NCERT Solutions
Class 11 - MathematicsChapter 5: LINEAR INEQUALITIES
Class 11 - Mathematics
EXERCISE 5.1
Question. 1
Solve \(24x<100\) when:
(i) \(x\) is a natural number.
(ii) \(x\) is an integer.
Answer:
(i) \(\{1,2,3,4\}\)
(ii) \(\{\ldots,-3,-2,-1,0,1,2,3,4\}\)
Question. 2
Solve \(-12x>30\) when:
(i) \(x\) is a natural number.
(ii) \(x\) is an integer.
Answer:
(i) No solution.
(ii) \(\{\ldots,-4,-3\}\)
Question. 3
Solve \(5x-3<7\) when:
(i) \(x\) is an integer.
(ii) \(x\) is a real number.
Answer:
(i) \(\{\ldots,-2,-1,0,1\}\)
(ii) \(( -\infty,2)\)
Question. 4
Solve \(3x+8>2\) when:
(i) \(x\) is an integer.
(ii) \(x\) is a real number.
Answer:
(i) \{ -1,0,1,2,3,\ldots \}
(ii) \((-2,\infty)\)
Question. 5
Solve the inequality for real \(x\):
\(4x+3\le 5x+7\).
Answer:
Solution set: \((-4,\infty)\)
Question. 6
Solve the inequality for real \(x\):
\(3x-7>5x-1\).
Answer:
Solution set: \(( -\infty,-3)\)
Question. 7
Solve the inequality for real \(x\):
\(3(x-1)\le 2(x-3)\).
Answer:
Solution set: \(( -\infty,-3] \)
Question. 8
Solve the inequality for real \(x\):
\(3(2-x)\ge 2(1-x)\).
Answer:
Solution set: \(( -\infty,4] \)
Question. 9
Solve the inequality for real \(x\):
\(x+\dfrac{x}{2}+\dfrac{x}{3}<11\).
Answer:
Solution set: \(( -\infty,6)\)
Question. 10
Solve the inequality for real \(x\):
\(\dfrac{x}{3}>\dfrac{x}{2}+1\).
Answer:
Solution set: \(( -\infty,-6)\)
Question. 11
Solve the inequality for real \(x\):
\(\dfrac{3(x-2)}{5}\le \dfrac{5(2-x)}{3}\).
Answer:
Solution set: \(( -\infty,2] \)
Question. 12
Solve the inequality for real \(x\):
\(\dfrac{1}{2}\left(\dfrac{3x}{5}+4\right)\ge \dfrac{1}{3}(x-6)\).
Answer:
Solution set: \(( -\infty,120] \)
Question. 13
Solve the inequality for real \(x\):
\(2(2x+3)-10<6(x-2)\).
Answer:
Solution set: \((4,\infty)\)
Question. 14
Solve the inequality for real \(x\):
\(37-(3x+5)\ge 9x-8(x-3)\).
Answer:
Solution set: \(( -\infty,2] \)
Question. 15
Solve the inequality for real \(x\):
\(\dfrac{x}{4}<\dfrac{5x-2}{3}-\dfrac{7x-3}{5}\).
Answer:
Solution set: \((4,\infty)\)
Question. 16
Solve the inequality for real \(x\):
\(\dfrac{2x-1}{3}\ge \dfrac{3x-2}{4}-\dfrac{2-x}{5}\).
Answer:
Solution set: \(( -\infty,2] \)
Question. 17
Solve the inequality \(3x-2<2x+1\) and represent the solution on the number line.
Answer:
Solution set: \(( -\infty,3)\)
Question. 18
Solve the inequality \(5x-3\ge 3x-5\) and represent the solution on the number line.
Answer:
Solution set: \([-1,\infty)\)
Question. 19
Solve the inequality \(3(1-x)<2(x+4)\) and represent the solution on the number line.
Answer:
Solution set: \((-1,\infty)\)
Question. 20
Solve the inequality \(\dfrac{x}{2}\ge \dfrac{5x-2}{3}-\dfrac{7x-3}{5}\) and represent the solution on the number line.
Answer:
Solution set: \(\left[ -\dfrac{2}{7},\infty \right)\)
Question. 21
Ravi obtained 70 and 75 marks in the first two unit tests. Find the minimum marks he should get in the third test to have an average of at least 60 marks.
Answer:
Minimum marks in third test: \(35\).
Question. 22
To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations, each of 100 marks. If Sunita’s marks in the first four examinations are 87, 92, 94 and 95, find the minimum marks that Sunita must obtain in the fifth examination to get Grade ‘A’ in the course.
Answer:
Minimum marks required in fifth examination: \(82\).
Question. 23
Find all pairs of consecutive odd positive integers, both of which are smaller than 10, such that their sum is more than 11.
Answer:
Pairs: \((5,7)\) and \((7,9)\).
Question. 24
Find all pairs of consecutive even positive integers, both of which are larger than 5, such that their sum is less than 23.
Answer:
Pairs: \((6,8)\), \((8,10)\) and \((10,12)\).
Question. 25
The longest side of a triangle is three times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.
Answer:
Minimum length of the shortest side: \(9\,\text{cm}\).
Question. 26
A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest.
What are the possible lengths of the shortest board if the third piece is to be at least 5 cm longer than the second?
Answer:
The length of the shortest board is greater than or equal to \(8\,\text{cm}\) but less than or equal to \(22\,\text{cm}\).