Chapter 7, Problem 16CT

To determine

To sketch: the graph of the given solution set of the system of in equalities.

Answer to Problem 16CT

The points of intersections of the curve and the line are $\left(1,4\right)\text{and }\left(-4,-16\right)$

Explanation of Solution

Given:

The given solution is

  $\{\begin{array}{l}y<-x^2+x+4\\ y>4x\end{array}$

Calculation:

Consider the system of inequalities

  $\{\begin{array}{l}y<-x^2+x+\mathrm{4...............}(1)\\ y>4x\mathrm{..........................}(2)\end{array}$

Consider the equations

  $\begin{array}{l}y=-x^2+x+\mathrm{4..................}(3)\\ y=4x\mathrm{..................}(4)\end{array}$

The points of intersection of the curve (3) and the line (4) are given by

  $\begin{array}{l}4x=-x^2+x+4\\ x^2+x-4=0\end{array}$

Factorize by splitting the middle term

  $\begin{array}{c}{x}^2+4x-x-4=0\\ x\left(x+4\right)-1\left(x+4\right)=0\\ \left(x-1\right)\left(x+4\right)=0\\ x=-4,1\end{array}$

Thus the points of intersections of the curve and the line are $\left(1,4\right)\text{and }\left(-4,-16\right)$

The graphs of the curve (3) and the line (4) and the vertices of the solution set of the inequalities is shown below