Illustrate the following on graph paper and shade the region which satisfies all the three inequalities at the same time : \(- x + 5y \leq 10 ; 3x - 4y \leq...

Rewrite the boundary equations as

\[ -x+5y=10\quad\Rightarrow\quad y=2+\frac{x}{5}, \]

\[ 3x-4y=8\quad\Rightarrow\quad y=-2+\frac{3x}{4}, \]

together with the vertical boundary \(x=-1\).

Use the following values to plot the two straight lines.

Plot the graph below. Draw the first two boundary lines solid, since their inequalities include equality. Draw \(x=-1\) as a broken line, since \(x>-1\) does not include the boundary.

Using \((0,0)\) as a test point:

\[0\leq10,\qquad 0\leq8,\qquad 0>-1.\]

Hence select the side containing \((0,0)\) for each inequality. Therefore the required region is

\[\boxed{\;x>-1,\qquad -2+\frac{3x}{4}\leq y\leq2+\frac{x}{5}\;}\]

The two sloping lines meet at

\[2+\frac{x}{5}=-2+\frac{3x}{4}\Rightarrow x=\frac{80}{11},\quad y=\frac{38}{11}.\]

Thus shade the triangular region between the two solid lines, to the right of the broken line \(x=-1\). The broken left-hand edge is not part of the region.