NCERT Exemplar solutions for Mathematics [English] Class 11 chapter 6 - Linear Inequalities [Latest edition]

Solve the inequality, 3x – 5 < x + 7, when x is a natural number.

Solve the inequality, 3x – 5 < x + 7, when x is a whole number.

Solve the inequality, 3x – 5 < x + 7, when x is an integer.

Solve the inequality, 3x – 5 < x + 7, when x is a real number.

Solve ${\displaystyle \frac{x-2}{x+5}>2}$.

Solve |3 – 4x| ≥ 9.

Solve 1 ≤ |x – 2| ≤ 3.

The cost and revenue functions of a product are given by C(x) = 20x + 4000 and R(x) = 60x + 2000, respectively, where x is the number of items produced and sold. How many items must be sold to realise some profit? 

Solve for x, |x + 1| + |x| > 3.

Solve for x, ${\displaystyle \frac{|x+3|+x}{x+2}>1}$.

Solve the following system of inequalities:

${\displaystyle \frac{x}{2x+1}\ge \frac{1}{4},\frac{6x}{4x-1}<\frac{1}{2}}$

Find the linear inequalities for which the shaded region in the given figure is the solution set.

If ${\displaystyle \frac{|x-2|}{x-2}\ge 0}$, then ______.

The length of a rectangle is three times the breadth. If the minimum perimeter of the rectangle is 160 cm, then ______.

Solutions of the inequalities comprising a system in variable x are represented on number lines as given below, then ______.

If |x + 3| ≥ 10, then ______.

State whether the following statement is True or False.

If x > y and b < 0, then bx < by

State whether the following statement is True or False.

If xy > 0, then x > 0, and y < 0

State whether the following statement is True or False.

If xy < 0, then x > 0, and y > 0

State whether the following statement is True or False.

If x > 5 and x > 2, then x ∈ (5, ∞)

State whether the following statement is True or False.

If |x| < 5, then x ∈ (–5, 5)

State whether the following statement is True or False.

Graph of x > –2 is

State whether the following statement is True or False.

Solution set of x – y ≤ 0 is

If x ≥ –3, then x + 5 ______ 2.

If –x ≤ –4, then 2x ______ 8.

If ${\displaystyle \frac{1}{x-2}<0}$, then x ______ 2.

If a < b and c < 0, then ${\displaystyle \frac{a}{c}}$ ______ ${\displaystyle \frac{b}{c}}$.

If |x − 1| ≤ 2, then –1 ______ x ______ 3

If |3x – 7| > 2, then x ______ ${\displaystyle \frac{5}{3}}$ or x ______ 3.

If p > 0 and q < 0, then p + q ______ p.

Solve for x, the inequality given below.

${\displaystyle \frac{4}{x+1}\le 3\le \frac{6}{x+1}}$, (x > 0)

Solve for x, the inequality given below.

${\displaystyle \frac{|x-2|-1}{|x-2|-2}\le 0}$

Solve for x, the inequality given below.

${\displaystyle \frac{1}{\left|x\right|-3}\le \frac{1}{2}}$

Solve for x, the inequality given below.

|x − 1| ≤ 5, |x| ≥ 2

Solve for x, the inequality given below.

${\displaystyle -5\le \frac{2-3x}{4}\le 9}$

Solve for x, the inequality given below.

4x + 3 ≥ 2x + 17, 3x – 5 < –2

A company manufactures cassettes. Its cost and revenue functions are C(x) = 26,000 + 30x and R(x) = 43x, respectively, where x is the number of cassettes produced and sold in a week. How many cassettes must be sold by the company to realise some profit?

The water acidity in a pool is considerd normal when the average pH reading of three daily measurements is between 8.2 and 8.5. If the first two pH readings are 8.48 and 8.35, find the range of pH value for the third reading that will result in the acidity level being normal.

A solution of 9% acid is to be diluted by adding 3% acid solution to it. The resulting mixture is to be more than 5% but less than 7% acid. If there is 460 litres of the 9% solution, how many litres of 3% solution will have to be added? 

A solution is to be kept between 40°C and 45°C. What is the range of temperature in degree fahrenheit, if the conversion formula is F = ${\displaystyle \frac{9}{5}}$ C + 32?

The longest side of a triangle is twice the shortest side and the third side is 2cm longer than the shortest side. If the perimeter of the triangle is more than 166 cm then find the minimum length of the shortest side.

In drilling world’s deepest hole it was found that the temperature T in degree celcius, x km below the earth’s surface was given by T = 30 + 25(x – 3), 3 ≤ x ≤ 15. At what depth will the temperature be between 155°C and 205°C?

Solve the following system of inequalities ${\displaystyle \frac{2x+1}{7x-1}>5,\frac{x+7}{x-8}>2}$

Find the linear inequalities for which the shaded region in the given figure is the solution set.

Find the linear inequalities for which the shaded region in the given figure is the solution set.

Show that the following system of linear inequalities has no solution x + 2y ≤ 3, 3x + 4y ≥ 12, x ≥ 0, y ≥ 1

Solve the following system of linear inequalities:

3x + 2y ≥ 24, 3x + y ≤ 15, x ≥ 4

Show that the solution set of the following system of linear inequalities is an unbounded region 2x + y ≥ 8, x + 2y ≥ 10, x ≥ 0, y ≥ 0.

If x < 5, then ______.

Given that x, y and b are real numbers and x < y, b < 0, then ______.

If –3x + 17 < –13, then ______.

If x is a real number and |x| < 3, then ______.

x and b are real numbers. If b > 0 and |x| > b, then ______.

If |x − 1| > 5, then ______.

If |x + 2| ≤ 9, then ______.

The inequality representing the following graph is ______.

Solution of a linear inequality in variable x is represented on number line given below ______. 

Solution of a linear inequality in variable x is represented on number line given below ______.

Solution of a linear inequality in variable x is represented on number line given below  ______.

Solution of a linear inequality in variable x is represented on number line given below ______.

State which of the following statement is True or False.

If x < y and b < 0, then ${\displaystyle \frac{x}{\text{b}}<\frac{y}{\text{b}}}$

State which of the following statement is True or False.

If xy > 0, then x > 0 and y < 0

State which of the following statement is True or False.

If xy > 0, then x < 0 and y < 0

State which of the following statement is True or False.

If xy < 0, then x < 0 and y < 0

State which of the following statement is True or False.

If x < –5 and x < –2, then x ∈ (–∞, –5)

If x < –5 and x > 2, then x ∈ (– 5, 2)

If x > –2 and x < 9, then x ∈ (– 2, 9)

If |x| > 5, then x ∈ (– ${\displaystyle \infty }$, – 5) ∪ [5, ${\displaystyle \infty }$)

If |x| ≤ 4, then x ∈ [– 4, 4]

Graph of x < 3 is

Graph of x ≥ 0 is

Graph of y ≤ 0 is

Solution set of x ≥ 0 and y ≤ 0 is

Solution set of x ≥ 0 and y ≤ 1 is

Solution set of x + y ≥ 0 is

If – 4x ≥ 12, then x ______ – 3.

If ${\displaystyle \frac{-3}{4}x\le –3}$, then x ______ 4.

If ${\displaystyle \frac{2}{x+2}>0}$, then x  ______ –2.

If x > – 5, then 4x ______ –20.

If x > y and z < 0, then – xz ______ – yz.

If p > 0 and q < 0, then p – q ______ p.

If |x + 2| > 5, then x ______ – 7 or x ______ 3.

If – 2x + 1 ≥ 9, then x ______ – 4.