Chapter 2- Inequalities

Topic : Linear inequalities

• In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality:
• < is less than
• > is greater than
• ≤ is less than or equal to
• ≥ is greater than or equal to
• ≠ is not equal to
• A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign.
Example 1
 Solve and graph the solution set of:   2x - 6 < 2
Add 6 to both sides.  2x-6 < 2
Divide both sides by 2.  2x < 8
Open circle at 4             x < 4
(since x can x < not equal 4) and an arrow to the left (because we want valuesless than 4).


Example 2
 Solve and graph the solution set of:  
 5 - 3x  13 + x
Subtract 5 from both sides.
Subtract x from both sides.
Divide both sides by -4, and don't forget tochange the direction of the inequality ! 
(We divided by a negative.)
 
5 - 3x  13 + x
    -3x  8 + x
    -4x  8
        x  -2
Closed circle at -2 (since x can equal  -2) and an arrow to the right (because we want values largerthan -2).
 

Example 3
Solve and graph the solution set of:  
3(2x + 4) > 4x + 10 
Multiply out the parentheses
Subtract 4x from both sides.
Subtract 12 from both sides.
Divide both sides by 2, but don't change the direction of the inequality, since we didn't divide by a negative.
 		3(2x + 4) > 4x + 10
  6x + 12 > 4x + 10
  2x + 12 > 10
       2x > -2
         x > -1 
Open circle at -1 (since x can not equal -1) and an arrow to the right (because we want values largerthan -1).





Sketching quadratic inequalities













Example: x2 − x − 6 < 0

x2 − x − 6 has these simple factors (because I wanted to make it easy!):
(x+2)(x−3) < 0

Firstly, let us find where it is equal to zero:
(x+2)(x−3) = 0
It is equal to zero when x = −2 or x = +3
because when x = −2, then (x+2) is zero
and when x = +3, then (x−3) is zero

So between −2 and +3, the function will either be
  • always greater than zero, or
  • always less than zero
We don't know which ... yet!
Let's pick a value in-between and test it:
At x=0:  x2 − x − 6  =  0 − 0 − 6
=  −6

So between −2 and +3, the function is less than zero.
And that is the region we want, so...
x2 − x − 6 < 0 in the interval (−2, 3)

Note: x2 − x − 6 > 0 on the interval (−∞,−2) and (3, +∞)

And here is the plot of x2 − x − 6:
  • The equation equals zero at −2 and 3
  • The inequality "<0" is true between −2 and 3.
x^2-x-6






1)  (x-3)/2 < -5  

2) -2 < (6-2x)/3 < 4  

3)  3x - 7 < 5

4) -4 ≤ 3x + 2 < 5

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