Absolute value in practice

The following exercises are solved in this step.

We invite you to try to solve them before watching the video.

In any case, you will find below a PDF file with the solutions.

Exercise 1.

Prove the triangle inequality (|x+y|le |x|+|y|) for all (x,yinmathbb R)

Exercise 2.

Write (f(x) = |x-1| – |4x + 8|) as an expression where there are no absolute values.

Exercise 3.

Express the maximum (maxlbrace a,brbrace) between the real numbers (a) and (b) in a formula involving just sums, products and the absolute value of real numbers. (Hint: what is the distance from (maxlbrace a,brbrace) to the midpoint of the segment joining (a) to (b)?)

Exercise 4.

Let (A=lbrace xinmathbb R:, xge -3rbrace, B=lbrace xinmathbb R:, x<6rbrace,) ( C=lbrace xin mathbb R:, |x-5|le 2rbrace). Find the set (Acap Bcap C).

Exercise 5. [Solved only in the PDF file]

Knowing that (sqrt 3=1.732050808…) give an estimate of (|sqrt 3-1.73205|).