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Question
Solve the following inequality and write the solution set using interval notation
`(2x)/(x - 4) ≤ 5`
Solution
`(2x)/(x - 4) ≤ 5`
∴ `(2x)/(x - 4) - 5 ≤ 0`
∴ `(2x - 5x + 20)/(x - 4) ≤ 0`
∴ `(20 - 3x)/(x - 4) ≤ 0`
∴ Since, `"a"/"b" ≤ 0`,
when a ≥ 0 and b < 0 or a ≤ 0 and b > 0
∴ either 20 – 3x ≥ 0 and x – 4 < 0 or 20 – 3x ≤ 0 and x – 4 > 0
Case I:
20 – 3x ≥ 0 and x – 4 < 0
∴ `x ≤ 20/3` and x < 4
∴ x < 4 ...(i)
Case II:
20 – 3x ≤ 0 and x – 4 > 0
∴ `x ≥ 20/3` and x > 4
∴ `x ≥ 20/3` ...(ii)
From (i) and (ii), we get
∴ `x ∈ (- ∞, 4) ∪ [20/3, ∞]`
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