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Question Number 174982 by MikeH last updated on 15/Aug/22

Answered by Rasheed.Sindhi last updated on 15/Aug/22

$$a+b⩽9\wedge \underset{\left(i\right)}{\underbrace{a⩾2}}\wedge \underset{\left(ii\right)}{\underbrace{b⩾-2}}$$$$\left(ii\right)-\left(i\right)\Rightarrow b-a⩾-2-2=-4$$$$Minimumvalueofb-ais-4$$

Commented by JDamian last updated on 15/Aug/22

that is not correct, sir. if a=3 and b=-2, then b-a = -5

Commented by MikeH last updated on 15/Aug/22

$$\mathrm{I}\mathrm{got}\mathrm{an}\mathrm{answer}\mathrm{which}{\mathrm{I}}^{\prime }\mathrm{m}\mathrm{not}\mathrm{sure}\mathrm{of}$$$${\mathrm{that}}^{\prime }\mathrm{s}\mathrm{why}\mathrm{I}\mathrm{posted}.\mathrm{it}\mathrm{goes}\mathrm{as}\mathrm{follows}$$$$$$$$a+b⩽9\mathrm{and}a⩾2\mathrm{and}b⩾-2$$$$\mathrm{now}b-a\mathrm{is}\mathrm{minimum}\mathrm{when}b\mathrm{is}\mathrm{minimum}$$$$b\mathrm{is}\mathrm{minimum}\mathrm{when}b=-2$$$$\Rightarrow a-2⩽9\Rightarrow a⩽11$$$$\mathrm{now}\min (b-a)=-2-11=-13.$$$$\mathrm{is}\mathrm{that}\mathrm{correct}\mathrm{sir}?$$

Commented by Rasheed.Sindhi last updated on 16/Aug/22

$$\mathbb{T}\mathrm{han}\mathbb{k}\mathrm{s}\mathrm{JDamian}\mathrm{sir}!$$

Answered by Frix last updated on 15/Aug/22

$$a+b⩽9\wedge a⩾2\wedge b⩾-2$$$$a=2+p^2\wedge b=-2+q^2$$$$a+b⩽9\Rightarrow p^2+q^2⩽9\Rightarrow q^2⩽9-p^2$$$$b-a⩾m\Rightarrow -p^2+q^2-4⩾m\Rightarrow q^2⩾m+p^2+4$$$$m+p^2+4⩽9-p^2$$$$m⩽5-2p^2$$$$m\mathrm{is}\min \mathrm{when}{p}^2\mathrm{is}\max$$$$p^2+q^2⩽9\Rightarrow \max \left(p^2\right)=9$$$$\Rightarrow q^2=0$$$$m=-13$$

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