**Solving Absolute Value Equations and Inequalities**

** **

Let *k *be a positive real number, and *p *and *q *be real numbers.

**1. **To solve │ ** ax **+

**│=**

*b*

*k***,**solve the compound equation

** ax **+

**=**

*b*

*k***or**

**+**

*ax***= –**

*b*

*k***.**

The solution set is usually of the form {*p, q*}, which includes two

** **

**2. **To solve │ ** ax **+

**│ >**

*b*

*k***,**solve the compound inequality

** ax **+

**>**

*b*

*k***or**

**+**

*ax***< –**

*b*

*k***.**

The solution set is of the form (-∞, *p*) U (*q*,∞), which consists of

two separate intervals.

** **

**3. **To solve │ ** ax **+

**│ <**

*b*

*k***,**solve the three-part inequality

– ** k **<

**+**

*ax***<**

*b*

*k***.**

The solution set is of the form ( *p, q*), a single interval.