2. Examples


This lesson will discuss a few examples relating to modulus.

Example 1 Evaluate the following.

(i) |–4| – 5          (ii) |4| + 5           (iii) 5 – |–4|        (iv) |–5| + |4|                   (v) |5| – |–4|

(vi) |–5 + 4|        (vii) ||4| – 5|     (viii) 3 – |4 – 5| (ix) |5 – 4| – 3                  (x) 1 – |2 – 3|

Solution

(i) As we saw in the previous lesson, |–4| = 4 [i.e., –(–4)].

Hence, |–4| – 5 = 4 – 5 = –1. Easy, right?

(ii) Following from the definition again, we have |4| = 4. Hence, |4| + 5 = 4 + 5 = 9.

Why don’t you try the remaining ones yourself and come back here to cross-check?

(iii) Here, 5 – |–4| = 5 – 4 = 1.

(iv) Two modulus operators to scare you away. |–5| + |4| = 5 + 4 = 9.

(v) In this one, |5| – |–4| = 5 – 4 = 1.

(vi) –5 + 4 = –1. Hence, |–5 + 4| = |–1| = 1.

(vii) This one looks tricky! First, I’ll simply |4| – 5. This equals 4 – 5 or –1. Hence, ||4| – 5| = |–1| = 1.

(viii) I hope you’ve got the hang of it. 3 – |4 – 5| = 3 – |–1| = 3 – 1 = 2.

(ix) Quite similar to the previous one. |5 – 4| – 3 = |1| – 3 = 1 – 3 = –2.

(x) Another one, similar to the previous two. 1 – |2 – 3| = 1 – |–1| = 1 – 1 = 0.

Ready for some more?

Example 2 Evaluate the following.

(i) |–5 – |–4 – |–3|||                     (ii) |6 – |7 – |8|||           (iii) ||8 – 9| – |–10||

(iv) ||–1| – |–2| – |–3||              (v) |–9| + ||–8| – 7|

Solution Hope you haven’t run away. These problems are pretty easy, just like the previous ones. Focus on the definition of |x|.

(i) |–5 – |–4 – |–3||| Hmm. Where do we begin?

In general, we’ll start with the innermost expression. After simplification, we move progressively towards the outer ones.

Here’s what I’m talking about:

Starting with the innermost modulus expression, we have |–5 – |–4 – |–3||| = |–5 – |–4 – 3||

Next, we have |–5 – |–4 – 3|| = |–5 – |–7|| = |–5 – 7|

Finally, we’ll get |–5 – 7|= |–12| = 12

I hope you got the idea. Why don’t you try the remaining ones and come back here to verify your answers?

(ii) Proceeding in a way similar to (i), we have |6 – |7 – |8||| = |6 – |7 – 8|| = |6 – | –1|| = |6 – 1| = |5| = 5

(iii) ||8 – 9| – |–10|| = ||8 – 9| – 10| = ||–1| – 10| = |1 – 10| = |–9| = 9

Note that we could have started with |8 – 9| as well.

(iv) ||–1| – |–2| – |–3|| = |1 – 2 – 3| = |–4| = 4. I ate some steps there.

(v) |–9| + ||–8| – 7| = 9 + |8 – 7| = 9 + |1| = 9 + 1 = 10

And, we’re done here. I hope you can now evaluate any scary expression involving modulus of numbers (such as |4 – |–7 – |–8|| – |–3| + |–1||).

Be aware that these examples were just to familiarise you with how |x| works when x is a constant.

The fun begins in the next lesson, where we’ll start with equations involving modulus.