3.7.2 Solving "Greater Than" Inequalities

|x|> a means that x is more than "a" units from the origin.
If a>0, we can solve the absolute value inequality |x|> a, by solving the compound statement:

x<-a{\rm \; or \;}x> a

Now that we know how to solve absolute value equations that involve the less than symbol, the next natural question is then to explore absolute value equations involving the greater than symbol.

What does |x|>1 even mean?

Try to graph the solution to this equation on the number line before looking at the solution below. Note that your solution won't just have 2 points!

Now, to be more than 1 unit away from the origin means that I'm either further from the origin in the negative sense or the positive sense. Notice that we didn't fill in the "holes" at the ends of the line segment because of the strict inequality.

How could we have thought about this problem algebraically? If |x|>1, then by our picture, we see that this means that x is less than -1 or x is greater than 1; you may recall this type of set when we first studied number lines. In other words, if we're trying to solve |x|>"positive number", then we can just set up two equations. We'll need our x to be either less than the negative version of the number OR bigger than it's positive counterpart. Why can't we use the connecting word AND here? "x<-1 and x>1" would mean that both statements are true at the same time: but no numbers satisfy this requirement so our picture would be blank.

Explore!

Solve |2x-1|>5 for x: or

Now our "2x-1" is either less than -5 OR bigger than 5. This gives us the two equations

2x-1<-5 \textrm{ or }2x-1>5

Adding 1 to both sides of each equation gives us that 2x<-4 \textrm{ or }2x>6

Dividing both sides by 2, we see that x<-2 \textrm{ or }x>3.

We can write this last set more formally as \left \{ x|x<-2 \textrm { or }x>3 \right \} which is read as "All of the x's with the property that x is less than -2 or x is bigger than 3."

In the next section, you can find a summary of all of the major concepts from this chapter.