Question

Solve each of the following system of equations in R. 16. $\frac{7x-1}{2}<-3,\frac{3x+8}{5}+11<0$

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Solution

$\frac{7x-1}{2}<-3\phantom{\rule{0ex}{0ex}}⇒7x-1<-6\phantom{\rule{0ex}{0ex}}⇒7x<-6+1\phantom{\rule{0ex}{0ex}}⇒x<\frac{-5}{7}\phantom{\rule{0ex}{0ex}}⇒x\in \left(-\infty ,\frac{-5}{7}\right)...\left(\mathrm{i}\right)\phantom{\rule{0ex}{0ex}}\mathrm{Also},\frac{3x+8}{5}+11<0\phantom{\rule{0ex}{0ex}}⇒\frac{3x+8+55}{5}<0\phantom{\rule{0ex}{0ex}}⇒3x+63<0\phantom{\rule{0ex}{0ex}}⇒3x<-63\phantom{\rule{0ex}{0ex}}⇒x<-21\phantom{\rule{0ex}{0ex}}⇒x\in \left(-\infty ,-21\right)...\left(\mathrm{ii}\right)\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{the}\mathrm{solution}\mathrm{to}\mathrm{the}\mathrm{given}\mathrm{set}\mathrm{of}\mathrm{inequations}\mathrm{is}\mathrm{the}\mathrm{intersection}\mathrm{of}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right).\phantom{\rule{0ex}{0ex}}\left(-\infty ,\frac{-5}{7}\right)\cap \left(-\infty -21\right)=\left(-\infty ,-21\right)\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{the}\mathrm{solution}\mathrm{to}\mathrm{the}\mathrm{given}\mathrm{set}\mathrm{of}\mathrm{inequations}\mathrm{is}\left(-\infty ,-21\right).$

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