Graph feasible region from inequalities

7. \begin{figure}[h] \end{figure} The region \(R _ { 1 }\), shown shaded in Figure 2, is defined by the inequalities $$0 \leqslant y \leqslant 2 \quad y \leqslant 10 - 2 x \quad y \leqslant k x$$ where \(k\) is a constant.
The line \(x = a\), where \(a\) is a constant, passes through the intersection of the lines \(y = 2\) and \(y = k x\) Given that the area of \(R _ { 1 }\) is \(\frac { 27 } { 4 }\) square units,

1. find
   1. the value of \(a\)
   2. the value of \(k\)
2. Define the region \(R _ { 2 }\), also shown shaded in Figure 2, using inequalities.

Show in a sketch the region of the \(x\)-\(y\) plane within which all three of the following inequalities hold. \(y \geqslant x^2\), \(x + y \leqslant 2\), \(x \geqslant 0\). You should indicate the region for which the inequalities hold by labelling the region \(R\). [3]

Sketch the region defined by the inequalities $$y \leq (1 - 2x)(x + 3) \text{ and } y - x \leq 3$$ Clearly indicate your region by shading it in and labelling it \(R\). [3 marks] \includegraphics{figure_4}

A region, R, is defined by \(x^2 - 8x + 12 \leq y \leq 12 - 2x\)

1. Sketch a graph to show the region R. Shade the region R.
2. Find the area of R [6 marks]

Show in a sketch the region of the \(x\)-\(y\) plane within which all three of the following inequalities are satisfied. $$3y \geqslant 4x \qquad y - x \leqslant 1 \qquad y \geqslant (x-1)^2$$ You should indicate the region for which the inequalities hold by labelling the region R. [4]

It is given that $$y = \frac{1}{x+1} + \frac{1}{x-1},$$ where \(x\) and \(y\) are real and positive, and \(i^2 = -1\).

1. Show that $$x = \frac{1 \pm \sqrt{1-y^2}}{y} \quad \text{and} \quad y \leqslant 1.$$ [4]
2. Deduce that $$xy < 2.$$ [2]
3. Indicate the region in the \(x\)-\(y\) plane defined by $$y \leqslant 1 \quad \text{and} \quad xy < 2.$$ [3]