Inequalities, Inequations, Line Segment Length, Slope, Line Equations, and Absolute Value Inequalities

1. What Is an Inequality?

An Inequality is a mathematical relationship that compares two numerical or algebraic expressions and shows whether one value is greater than, less than, or equal to another.

Inequality Symbols:

• > Greater than
• < Less than
• ≥ Greater than or equal to
• ≤ Less than or equal to

Example:

5 > 3
x + 2 ≤ 7

2. What Is an Inequation?

An Inequation is a type of Inequality that includes variables. The goal is to find all values of the variable that make the inequality true.

Examples of Inequations:

x - 3 > 2
2x ≤ 8

3. Length of a Line Segment

The Length of a Line Segment represents the distance between two points on a number line or on the coordinate plane.

Formula on a Number Line:

|x2 - x1|

Formula on the Coordinate Plane:

√((x2 - x1)^2 + (y2 - y1)^2)

4. Slope of a Line

The Slope of a line measures its steepness and direction and is usually represented by the letter m.

Slope Formula:

m = (y2 - y1) / (x2 - x1)

Important Notes:

• If m > 0, the line is increasing
• If m < 0, the line is decreasing
• If m = 0, the line is horizontal

5. Equation of a Line

An Equation of a Line is a mathematical expression that describes all the points lying on a straight line.

Common Form of a Line Equation:

y = mx + b

• m: Slope of the line
• b: Y-intercept

Example:

y = 2x + 1

6. Absolute Value

Absolute Value represents the distance of a number from zero on the number line and is always non-negative.

Definition of Absolute Value:

|x| = x     if x ≥ 0
|x| = -x    if x < 0

7. Solving Absolute Value Inequalities

To solve Absolute Value Inequalities, the inequality must be converted into two simpler inequalities.

Case 1: |x| < a

-a < x < a

Case 2: |x| > a

x > a  or  x < -a

Example:

|x| < 3
-3 < x < 3

Conclusion

Concepts such as Inequality, Inequation, Line Segment Length, Slope, Line Equation, and Absolute Value Inequalities are fundamental topics in mathematics. Mastering these concepts helps students better understand algebra and geometry and prepares them for more advanced mathematical studies.