Exercise 60 Page 108

Let's start by expressing the given sentence as an inequality. We are investigating if the this inequality is true for any value of $x$ and $c.$ First, we will examine the cases for different values of $c.$ Then, we will look at the situations for different $x$ values in each case.

Positive $c$

If $c$ is positive, say $c=1,$ then the left-hand side of the inequality will be always positive since an absolute value is always non-negative. In this case, the statement is true for any value of $x.$

c=0

Let's take $c=0.$ Since absolute value represents a non-negative number, for any positive or negative value of $x,$ the result becomes greater than $0.$ However, when we pick $x=0,$ then the result becomes $0.$ In this case, the given statement is false for $x=0.$

Negative $c$

Let's take negative $c,$ say $c=\text{-}1.$ Picking different $x$ values gives us different results. Let's choose $x$ to be $3.$ In this case, the result is positive. But what happens if we pick a different value for $x?$ Let's now choose $x$ to be $0.$ We get zero. Therefore, for $c<0$ and $x=0$ the statement $\left|x-c\right|-c>0$ is false.

Conclusion

The conclusion is that the statement is sometimes true.