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India
Class IX

Absolute Values and Number Lines

Lesson

To find the absolute value of a number is to find its magnitude, ignoring whether the number is positive or negative. The mathematical shorthand for absolute value is a pair of vertical lines enclosing a number or an expression.

So, $|-3.1|$|3.1| means the magnitude of the number $-3.1$3.1, which is $3.1$3.1.  You can think of it as the distance from $0$0, to $-3.1$3.1 on a number line.  The distance is $3.1$3.1 units from zero.  Notice how this is the same distance as $3.1$3.1 is from zero on a number line.  So $|3.1|=|-3.1|=3.1$|3.1|=|3.1|=3.1.  

Example 1
Evaluate (a)  $|-5|$|5|
  (b)  $|-5+9|$|5+9|
  (c)  $|-5|+|9|$|5|+|9|


 

 

 

The absolute value of a negative number is its negative. So, $|-5|=-(-5)=5$|5|=(5)=5.

We evaluate the sum $-5+9$5+9 before applying the absolute value function. So, $|-5+9|=|4|=4$|5+9|=|4|=4.

We evaluate separate absolute value terms before adding them. So, $|-5|+|9|=5+9=14$|5|+|9|=5+9=14.

Note that $|-5+9|<|-5|+|9|$|5+9|<|5|+|9|.

 

Example 2

We can show on a number line that if $|x||x|<n, then $x$x is strictly between $-n$n and $n$n. For example, we investigate the meaning of $|x|<4$|x|<4.

The table shows that absolute values less than $4$4 only occur for numbers between $-4$4 and $4$4.

The numbers $-4$4 and $4$4 themselves are not included because it is not true that $|-4|<4$|4|<4 and it is not true that $|4|<4$|4|<4. However, similar statements using $\le$ instead of $<$< would be true and the end-points would be included in the interval.

 

Number lines

We can also visualise absolute values on a number line.

$|x|\le2$|x|2 means  show me the values of $x$x, so that all the distances from $0$0 are less than or equal to $2$2.  This we can see on the number line is this

$|x|<2$|x|<2 would be similar to above except this time we are strictly less than $2$2, so we use open points to mark the end of the interval. 

$|x|>2$|x|>2 means we are now interested in all the values of $x$x that have a distance of more than $2$2 on the number line.  This means we have $2$2 sections, all those points to the right of $2$2, and all the points to the left of $-2$2.  

$|x|\ge2$|x|2 means the same as above, except for now we want greater than or equal to a distance of $2$2, so we use a closed point to start the intervals.  

Outcomes

9.NS.RN.1

Review of representation of natural numbers, integers, rational numbers on the number line. Representation of terminating/non-terminating recurring decimals, on the number line through successive magnification. Rational numbers as recurring/terminating decimals.

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