Chapter 4.9, Problem 40PPSE

Solution

To Describe: And correct the error

The statement of because the shaded regions do not intersect, there are no solutions to the system is false.

Given:

Given system of inequalities:

  $\left\{\begin{array}{l}y\le x^2-4x+6\hfill \\ y\ge x^2-4x+2\hfill \end{array}\right.$

Theory: Because the shaded regions do not intersect, there are no solutions to the system.

Explanation:

It is not possible for the parabolas for quadratic functions to intersect the parabolas for quadratic inequalities given above.

It indicates quadratic equations do not have a solution.

Although this does not indicate that the pair of inequalities do not have a solution.

If there exists a region with points whose $x,y$ coordinates satisfy a pair of inequalities, then the pair of inequalities have a solution.

For a given pair of inequalities, the solution is the set of values of $x,y$ that satisfy both inequalities.

  $\begin{array}{c}y\le x^2-4x+6\\ y\ge x^2-4x+2\end{array}$

The Blue shaded part below includes the points whose coordinates $x,y$ that satisfies: $y\le x^2-4x+6$

  

The Black shaded part below includes the points whose coordinates $x,y$ which satisfy:

  $y\ge x^2-4x+2$

  

There are a set of points whose $x,y$ coordinates satisfy both inequality conditions in the dark blue region bounded by the two curves:

These two parabolas also have solutions at the points on their boundaries.