maths

LINEAR INEQUALITIES
MAIN CONCEPTS AND RESULTS
* Two real numbers or two algebraic expressions related by the symbol „<‟, „>‟, „≤‟ or „≥‟ form an
inequality.
* Numerical inequalities : 3 < 5; 7 > 5
* Literal inequalities : x < 5; y > 2; x ≥ 3, y ≤ 4
* Double inequalities : 3<5<7 ,2<y<4
* Strict inequalities : ax + b < 0 , ax + b > 0, ax2 + bx + c > 0
* Slack inequalities : ax + by ≤ c , ax + by ≥ c , ax2 + bx + c ≤ 0
* Linear inequalities : ax + b < 0, ax + b ≥ 0
* Quadratic inequalities : ax2 + bx + c > 0, ax2 + bx + c ≤ 0
** Algebraic Solutions of Linear Inequalities in One Variable and their Graphical
Representation




** Graph of system of linear inequalities, 2x - 6y < 12, 3x + 4y < 12 and 4x + 2y ≥ 8.




**Graph the system of linear inequalities. 2x - 3y < 6 , - x + y ≤ 4 , 2x + 4y < 8




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, II. Some Illustrations/ Examples with solutions
1. The length of a rectangle is three times the breadth. If the minimum perimeter of the
rectangle is 160 cm, then

(a) breadth > 20 cm (b) length < 20 cm (c) breadth x ≥ 20 cm (d) length ≤ 20 cm

Correct option: (c) breadth x ≥ 20 cm

Solution:

Let x be the breadth of a rectangle.

So, length = 3x

Given that the minimum perimeter of a rectangle is 160 cm.

Thus, 2 (3x + x) ≥ 160

⇒ 4x ≥ 80

⇒ x ≥ 20

2.If – 3x + 17 < – 13, then

(a) x∈ (10, ∞) (b) x ∈ [10, ∞) (c) x ∈ (–∞, 10] (d) x ∈ [– 10, 10)

Correct option: (a) x ∈ (10, ∞)

Solution:

Given,

-3x + 17 < -13

Subtracting 17 from both sides,

-3x + 17 – 17 < -13 – 17

⇒ -3x < -30

⇒ x > 10 {since the division by negative number inverts the inequality sign}

⇒ x ∈ (10, ∞)

3. The interval form of x ≤ -2 is

(a) x∈ (–∞, – 2) (b) x ∈ (–∞, – 2] (c) x ∈ (– 2, ∞] (d) x ∈ [– 2, ∞)

Correct option: (b) x ∈ (–∞, – 2]

4.If |x −1| > 5, then

(a) x∈ (– 4, 6) (b) x ∈ [– 4, 6] (c) x ∈ (–∞, – 4) U (6, ∞) (d) x ∈ [–∞, – 4) U [6, ∞)

Correct option: (c) x ∈ (–∞, – 4) ∪ (6, ∞)

Solution:

|x – 1| > 5

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