Comprehensive Guide to Inequalities
Inequalities are mathematical expressions that compare quantities showing their relative sizes. Unlike equations that show equality (=), inequalities show relationships like greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤).
1. Basic Inequality Symbols
| Symbol | Meaning | Example |
|---|---|---|
| > | Greater than | 5 > 3 |
| < | Less than | 3 < 5 |
| ≥ | Greater than or equal to | x ≥ 4 |
| ≤ | Less than or equal to | x ≤ 4 |
| ≠ | Not equal to | x ≠ 0 |
2. Properties of Inequalities
Key Properties:
- Addition/Subtraction Property: Adding or subtracting the same value from both sides of an inequality does not change the inequality.
- Multiplication/Division Property: Multiplying or dividing both sides of an inequality by a positive number preserves the inequality. But when multiplying or dividing by a negative number, the inequality sign must be reversed.
- Transitive Property: If a > b and b > c, then a > c.
- Additive Inverse: If a > b, then -a < -b (the inequality sign is reversed).
Important: Always remember to reverse the inequality sign when multiplying or dividing both sides by a negative number!
3. Types of Inequalities
3.1 Linear Inequalities
Linear inequalities involve variables with a degree of 1, like ax + b < c.
Example 1: Solve 2x + 3 > 7
Solution:
Therefore, the solution is x > 2, which means all values of x greater than 2 satisfy the inequality.
3.2 Quadratic Inequalities
Quadratic inequalities involve variables with a maximum degree of 2, like ax² + bx + c > 0.
Example 2: Solve x² - 5x + 6 > 0
Solution:
First, we need to find where x² - 5x + 6 = 0 by factoring:
These critical points divide the number line into three regions: x < 2, 2 < x < 3, and x > 3.
Now we test a value in each region to determine where the inequality is satisfied.
For x = 1 (in the region x < 2):
For x = 2.5 (in the region 2 < x < 3):
For x = 4 (in the region x > 3):
Therefore, the solution is x < 2 or x > 3.
3.3 Rational Inequalities
Rational inequalities involve fractions with variables, like (ax + b)/(cx + d) > 0.
Example 3: Solve (x - 2)/(x + 1) ≤ 0
Solution:
For a rational expression to be less than or equal to zero, either:
- The numerator ≤ 0 and the denominator > 0, or
- The numerator ≥ 0 and the denominator < 0
Critical points: x = 2 (where numerator = 0) and x = -1 (where denominator = 0)
The denominator x + 1 = 0 when x = -1. This is an excluded value.
Let's analyze the regions:
- For x < -1: numerator < 0, denominator < 0, so fraction > 0 ✗
- For -1 < x < 2: numerator < 0, denominator > 0, so fraction < 0 ✓
- For x > 2: numerator > 0, denominator > 0, so fraction > 0 ✗
Therefore, the solution is -1 < x ≤ 2 (The equal sign is included for x = 2 where the fraction equals 0).
3.4 Absolute Value Inequalities
Absolute value inequalities involve the absolute value of expressions, like |ax + b| < c.
Remember:
- |x| < a means -a < x < a (values between -a and a)
- |x| > a means x < -a or x > a (values outside the interval [-a, a])
Example 4: Solve |2x - 6| < 4
Solution:
Using the property |x| < a ⟹ -a < x < a:
Therefore, the solution is 1 < x < 5 or x ∈ (1, 5).
Example 5: Solve |x - 3| ≥ 5
Solution:
Using the property |x| ≥ a ⟹ x ≤ -a or x ≥ a:
Therefore, the solution is x ≤ -2 or x ≥ 8.
3.5 Systems of Inequalities
Systems of inequalities involve multiple inequality constraints that must be satisfied simultaneously.
Example 6: Solve the system { x + y ≤ 6 x - y ≥ 2 x ≥ 0 y ≥ 0 }
Solution:
This system defines a region in the coordinate plane bounded by the lines:
- x + y = 6 (with the region below this line)
- x - y = 2 (with the region above this line)
- x = 0 (with the region to the right)
- y = 0 (with the region above)
To find the vertices of this region, we solve the system of equations formed by these lines:
From x + y = 6 and x - y = 2:
From x + y = 6 and y = 0:
From x - y = 2 and y = 0:
The solution is the triangular region with vertices at (2, 0), (6, 0), and (4, 2).
4. Methods to Solve Inequalities
4.1 Algebraic Method
This is the most common method, involving algebraic manipulation to isolate the variable.
Example 7: Solve 3(x - 2) - 4 ≥ 2x + 5
Solution:
Therefore, the solution is x ≥ 15.
4.2 Graphical Method
The graphical method involves plotting the related functions and identifying where one is greater than the other.
For an inequality like f(x) > g(x), you can:
- Graph both f(x) and g(x)
- Find intersection points where f(x) = g(x)
- Determine regions where f(x) > g(x)
4.3 Test Point Method
Especially useful for quadratic and rational inequalities, this method involves:
- Finding critical points (zeros of the expression or points where the expression is undefined)
- Using these points to divide the number line into regions
- Testing one value in each region to determine where the inequality is satisfied
Example 8: Solve (x² - 4)/(x - 1) > 0
Solution:
Step 1: Find critical points
These points divide the number line into four regions: x < -2, -2 < x < 1, 1 < x < 2, and x > 2
Step 2: Test a value in each region
Therefore, the solution is -2 < x < 1 or x > 2.
4.4 Interval Method
The interval method is similar to the test point method but uses a more structured approach with a number line and interval notation.
5. Applications of Inequalities
5.1 Optimization Problems
Linear programming problems use systems of linear inequalities to maximize or minimize an objective function.
Example 9: A company manufactures two products, A and B. Each product A requires 2 hours of labor and 3 units of raw material. Each product B requires 3 hours of labor and 2 units of raw material. The company has at most 60 hours of labor and 60 units of raw material available. If product A yields a profit of $40 and product B yields a profit of $30, how many of each product should be manufactured to maximize profit?
Solution:
Let x = number of product A and y = number of product B.
Constraints:
Profit function to maximize: P = 40x + 30y
Solving this system, we find the maximum profit occurs at the intersection of 2x + 3y = 60 and 3x + 2y = 60.
5.2 Range of Functions
Inequalities help determine the range of functions and the domain restrictions.
Example 10: Find the range of f(x) = 2x² + 4x - 3.
Solution:
This is a quadratic function with a positive leading coefficient, so it has a minimum value.
Since (x + 1)² ≥ 0 for all real x, the minimum value is -5 (when x = -1).
Therefore, the range of f(x) is f(x) ≥ -5 or [-5, ∞).