1. INEQUALITIES

2. Introduction
An inequality is similar to an equation except that the statement is that two expressions have a relationship other than equality, such as <, , > or .

3. Rules for inequalities
Rule 1: An equal quantity may be added to (or subtracted from) both sides of an inequality without changing the inequality
Example: Solve x – 3 > 5
Solution:
x – > (add 3 to both sides)
Thus, the solution is x > 8

4. Rules for inequalities (cont)
Rule 2: An equal positive quantity may multiply (or divide) both sides of an inequality without changing the in equality
Example: Solve 2x – 1 > 7
Solution: 2x – > (add 1 to both sides)
 2x > 8
 (divide both sides by 2)
 x > 4 (the inequality remain the same)

5. Rules for inequalities (cont)
Rule 3: If both sides of an inequality are multiplied (or divided) by a negative quantity then the inequality is reversed.
Example: Solve 3 – 2x > -5
Solution: 3 – 3 – 2x > -5 – 3 (subtract 3 from both sides)
 -2x < -8
 (divide both sides by -2)
(Apply rule 3, the inequality reversed)
 x > 4

6. Solving linear inequalities
Example 1: Solve the following inequalities.
7x – 6 > 3 + 2x
2 – 6x  4x + 3
3x < 9x + 4
4  3x – 2 < 13
x – 10 < 2x – 2 < x

7. Solving Quadratic Inequalities
Quadratic inequalities need to handle with care.
Example 2: Solve x2 – x – 12 > 0
Step 1 : Factorize the quadratic expression
 (x + 3) (x – 4) > 0
Step 2: The critical points are: 4 and 3.
Step 3 : Sketch a sign change diagram - +
(x + 3) +
(x – 4) - - +
The solution is
x < -3 or x > 4 + -3 - 4 +

8. Example 3: Solve 2x2  9x + 5.
Step 1: Rearrange the inequality so that the expressions are on the left side and zero on the right side.
2x2 – 9x - 5  0
Step 2: Factorize the quadratic expression
(2x + 1) (x – 5)  0
Step 3. The critical points are: 5 and -1/2
Step 4: Draw the sign change diagram
The solution is
-1/2  x  5 + - +
-1/2 5

9. - + + Example 4: Solve (2 – x)(x + 3) > 0
Step 1: The critical points are: 2 and - 3
Step 2: Make sure the coefficient of x is positive.
 (2 – x) = -x + 2 = -(x – 2)
Hence (2 – x)(x + 3) >0
-(x – 2)(x + 3) > 0
Multiple both sides with -1,  -1  -(x – 2)(x + 3) < -1  0
 (x – 2) (x + 3) < 0
Step 4: Draw a sign change diagram
Note that the inequality is reversed
The solution is
-3 < x < 2 + - + -3 2

10. Solving Absolute Values Inequalities
The absolute value of a real number x can be thought of as the distance from 0 to x on a real number line.
|2| = 2 and |-2|= 2
Absolute value, denoted as |x|, is defined as follows
|-x|
|x| -x x

11. Solving Absolute Values Inequalities
In a statement such as x < 2, it means that x must lie between 2 and -2. We can write this as
|x|< 2  -2 < x < 2
Similarly, |x - 4|< 2 represents all points whose distance from point 4 is less than 2.
|x - 4|< 2 -2 < x – 4 < 2  2 < x < 6
The points lie in the interval 2 < x < 6
< 2
< 2 x -x 4

12. Solving Absolute Values Inequalities
If |x| > 5, this means the distance from 0 to x is greater than 5.
This also indicates than the point x can be greater than 5 or less than -5.
|x| > 5  x > 5 or x < -5 -x x -5 5

13. Solving Absolute Values Inequalities
Properties :
|x| = a if and only if x = ± a
|x| < a if and only if –a < x < a
|x|  a if and only if –a  x  a
|x|  a if and only if x  a or x  - a
|x| > a if and only if x > a or x < - a

14. Exercises: Solve the following inequalities