5.
The absolute value of a number is defined as the distance of the number from zero on a number line. Thus, |a + 5| is equal to the distance of (a + 5) from zero on a number line. Since distance is always non-negative, |a + 5| is always non-negative.
On the other hand, the expression −5 − a is always negative, since the sum of a negative number (-5) and any number (a) is always negative.
Therefore, for the equation |a + 5| = −5 − a to be true, the absolute value of (a + 5) would have to be negative, which is impossible since absolute value is always non-negative.
For the equation |a + 5| = a + 5 to be true, a must be greater than or equal to -5, because when a is greater than or equal to -5, (a + 5) is already non-negative, so the absolute value of (a + 5) is equal to (a + 5).
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