We want to compare two numbers a and b and explain how the following inequalities can both be true statements.
a> b and |a|< |b|
First, let's recall that the absolute value of a number is the non-negative value of that number. This means that the absolute value of a negative number will be positive because when taking an absolute value of a negative number, we erase the negative sign. Let's take a look at some examples.
Absolute Value
Simplify
|-1|
1
|-3|
3
|-100|
100
Now, we want to find two numbers a and b in which one is greater than the other but the opposite is true if we compare their absolute values. This suggests that a should be non-negative and b should be negative. Let's check if that condition is enough to satisfy both statements.
a
b
a> b
|a|< |b|
1
-5
1> -5 âś“
| 1| < | -5| âś“
3
-1
3> -1 âś“
| 3| ≮ | -1| *
10
-15
10> -15 âś“
| 10| < | -15| âś“
0
-2
0> -2 âś“
| 0| < | -2| âś“
We can see that when a is equal to 0 and b is a negative number, both statements are true. However, in case of positive a and negative b, the distance from b to 0 on the number line must greater than the distance from a to 0.
Notice that if both a and b are positive, then their absolute values are exactly the same as their positive values. As such, it is impossible for two positive numbers to satisfy both inequalities at the same time.
