Linear magnification Shown in the figure is a simple magnifier consisting of a convex lens...

Linear magnification Shown in the figure is a simple magnifier consisting of a convex lens. The object to be magnified is positioned so that the distance p from the lens is less than the focal length f. The linear magnification M is the ratio of the image size to the object size. It is shown in physics that M = f/(f − p). If, how far should the object be placed from the lens so that its image appears at least three times as large? (Compare with Example.)

EXERCISE 85

EXAMPLE

Using a lens formula

As illustrated in Figure 8, if a convex lens has focal length f centimeters and if an object is placed a distance p centimeters from the lens with p > f, then the distance q from the lens to the image is related to p and f by the formula

FIGURE 8

If f = 5 cm, how close must the object be to the lens for the image to be more than 12 centimeters from the lens?

SOLUTION Since f = 5, the given formula may be written as

We wish to determine the values of q such that q > 12. Let us first solve the equation for q:

To solve the inequality q > 12, we proceed as follows:

FIGURE 9

Combining the last inequality with the fact that p is greater than 5, we obtain the solution

If a point X on a coordinate line has coordinate x, as shown in Figure 9, then X is to the right of the origin O if x > 0 and to the left of O if x < 0. From Section 1.1, the distance d(O, X) between O and X is the nonnegative real number given by

It follows that the solutions of an inequality such as | x \ < 3 consist of the coordinates of all points whose distance from O is less than 3. This is the open interval (–3, 3) sketched in Figure 10. Thus,

FIGURE 10

Similarly, for | x | > 3, the distance between O and a point with coordinate x is greater than 3; that is,

FIGURE 11

The graph of the solutions to |x | > 3 is sketched in Figure 11. We often use the union symbol ⋃and write

to denote all real numbers that are in either (−∞, − 3) or (3, ∞). The notation

represents the set of all real numbers except 2.

The intersection symbol ∩ is used to denote the elements that are common to two sets. For example,

since the intersection of (−∞, 3) and (−3, ∞) consists of all real numbers x such that both x < 3 and x > − 3.

The preceding discussion may be generalized to obtain the following properties of absolute values.