Absolute Value Inequalities

[k.future.surgeon]

When solving absolute inequalities, why do you flip the sign of the second inequality?

Ex:
|x|>5 x>5



x<-5 <----- second equation

 

[mmm4444bot]

For a quick answer, I'd say it's due to the behavior of |x|, as observed by the shape and symmetry of the absolute-value graph.

Look at a graph of y = |x|

Graphically speaking, |x| > 5 refers to the parts of the graph which lie above the horizontal line y=5.

Can you see two parts, where y > 5?

One part corresponds to x-values greater than +5

One part corresponds to x-values less than -5

This is why we must have:

x > 5
or
x < -5

in order to ensure that |x| > 5

---

Two algebraic rules (for eliminating absolute-value symbols) are:

Given |x| > c

where symbol c is any positive constant, we may write:

x < -c
or
x > c

---

Given |x| < c

where again c represents a positive constant, we may write:

-c < x < c

You can see why this second rule makes sense, by also looking at the graph of y=|x|. :cool:

 

[mmm4444bot]

Here's a different explanation.

Have you heard |x| described as representing a distance from zero on the Real number line?

This is another way of thinking about |x|.

To say |x| > 5 means x-values (i.e., points on the Real number line) which are more than 5 units away from zero.

But we can measure away from zero in two different directions.

If we measure to the right (i.e., in the positive direction), then the points on the Real number line located more than 5 units away are x>5.

If we measure to the left (i.e., in the negative direction), then the points located more than 5 units away from zero are x<-5.

This interpretation also works for |x| < 5

Thinking of |x| as a distance from zero, |x| < 5 means points that are less than 5 units away from zero. Well, all points less than 5 units from zero (measured in either direction) must be the points in between -5 and 5.

In other words, -5<x<5

 

[Dr.Peterson]

When solving absolute inequalities, why do you flip the sign of the second inequality?

Ex:
|x|>5 x>5

x<-5 <----- second equation

Another way to think of it is that |x| > 5 means that x is more than 5 units away from 0 in either direction. So either x is more than 5 units greater than 0: x > 5; or x is more than 5 units less than 0: x < -5.