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Question Number 101363 by bemath last updated on 02/Jul/20
The solution set of inequality (((√((3x−7)^2 ))−2)/(x−3)) ≤ ((3−(√x^2 ))/(x−3)) is __ (A) (−∞, (1/2)] (D) [(1/2), ∞) (B) [(1/2),1 ] (E) (−∞,(2/3)] (C) (−∞,1 ]
Thesolutionsetofinequality(3x−7)2−2x−3⩽3−x2x−3is__(A)(−∞,12](D)[12,∞)(B)[12,1](E)(−∞,23](C)(−∞,1]
Answered by 1549442205 last updated on 02/Jul/20
⇔((∣3x−7∣−2)/(x−3))−((3−∣x∣)/(x−3))≤0(1) i)if x≥(7/3) then (1)⇔((3x−7−2−(3−x))/(x−3))≤0 ⇔((4x−12)/(x−3))≤0⇔((4(x−3))/(x−3))≤0⇔4≤0 ⇒(1) has no solutions ii)if 0≤x<(7/3) then (1)⇔((7−3x−2−(3−x))/(x−3)) ⇔((2−2x)/(x−3))≤0⇔((1−x)/(x−3))≤0⇔x∈{(−∞;1]∪(3;+∞)}∩[0;(7/3)) ⇔x∈[0;1] iii)if x<0 then (1)⇔((7−3x−2−(3+x))/(x−3))≤0 ⇔((2−4x)/(x−3))≤0⇔((1−2x)/(x−3))≤0⇔(−∞;(1/2)]∪(3;+∞)∩(−∞;0) ⇔(−∞;0) Combinating three above cases we get The solutions of given inequality is x∈(−∞;1],so choose answer C
⇔∣3x−7∣−2x−3−3−∣x∣x−3⩽0(1)i)ifx⩾73then(1)⇔3x−7−2−(3−x)x−3⩽0⇔4x−12x−3⩽0⇔4(x−3)x−3⩽0⇔4⩽0⇒(1)hasnosolutionsii)if0⩽x<73then(1)⇔7−3x−2−(3−x)x−3⇔2−2xx−3⩽0⇔1−xx−3⩽0⇔x∈{(−∞;1]∪(3;+∞)}∩[0;73)⇔x∈[0;1]iii)ifx<0then(1)⇔7−3x−2−(3+x)x−3⩽0⇔2−4xx−3⩽0⇔1−2xx−3⩽0⇔(−∞;12]∪(3;+∞)∩(−∞;0)⇔(−∞;0)CombinatingthreeabovecaseswegetThesolutionsofgiveninequalityisx∈(−∞;1],sochooseanswerC
Commented by 1549442205 last updated on 02/Jul/20
Thank you.You are welcome sir.
Thankyou.Youarewelcomesir.
Commented by bemath last updated on 02/Jul/20
thank you. agree
thankyou.agree
Commented by Rasheed.Sindhi last updated on 02/Jul/20
⇔((∣3x−7∣−2)/(x−3))−((3−∣x∣)/(x−3))≤0(1) i)if x≥(7/3) then (1)⇔((3x−7−2−(3−x))/(x−3))≤0 ⇔((4x−12)/(x−3))≤0⇔((4(x−3))/(x−3))≤0⇔4≤0 At x=(7/3),((4(x−3))/(x−3))≤0⇏4≤0 But x=(7/3),((4(x−3))/(x−3))≤0⇏4≥0 ∵ x−3<0 ⇒(1) has no solutions ii)if 0≤x<(7/3) then (1)⇔((7−3x−2−(3−x))/(x−3)) ..................... .......... .....
⇔∣3x−7∣−2x−3−3−∣x∣x−3⩽0(1)i)ifx⩾73then(1)⇔3x−7−2−(3−x)x−3⩽0⇔4x−12x−3⩽0⇔4(x−3)x−3⩽0⇔4⩽0Atx=73,4(x−3)x−3⩽0⇏4⩽0Butx=73,4(x−3)x−3⩽0⇏4⩾0∵x−3<0⇒(1)hasnosolutionsii)if0⩽x<73then(1)⇔7−3x−2−(3−x)x−3………………………….…..

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