Show on a graph, the area which gives the solution set of the inequalities: \(y - 2x \leq 4 ; 3y + x \geq 6 ; y \geq 7x - 9\).
Question 1 Report
Show on a graph, the area which gives the solution set of the inequalities: \(y - 2x \leq 4 ; 3y + x \geq 6 ; y \geq 7x - 9\).
Answer Details
Rewrite the inequalities in terms of y:
\[y\leq 2x+4,\qquad y\geq 2-\frac{x}{3},\qquad y\geq 7x-9.\]
Draw the three boundary lines as solid lines, since equality is included. Their pairwise intersections are:
\[\begin{aligned}2x+4&=2-\frac{x}{3} &\Rightarrow&\ x=-\frac67,\ y=\frac{16}{7};\\[2mm]2-\frac{x}{3}&=7x-9 &\Rightarrow&\ x=\frac32,\ y=\frac32;\\[2mm]2x+4&=7x-9 &\Rightarrow&\ x=\frac{13}{5},\ y=\frac{46}{5}.\end{aligned}\]
The required solution set is the common region below \(y=2x+4\) and above both \(y=2-\frac{x}{3}\) and \(y=7x-9\). It is the shaded triangular region shown.
Hence, the solution region is the closed triangle with vertices
\[\boxed{\left(-\frac67,\frac{16}{7}\right),\quad\left(\frac32,\frac32\right),\quad\left(\frac{13}{5},\frac{46}{5}\right).}\]
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