Solve: 2|x + 7|−4 ≥ 0
Express the answer in set-builder notation.
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Express the answer in interval notation..
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[–9, –5]
(–9, 5)
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Answers
Answer:
{ x | x ≤ -9 or x ≥ -5 ∀ x ∈ R}
(-∞, 9] ∪[-5, ∞)
Step-by-step explanation:
Given inequality,
2|x + 7|−4 ≥ 0
2|x + 7| ≥ 4
|x + 7| ≥2
⇒ ± (x + 7) ≥ 2, ( ∵ |x| = x if x > 0 while |x|=-x if x < 0 )
⇒ -(x + 7) ≥ 2 or x + 7 ≥ 2
⇒ x + 7 ≤ -2 or x + 7 ≥ 2 ( a < b ⇒ ca > cb if c < 0 )
⇒ x ≤ -2 - 7 or x ≥ 2 - 7 ( a < b ⇒ a + c < b + c ∀ c ∈R )
⇒ x ≤ -9 or x ≥ -5
⇒ { x | x ≤ -9 or x ≥ -5 ∀ x ∈ R}
or x ∈ (-∞, 9] ∪[-5, ∞)
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Given : 2|x + 7|−4 ≥ 0
To Find : Express the answer in interval notation.
Solution:
2|x + 7| − 4 ≥ 0
| x | = x if x ≥ 0
= - x if x < 0
|x + 7| = (x + 7) if x + 7 ≥ 0 => x ≥ -7
2(x + 7) - 4 ≥ 0
=> 2x + 14 - 4 ≥ 0
=> 2x + 10 ≥ 0
=> 2x ≥ -10
=> x ≥ -5
x ≥ -7
Hence x ≥ -5
|x + 7| = -(x + 7) if x + 7 < 0 => x < -7
2{-(x + 7)} - 4 ≥ 0
=> -2x - 14 - 4 ≥ 0
=> -2x -18 ≥ 0
=> -18 ≥ 2x
=> -9 ≥ x
=> x ≤ - 9
x < -7
Hence x ≤ - 9
x ≤ - 9 and x ≥ -5
x ∈ (-∞ , - 9] ∪ [-5 , ∞)
x ∈ R - (-9 , - 5)
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