Linear Absolute Value Equations and Inequalities

Recall the definition of the absolute value of X :

Therefore, when an absolute value appears in an equation, we must account for both the positive and negative value of X.

Section

Rules

Examples	Explanation
	This represents the equations x=2 and -x=2 . The solution set is {2, -2} .

The solution set of absolute value equations are often graphed on a number line. The filled circles represent the solution points (x=2, x=-2 from the previous example).

Linear Absolute Value Inequalities

These problems are very similar to absolute value equations. The difference is that, whereas the solution set of absolute value equations is usually a set of discrete points, the solution set of absolute value inequalities is often a range of values.

Examples	Explanation
	2x+3>5 and  2x+3>-5 are the inequalities. The solution is -4<X<1 .

It is important to remember to flip the greater than or lesser than sign in the case of the negative absolute value, as we did for 2X+3>-5 in this example.

Rules

   is equivalent to -a

   is equivalent to -a ≤x≤a

Try these exercises

Solve

1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


10. 

Answers to questions

1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 
   


9. 
   


10.