### Find greatest value of shaded regions given a set of simultaneous inequalities

Question Sample Titled 'Find greatest value of shaded regions given a set of simultaneous inequalities'

Consider the following system of inequalities:

 ${x}\ge{4}$ ${y}\ge{0}$ ${2}{x}+{5}{y}\le{43}$ ${3}{x}-{5}{y}\le{27}$

 Let ${D}$ be the region which represents the solution of the above system of inequalities. If ${\left({x},{y}\right)}$ is a point lying in ${D}$ , then the grestest value of ${4}{x}-{2}{y}+{2}$ is 

A
${52}$ .
B
${4}$ .
C
${18}$ .
D
${69}$ .

 ${x}\ge{4}$ ${y}\ge{0}$ ${2}{x}+{5}{y}\le{43}$ ${3}{x}-{5}{y}\le{27}$ The solution region is drawn and shaded below.

${x}$${y}$${\left({4},{7}\right)}$${\left({14},{3}\right)}$${4}$${9}$

 The intersection points are ${\left({4},{7}\right)}$ and ${\left({14},{3}\right)}$ .  At ${\left({4},{0}\right)}$, ${4}{x}-{2}{y}+{2}$ $={4}{\left({4}\right)}-{2}{\left({0}\right)}+{2}={18}$ At ${\left({9},{0}\right)}$, ${4}{x}-{2}{y}+{2}$ $={4}{\left({9}\right)}-{2}{\left({0}\right)}+{2}={38}$ At ${\left({4},{7}\right)}$, ${4}{x}-{2}{y}+{2}$ $={4}{\left({4}\right)}-{2}{\left({7}\right)}+{2}={4}$ At ${\left({14},{3}\right)}$, ${4}{x}-{2}{y}+{2}$ $={4}{\left({14}\right)}-{2}{\left({3}\right)}+{2}={52}$ ∴   The required grestest value is ${52}$ .

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