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Unit 5: Linear Equations and Inequalities — Review Exercise

9th Class Mathematics · Unit 5: Linear Equations and Inequalities

R5.2.Solve and represent their solutions on real line. (i) (x+5)/3 = 1 - x (ii) (2x+1)/3 + 1/2 = 1 - (x-1)/3 (iii) 3x + 7 < 16 (iv) 5(x-3) ≥ 26x - (10x+4)

Given

(i) x+53 = 1-x

(i) x+5=3-3x ⇒ x+3x=3-5 ⇒ 4x=-2

Result

(i) x=-12

Given

(ii) 2x+13 + 12 = 1 - x-13

(ii) 2(2x+1)+3 = 6-2(x-1) ⇒ 4x+2+3=6-2x+2 ⇒ 6x=3

Result

(ii) x=12

Given

(iii) 3x+7<16

(iii) 3x<9

Result

(iii) x<3

Given

(iv) 5(x-3) ge 26x-(10x+4)

(iv) 5x-15 ge 16x-4 ⇒ 5x-16x ge -4+15 ⇒ -11xge11

Result

(iv) xle -1

R5.3.Find the solution region of the following linear inequalities: (i) 3x - 4y ≤ 12; 3x + 2y ≥ 3 (ii) 2x + y ≤ 4; x + 2y ≤ 6

Given

(i) 3x-4yle12;quad 3x+2yge3

Working

(i) 3x-4y=12: (0,-3),(4,0)qquad 3x+2y=3: left(0,tfrac32right),(1,0)

Result

(i) 0<12 true (towards); 0>3 false (away)

Given

(ii) 2x+yle4;quad x+2yle6

Working

(ii) 2x+y=4: (0,4),(2,0)qquad x+2y=6: (0,3),(6,0)

Result

(ii) 0<4 true (towards); 0<6 true (towards)

R5.4.Find the maximum value of g(x,y) = x + 4y subject to constraints x + y ≤ 4, x ≥ 0 and y ≥ 0.

Given

Constraint x+yle4;quad xge0, yge0

Formula

Associated equation x+y=4

Working

Intercepts x=0 ⇒ (0,4);quad y=0 ⇒ (4,0)

Corner points A(0,0), B(0,4), C(4,0)

g(A)=0;quad g(B)=0+4(4)=16;quad g(C)=4+4(0)=4

Result

Maximum g=x+4y is maximum at (0,4)

R5.5.Find the minimum value of f(x,y) = 3x + 5y subject to constraints x + 3y ≥ 3, x + y ≥ 2, x ≥ 0, y ≥ 0.