Find the solution set of each of the following inequations : (i) 2 x - 3 7, x ∈ N (ii) 10 4x - 5 21, x ∈ N (iii) 2 - x 4x - 7 11 - 2x, x ∈ Z (iv) 4 - 2x 3x + 19 42 - 5x, x ∈ Z (v) -5 - 3 , x ∈ Z (vi) 9 - x 5x - 11 17 - , x ∈ Z

Question

KnowledgeBoat · Accepted Answer

(i) 2 x - 3 7, x ∈ N We have: 2 x - 3 7 Let's separate the inequalities: Case 1: 2 x - 3 ⇒ 2 + 3 x [Adding 3 on both sides] ⇒ 5 x ⇒ x 5 Natural numbers greater than 5 are {6, 7, 8, 9, 10, ...} ∴ Solution set A = {6, 7, 8, 9, 10, ...} Case 2: x - 3 7 ⇒ x 7 + 3 [Adding 3 on both sides] ⇒ x 10 Natural numbers less than 10 are {1, 2, 3, 4, 5, 6, 7, 8, 9} ∴ Solution set B = {1, 2, 3, 4, 5, 6, 7, 8, 9} Final solution set = A ∩ B = {6, 7, 8, 9} (ii) 10 4x - 5 21, x ∈ N We have: 10 4x - 5 21 Let's separate the inequalities: Case 1: 10 4x - 5 ⇒ 10 + 5 4x [Adding 5 on both sides] ⇒ 15 4x ⇒ x [Dividing both sides by 4] ⇒ 3.75 x ⇒ x 3.75 Natural numbers greater than 3.75 are {4, 5, 6, 7, ...} ∴ Solution set A = {4, 5, 6, 7, ...} Case 2: 4x - 5 21 ⇒ 4x 21 + 5 [Adding 5 on both sides] ⇒ 4x 26 ⇒ x ⇒ x 6.5 Natural numbers less than 6.5 are {1, 2, 3, 4, 5, 6} ∴ Solution set B = {1, 2, 3, 4, 5, 6} Final solution set = A ∩ B = {4, 5, 6} (iii) 2 - x 4x - 7 11 - 2x, x ∈ Z We have: 2 - x 4x - 7 11 - 2x Let's separate the inequalities: Case 1: 2 - x 4x - 7 ⇒ 2 + 7 4x + x [Adding 7 and x on both sides] ⇒ 9 5x ⇒ x [Dividing both sides by 5] ⇒ 1.8 x ⇒ x 1.8 Integers greater than 1.8 are {2, 3, 4, 5, ...} ∴ Solution set A = {2, 3, 4, 5, ...} Case 2: 4x - 7 11 - 2x ⇒ 4x + 2x 11 + 7 [Adding 7 and 2x on both sides] ⇒ 6x 18 ⇒ x ⇒ x 3 Integers less than 3 are {..., -1, 0, 1, 2} ∴ Solution set B = {..., -1, 0, 1, 2} Final solution set = A ∩ B = {2} (iv) 4 - 2x 3x + 19 42 - 5x, x ∈ Z We have: 4 - 2x 3x + 19 42 - 5x Let's separate the inequalities: Case 1: 4 - 2x 3x + 19 ⇒ 4 - 19 3x + 2x [Subtracting 19 and adding 2x on both sides] ⇒ -15 5x ⇒ x [Dividing both sides by 5] ⇒ -3 x ⇒ x -3 Integers greater than -3 are {-2, -1, 0, 1, 2, ...} ∴ Solution set A = {-2, -1, 0, 1, 2, ...} Case 2: 3x + 19 42 - 5x ⇒ 3x + 5x 42 - 19 [Subtracting 19 and adding 5x on both sides] ⇒ 8x 23 ⇒ x ⇒ x 2.875 Integers less than 2.875 are {..., -2, -1, 0, 1, 2} ∴ Solution set B = {..., -2, -1, 0, 1, 2} Final solution set = A ∩ B = {-2, -1, 0, 1, 2} (v) -5 - 3 , x ∈ Z We have: -5 - 3 Let's separate the inequalities: Case 1: -5 - 3 ⇒ -5 + 3 ⇒ -2 ⇒ -2 x 2 x [Multiplying 2 on both sides] ⇒ -4 x ⇒ x -4 Integers greater than -4 are {-3, -2, -1, 0, ...} ∴ Solution set A = {-3, -2, -1, 0, ...} Case 2: Integers less than 11 are {..., 8, 9, 10} ∴ Solution set B = {..., 8, 9, 10} Final solution set = A ∩ B = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (vi) 9 - x 5x - 11 17 - , x ∈ Z We have: 9 - x 5x - 11 17 - Let's separate the inequalities: Case 1: Integers greater than 3.52.. are {4, 5, 6, ...} ∴ Solution set A = {4, 5, 6, ...} Case 2: Integers less than 5.33.. are {..., 3, 4, 5} ∴ Solution set B = {..., 3, 4, 5} Final solution set = A ∩ B = {4, 5}

Mathematics

Find the solution set of each of the following inequations : (i) 2 x - 3 7, x ∈ N (ii) 10 4x - 5 21, x ∈ N (iii) 2 - x 4x - 7 11 - 2x, x ∈ Z (iv) 4 - 2x 3x + 19 42 - 5x, x ∈ Z (v) -5 - 3 , x ∈ Z (vi) 9 - x 5x - 11 17 - , x ∈ Z

Linear Inequations

Related Questions

If a > b and m < 0, then which of the following is correct :
am < bm
am = bm
am > bm
am and bm cannot be compared

Which one of the following is a solution to the inequality 3x - 5 < 6?
3
4
5
6

Mathematics

Find the solution set of each of the following inequations : (i) 2 x - 3 7, x ∈ N (ii) 10 4x - 5 21, x ∈ N (iii) 2 - x 4x - 7 11 - 2x, x ∈ Z (iv) 4 - 2x 3x + 19 42 - 5x, x ∈ Z (v) -5 - 3 , x ∈ Z (vi) 9 - x 5x - 11 17 - , x ∈ Z

Linear Inequations

Related Questions

If a > b and m < 0, then which of the following is correct :am < bmam = bmam > bmam and bm cannot be compared

Which one of the following is a solution to the inequality 3x - 5 < 6?3456

If a > b and m < 0, then which of the following is correct :
am < bm
am = bm
am > bm
am and bm cannot be compared

Which one of the following is a solution to the inequality 3x - 5 < 6?
3
4
5
6