Question Number 44502 by Tip Top last updated on 30/Sep/18
$$\mathrm{If}\:\mathrm{a}>\mathrm{b},\mathrm{and}\:\mathrm{c}>\mathrm{d},\mathrm{prove}\:\mathrm{that}\:\mathrm{a}−\mathrm{c}\:\mathrm{may}\:\mathrm{be}\:\mathrm{greater}\:\mathrm{than}, \\ $$ $$\mathrm{equal}\:\mathrm{to}\:\mathrm{or}\:\mathrm{less}\:\mathrm{than}\:\mathrm{b}−\mathrm{d}. \\ $$ $$ \\ $$
Answered by MrW3 last updated on 30/Sep/18
$${a}={b}+\delta,\:\delta>\mathrm{0} \\ $$ $${c}={d}+\varrho,\:\varrho>\mathrm{0} \\ $$ $${a}−{c}=\left({b}−{d}\right)+\left(\delta−\varrho\right) \\ $$ $$\left({a}−{c}\right)−\left({b}−{d}\right)=\left(\delta−\varrho\right) \\ $$ $$ \\ $$ $${if}\:\delta>\varrho,\:\left({a}−{c}\right)−\left({b}−{d}\right)>\mathrm{0} \\ $$ $$\Rightarrow{a}−{c}\:{is}\:{greater}\:{than}\:{b}−{d} \\ $$ $$ \\ $$ $${if}\:\delta=\varrho,\:\left({a}−{c}\right)−\left({b}−{d}\right)=\mathrm{0} \\ $$ $$\Rightarrow{a}−{c}\:{is}\:{equal}\:{to}\:{b}−{d} \\ $$ $$ \\ $$ $${if}\:\delta<\varrho,\:\left({a}−{c}\right)−\left({b}−{d}\right)<\mathrm{0} \\ $$ $$\Rightarrow{a}−{c}\:{is}\:{less}\:{than}\:{b}−{d} \\ $$
Commented byTip Top last updated on 30/Sep/18
$$ \\ $$ $$ \\ $$ $$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Sir} \\ $$