Modulus inequalities are mathematical inequalities that involve the absolute value function. The absolute value of a number represents its distance from zero on the number line, always giving a non-negative result. For example, both positive 3 and negative 3 have an absolute value of 3, since they are both 3 units away from zero.
There are two basic types of modulus inequalities. Type one is absolute value of x less than a, which has the solution negative a less than x less than a. Type two is absolute value of x greater than a, which gives x less than negative a or x greater than a. For example, absolute value of x less than 3 means x is between negative 3 and 3.
To solve complex modulus inequalities like absolute value of 2x minus 1 greater than or equal to 3, we consider two cases. Case one: when 2x minus 1 is non-negative, the inequality becomes 2x minus 1 greater than or equal to 3, giving x greater than or equal to 2. Case two: when 2x minus 1 is negative, we get negative of 2x minus 1 greater than or equal to 3, which gives x less than or equal to negative 1. The final solution is x less than or equal to negative 1 or x greater than or equal to 2.
Modulus inequalities have a clear graphical interpretation. The expression absolute value of x minus a less than b means the distance from x to a is less than b. For example, absolute value of x minus 2 less than 1 means the distance from x to 2 is less than 1, which gives the solution 1 less than x less than 3. In general, absolute value of x minus a less than b is equivalent to a minus b less than x less than a plus b.
To summarize what we've learned about modulus inequalities: They involve absolute values and represent distance relationships on the number line. The two main types have distinct solution patterns, with complex forms requiring careful case analysis. The graphical interpretation helps us visualize solutions as intervals, making these mathematical tools essential for many applications.