The solution of the inequality, |x + 6| ≥ 5 is given as follows,
(A) x > -1
(B) It follows that x either belongs to the interval (-∞, -11) or to the interval (-1,∞) for the given absolute inequality.
(A) Solving the Given Inequality:
|x + 6| ≥ 5
x+6 ≥ 5 or x+6 ≤ (-5)
x+6-6 ≥ 5-6 or x+6-6 ≤ (-5)-6
x ≥ -1 or x ≤ -11
Hence the solution of the given inequality is given as,
x ∈ (-∞, -11] or x ∈ [-1, ∞)
(B) Description of the Solution
From the given absolute inequality, it can be concluded that x either belongs to the interval (-∞, -11] or to the interval [-1, ∞). Hence, on the number line, the solution of the inequality will be shown as a shaded region from -∞ to -11 and from -1 to ∞. The area between -1 and -11 will remain unshaded.
In the deduced solution, the interval is either open or closed on both sides. Open interval means the extreme value is not includes. On the other hand, closed interval indicates inclusion of the extreme point in the solution of the inequality.
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