Solve each of the following linear inequalities in one variables.
Problem 1 :
5x - 3 < 3x + 1
(i) when x is a real number
(ii) when x is an integer
(iii) when x is a natural number
Problem 2 :
3x + 17 ≤ 2(1 - x)
Problem 3 :
2(2x + 3) - 10 ≤ 6(x - 2)
Problem 4 :
3x - 7 > x + 1
Problem 5 :
-(x - 3) + 4 < 5 - 2x
Problem 6 :
Problem 7 :
Problem 8 :
Problem 9 :
Problem 10 :
1. Answer :
5x - 3 < 3x + 1
2x - 3 < 1
2x < 4
x < 2
(i) When x is a real number, the solution set is
(-∞, 2)
(ii) When x is an integer, the solution set is
{...............,-4, -3, -2, - 1, 0, 1}
(iii) When x is a natural number, the solution set is
{1}
2. Answer :
3x + 17 ≤ 2(1 - x)
3x + 17 ≤ 2 - 2x
5x + 17 ≤ 2
5x ≤ -15
x ≤ -3
3. Answer :
2(2x + 3) - 10 ≤ 6(x - 2)
4x + 6 - 10 ≤ 6x - 12
4x - 4 ≤ 6x - 12
-2x - 4 ≤ -12
-2x ≤ -8
x ≥ 4
4. Answer :
3x - 7 > x + 1
2x - 7 > 1
2x > 8
x > 4
5. Answer :
-(x - 3) + 4 < 5 - 2x
-x + 3 + 4 < 5 - 2x
-x + 7 < 5 -2x
x + 7 < 5
x < -2
6. Answer :
x + 10 < -4
x < -14
7. Answer :
3x - 2 < -6
3x < -4
x < -⁴⁄₃
8. Answer :
3(x - 2) + 2(x + 1) > 12
3x - 6 + 2x + 2 > 12
5x - 4 > 12
5x > 16
x > ¹⁶⁄₅
9. Answer :
3(2x - 1) + 4(3x + 2) ≥ 12x + 12
6x - 3 + 12x + 8 ≥ 12x + 12
18x + 5 ≥ 12x + 12
16x + 5 ≥ 12
16x ≥ 7
x ≥ ⁷⁄₁₆
10. Answer :
-4 < 3x - 1 < 4
-3 < 3x < 4
-1 < x < ⁴⁄₃
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