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Solve-2-x-2-x-1-lt-1-

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Question Number 16519 by tawa tawa last updated on 23/Jun/17
Solve: ∣2 − x∣ − 2 ∣x + 1∣ < 1
Solve:∣2−x∣−2∣x+1∣<1
Answered by ajfour last updated on 23/Jun/17
Commented by ajfour last updated on 23/Jun/17
∣2−x∣−2∣x+1∣<1 whenever y=∣2−x∣ is below y= 2∣x+1∣+1 this occurs for x<x_1 (see graph) and x>x_2 to obtain x_1 (wbich is <−1): 2−x_1 = 1−2(x_1 +1) x_1 = −3 to obtain x_2 (−1<x_2 <2) 2−x_2 = 1+2(x_2 +1) 3x_2 = −1 ⇒ x_2 = −1/3 so, for given condition to remain true x ∉ [−3, ((−1)/3)] that is x∈ (−∞, −3)∪(((−1)/3), ∞).
∣2−x∣−2∣x+1∣<1whenevery=∣2−x∣isbelowy=2∣x+1∣+1thisoccursforx<x1(seegraph)andx>x2toobtainx1(wbichis<−1):2−x1=1−2(x1+1)x1=−3toobtainx2(−1<x2<2)2−x2=1+2(x2+1)3x2=−1⇒x2=−1/3so,forgivenconditiontoremaintruex∉[−3,−13]thatisx∈(−∞,−3)∪(−13,∞).
Commented by tawa tawa last updated on 23/Jun/17
God bless you sir. i really appreciate.
Godblessyousir.ireallyappreciate.
Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 24/Jun/17
1)x≥2⇒−(2−x)−2(x+1)<1⇒ −x−4<1⇒−x<−5⇒x>5⇒x∈[5,∞) 2) −1≤x<2⇒2−x−2(x+1)<1⇒ −3x<1⇒x>−(1/3)⇒x∈[−(1/3),2) ⇒x∈[−(1/3),2)∪[5,∞).
1)x⩾2⇒−(2−x)−2(x+1)<1⇒−x−4<1⇒−x<−5⇒x>5⇒x∈[5,∞)2)−1⩽x<2⇒2−x−2(x+1)<1⇒−3x<1⇒x>−13⇒x∈[−13,2)⇒x∈[−13,2)∪[5,∞).
Commented by tawa tawa last updated on 24/Jun/17
God bless you sir.
Godblessyousir.
Commented by ajfour last updated on 24/Jun/17
if x=3 >2 ∣2−x∣= 1 2∣x+1∣=8 ∣2−x∣−2∣x+1∣=−7 <1 if x=−4 ∣2−x∣= 6 2∣x+1∣ = 6 ∣2−x∣−2∣x+1∣=0 < 1 if x→−∞ ∣2−x∣−2∣x+1∣= 2−x+2(x+1) = x+4 = −∞+4 <1
ifx=3>2∣2−x∣=12∣x+1∣=8∣2−x∣−2∣x+1∣=−7<1ifx=−4∣2−x∣=62∣x+1∣=6∣2−x∣−2∣x+1∣=0<1ifx→−∞∣2−x∣−2∣x+1∣=2−x+2(x+1)=x+4=−∞+4<1
Commented by tawa tawa last updated on 24/Jun/17
God bless you sir.
Godblessyousir.

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