2.5 – Absolute Value Inequalities
Absolute Value Inequalities
When an inequality has an absolute value in it, we must treat it as the previous section and eliminate the absolute value before we can find solutions. We consider two different consequences of having an absolute value with an inequality.
Consider the graph of the solutions to the absolute inequality 

For the absolute value of an unknown number to be greater than 4, the number must be further away from zero than 4. That means that the unknown number could be greater than 4, or could be less than . Graphically:
We see that if the absolute value is set larger than a number, we have the union of two sets. This means that we can rewrite the expression as the conjunction This will be useful in solving absolute value inequalities of this type! 
Consider the graph of the solutions to the absolute inequality 

For the absolute value of the unknown number to be less than 4, that means that the unknown must be less than 4 units away from zero. Graphically this looks like:
We can see that if an absolute value is set less than a number, we have an intersection. This means that we can rewrite the expression as the conjunction 
Solving Absolute Value Inequalities 


Solve. Express solutions interval notation. 

The solution in interval notation is 
Solve. Express solutions graphically and in interval notation. 

Because the inequality is set less than (less than or equal to) we can create a tripartite: Again, note the construction of the tripartite. On the left, we have the negative 2 less than the expression from the absolute value, which is in turn less than 2. We solve the tripartite as one: In interval notation: 
Solve the absolute value inequality, graph and write the solution in interval notation 

First, we will isolate the absolute value: Now that the absolute value is isolated, we can set up our equations: And graph: In interval notation: 
Solve, graph and write solutions in interval notation 

First, we isolate the absolute value: This inequality can be written as a tripartite, and solved: In interval notation: 
Solve, graph and write solutions in interval notation 

And we stop here! Since the absolute value is less than a negative number, there will be no solutions. The solution set to this problem is empty, so we indicate this with the emptyset symbol: 
Solve, graph and write the solution in interval notation 

And we may also stop here. Since the absolute value is always a positive number, setting it greater than a negative number will result in a true statement regardless of what value of we use. The solution to this equation is all real numbers: The solution in interval notation is: or simply 