12.2.2 Linear Inequalities, PT3 Focus Practice


Question 6:
List all the integer values of x which satisfy the following linear inequalities:
–2 < 3x + 1 ≤ 10

Solution:
–2 < 3x + 1
–3 < 3x
x > –1
x = 0, 1, 2, 3, …

3x + 1 ≤ 10
3x ≤ 9
x ≤ 3
x = 3, 2, 1, 0, …

Therefore x = 0, 1, 2, 3


Question 7:
List all the integer values of x which satisfy the following linear inequalities:
–5 < 2x – 3 ≤ 1

Solution:
–5 < 2x – 3
–5 + 3 < 2x
2x > –2
x > –1
x = 0, 1, 2, 3, …  

2x – 3 ≤ 1
2x ≤ 4
x ≤ 2
x = 2, 1, 0, –1, …

Therefore x = 0, 1, 2


Question 8:
Given that 3< x2 <4 and x is an integer. List all the possible values of x.

Solution:

3< x2 <4 3 2 <x2< 4 2 9<x2 x>11   or   x2<16 x<18 11<x<18 x=12, 13, 14, 15, 16, 17


Question 9:
Find the biggest and the smallest integer of x that satisfy
3x + 2 ≥ –4 and 4 – x > 0.

Solution:
3x + 2 ≥ –4
3x ≥ –4 – 2
3x ≥ –6
x ≥ –2

4 – x > 0
x > –4
x < 4

Smallest integer of x is –2, and the biggest integer of x is 3.



Question 10:
If xhy satisfy the two inequalities 7 h 2 5 and  3( h+2 )20+h, find the values of x and y.

Solution:

7 h 2 5 h 2 57 h4 h4 3( h+2 )20+h 3h+620+h 2h14 h7 4h7 x=4,y=7

12.2.1 Linear Inequalities, PT3 Focus Practice


12.2.1 Linear Inequalities, PT3 Focus Practice

Question 1:
Draw a number line to represent the solution for the linear inequalities –3 < 5 – x  ≤ 4.
 
Solution:
–3 < 5 – x   and   5 – x  ≤ 4
x < 5 + 3   and   x  ≤ 4 – 5
x < 8   and   x ≤ –1 → x  ≥ 1

Thus, the solution is 1 ≤ x < 8

Question 2:
Solve the following simultaneous linear inequalities.
3x51    and    2 1 3 x<3

Solution:
3x – 5 ≤ 1
3x  ≤ 1 + 5
3x  ≤ 6
x  ≤ 2

2 1 3 x<3 6x<9   Multiply by 3   x<96 x<3 x>3   Multiply by 1  
The solution is –3 < x  ≤ 2.



Question 3:
The solution for the inequality 2 + < 3x – 4 is
  
Solution:
2 + x < 3x – 4  
x – 3x < –4 – 2
–2x < –6
x < –3
x > 3


Question 4:
The solution for the inequality –2 (6y + 3) < 3 (4 – 2y) is

Solution:
–2 (6y + 3) < 3 (4 – 2y)
–12y – 6 < 12 – 6y
–12y + 6< 12 + 6
–6y < 18
y < 3
y > –3 


Question 5:
Solve each of the following inequalities.
(a) 3x + 4 > 5x – 10
(b)   –3  ≤ 2x + 1 < 7

Solution:
(a)
3x + 4 > 5x – 10
3x – 5x > –10 – 4
–2x > –14
x > –7
x < 7

(b)
–3 ≤ 2x + 1 < 7
–3 ≤ 2x + 1   and   2x + 1 < 7
–2x  ≤ 1 + 3   and   2x < 7 – 1
–2x  ≤ 4 and   2x < 6
≥ –2   and   x < 3

The solution is –2 ≤ x < 3.

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) 32x<1 (b)  52x 3 7

Solution:
(a)
     32x<1 32x3<13        2x<2            2x>2            2x 2 > 2 2              x>1

(b)
         52x 3 7 ( 52x 3 )×37×3          52x21            2x16                2x16                2x 2 16 2                  x8



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 inequality which satisfies both inequalities.