12.1 Linear Inequalities


12.1 Linear Inequalities

12.1.1 Inequalities
1.   To write the relationship between two quantities which are not equal, we use the following inequality signs:
  > greater than
  < less than
  ≥ greater than or equal to
  ≤ less than or equal to

2.   7 > 4 also means 4 < 7. 7 > 4 and 4 < 7 are equivalent inequalities.


   12.1.2  Linear Inequalities in One Unknown
1.   An inequality in one unknown to the power of 1 is called a linear inequality.
   Example: 2x + 5 > 7

2.  
A linear inequality can be represented on a number line.
  Example:


12.1.3 Computation on Inequalities
1.   When a number is added or subtracted from both sides of an inequality, the condition of the inequality is unchanged.
Example:
Given 5 > 3
Then, 5 + 2 > 3 + 2 ← (symbol ‘>’ remains)
Hence, 7 > 5


2.   When both sides of an inequality are multiplied or divided by the same positive number, the condition of the inequality is unchanged.
Example:
Given 4x ≤ 16
Then, 4x ÷ 4 ≤ 16 ÷ 4 ← (symbol ‘≤’ remains)
Hence, x ≤ 4


3.   When both sides of an inequality are multiplied or divided by the same negative number, the inequality is reversed.
Example:
Given –3 > –5
Hence, 3 < 5
Given –5y > –10
Then, –5y ÷ 5 > –10 ÷ 5
 –y > –2
Hence, y < 2


12.1.4 Solve Inequalities in One Variable
To solve linear inequalities in one variable, use inverse operation to make the variable as the subject of the inequality.

Example:
Solve the following linear inequalities.
(a) 3 2 x < 1 (b) 5 2 x 3 7


Solution:
(a)
3 2 x < 1 3 2 x 3 < 1 3 2 x < 2 2 x > 2 2 x 2 > 2 2 x > 1

(b)
5 2 x 3 7 ( 5 2 x 3 ) × 3 7 × 3 5 2 x 21 2 x 16 2 x 16 2 x 2 16 2 x 8


12.1.5 Simultaneous Linear Inequalities in One Variable
1.   The common values of two simultaneous inequalities are values which satisfy both linear inequalities.
The common values of the simultaneous linear inequalities x ≤ 3 and x > –1 is –1 < x ≤ 3.

2.   To solve two simultaneous linear inequalities is to find a single equivalent inequalitywhich satisfies both inequalities.
 

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