Suppose that |x-2|<0.01 and |y-3|<0.04 .Use the Triangle Inequality to show

Question

that |(x+y)-5|<0.05 represent your solution on a number line.

Answer 1

By the Triangle Inequality, we have:

|(x+y)-5| <= |x-2| + |y-3|

Since we know that |x-2|<0.01 and |y-3|<0.04, we can substitute:

|(x+y)-5| <= 0.01 + 0.04

Simplifying:

|(x+y)-5| < 0.05

This shows that if |x-2|<0.01 and |y-3|<0.04, then |(x+y)-5|<0.05. On a number line, this means that the solutions lie within a shaded interval of length 0.05, centered at 5.