Solving inequalities with modulus (absolute value) involves considering two separate cases to account for both positive and negative possibilities within the absolute value. Here's a breakdown of the method:
Understanding Absolute Value
The absolute value of a number, denoted by |x|, is its distance from zero. This means |x| is always non-negative. For example, |3| = 3 and |-3| = 3.
General Approach
When solving inequalities of the form |x| < a or |x| > a (where 'a' is a positive number), you need to split the problem into two separate inequalities:
Case 1: The Expression Inside the Modulus is Positive or Zero
Keep the expression inside the absolute value the same and solve the inequality.
Case 2: The Expression Inside the Modulus is Negative
Change the sign of the expression inside the absolute value and reverse the inequality sign.
Specific Inequality Types and Solutions
Let's consider different types of absolute value inequalities:
1. |x| < a (Less Than)
This means the distance of 'x' from zero is less than 'a'. This translates to:
- -a < x < a
Example: Solve |x| < 3
- -3 < x < 3
The solution is all x values between -3 and 3 (excluding -3 and 3).
2. |x| ≤ a (Less Than or Equal To)
This is similar to the previous case, but includes the endpoints:
- -a ≤ x ≤ a
Example: Solve |x| ≤ 5
- -5 ≤ x ≤ 5
The solution is all x values between -5 and 5 (including -5 and 5).
3. |x| > a (Greater Than)
This means the distance of 'x' from zero is greater than 'a'. This translates to two separate inequalities:
- x < -a OR x > a
Example: Solve |x| > 2
- x < -2 OR x > 2
The solution is all x values less than -2 or greater than 2.
4. |x| ≥ a (Greater Than or Equal To)
Similar to the previous case, but includes the endpoints:
- x ≤ -a OR x ≥ a
Example: Solve |x| ≥ 4
- x ≤ -4 OR x ≥ 4
The solution is all x values less than or equal to -4 or greater than or equal to 4.
Solving More Complex Absolute Value Inequalities
When the expression inside the absolute value is more complex (e.g., |2x + 1|), the same principles apply. You still need to consider both positive and negative cases:
Example: Solve |2x + 1| < 5
-
Case 1: (2x + 1) is positive or zero
2x + 1 < 5
2x < 4
x < 2 -
Case 2: (2x + 1) is negative
-(2x + 1) < 5
-2x - 1 < 5
-2x < 6
x > -3 (Remember to flip the inequality sign when dividing by a negative number)
Combining both cases, the solution is -3 < x < 2.
Example: Solve |3x - 2| ≥ 4
-
Case 1: (3x - 2) is positive or zero
3x - 2 ≥ 4
3x ≥ 6
x ≥ 2 -
Case 2: (3x - 2) is negative
-(3x - 2) ≥ 4
-3x + 2 ≥ 4
-3x ≥ 2
x ≤ -2/3
Combining both cases, the solution is x ≤ -2/3 OR x ≥ 2.
Key Steps Summarized:
- Isolate the absolute value expression. Get the |expression| by itself on one side of the inequality.
- Identify the type of inequality (<, ≤, >, ≥).
- Split the inequality into two cases:
- The expression inside the absolute value is positive or zero (keep the inequality sign the same).
- The expression inside the absolute value is negative (reverse the inequality sign and negate the expression).
- Solve each inequality separately.
- Combine the solutions appropriately:
- For "<" or "≤" inequalities, the solution is an "and" statement (the intersection of the two solutions).
- For ">" or "≥" inequalities, the solution is an "or" statement (the union of the two solutions).
By following these steps, you can effectively solve a wide range of inequalities involving absolute values.
