Let A 3 B1 C6 and D5 denote points on a line with their coor

Let A [-3], B[-1], C[6], and D[5] denote points on a line, with their coordinates. If P[x] denotes any other point on AB bar, find the range of x allowable if P elementof AB bar P elementof BD bar P elementof DC bar P elementof AB bar union CD bar.

Solution

This is a simple problem of finding the define the range of any point P under various conditions.

To solve this problem ,we should start by defining the given points .

Let us assume that the given line is x axis and the points A, B ,C and D lie on the x axis thus

A(-3,0) ,B(-1,0) ,C(6,0) and D(5,0).

Now please focus on point P. This point P is such that it lies anywhere from point A to point B only. ie from (-3,0) to (-1,0)

Now let us check each case

i) when p belongs to AB

clearly in this case P can be any value from (-3,0 ) to (-1,0) thus its range will be [-3,1] .

ii) when p belong to BD

In this case the cordinate ranges from (-1,0) to (5,0) . But P can move only upto B which is (-1,0) . Hence for range defined by BD ,P can only take one value which is (-1,0) hence the range of P for it to be valid in BD is [-1].

iii). When P belongs to DC

the range of DC is from (5,0) to (6,0) .Clearly P can only move upto (-1,0) which does not fall in the range of DC hence P does not have any value for which it can be defined in DC .

iv).P belongs to the union of AB and CD

When we consider union of AB and CD then we can see that P can belong to any value in AB but for no value in CD (from previous parts of the question)

thus for the union of AB and CD ,P will have range of (-3,0 ) to (-1,0) or simply [-3,1].