RD Sharma solutions for Mathematics [English] Class 11 chapter 15 - Linear Inequations [Latest edition]

Solve: 12x < 50, when x ∈ R 

Solve: 12x < 50, when  x ∈ Z 

Solve: 12x < 50, when x ∈ N 

Solve: −4x > 30, when  x ∈ R 

Solve: −4x > 30, when x ∈ Z 

Solve: −4x > 30, when x ∈ N 

Solve: 4x − 2 < 8, when x ∈ R 

Solve: 4x − 2 < 8, when x ∈ Z 

Solve: 4x − 2 < 8, when x ∈ N 

3x − 7 > x + 1 

x + 5 > 4x − 10 

3x + 9 ≥ −x + 19 

\[2\left( 3 - x \right) \geq \frac{x}{5} + 4\]

\[\frac{3x - 2}{5} \leq \frac{4x - 3}{2}\] 

−(x − 3) + 4 < 5 − 2x

\[\frac{x}{5} < \frac{3x - 2}{4} - \frac{5x - 3}{5}\]

\[\frac{2\left( x - 1 \right)}{5} \leq \frac{3\left( 2 + x \right)}{7}\]

\[\frac{5x}{2} + \frac{3x}{4} \geq \frac{39}{4}\]

\[\frac{x - 1}{3} + 4 < \frac{x - 5}{5} - 2\]

\[\frac{2x + 3}{4} - 3 < \frac{x - 4}{3} - 2\]

\[\frac{5 - 2x}{3} < \frac{x}{6} - 5\] 

\[\frac{4 + 2x}{3} \geq \frac{x}{2} - 3\]

\[\frac{2x + 3}{5} - 2 < \frac{3\left( x - 2 \right)}{5}\]

\[x - 2 \leq \frac{5x + 8}{3}\] 

\[\frac{6x - 5}{4x + 1} < 0\]

\[\frac{2x - 3}{3x - 7} > 0\] 

\[\frac{3}{x - 2} < 1\]

\[\frac{1}{x - 1} \leq 2\]

\[\frac{4x + 3}{2x - 5} < 6\] 

\[\frac{5x - 6}{x + 6} < 1\]

\[\frac{5x + 8}{4 - x} < 2\]

\[\frac{x - 1}{x + 3} > 2\]

\[\frac{7x - 5}{8x + 3} > 4\]

\[\frac{x}{x - 5} > \frac{1}{2}\] 

Solve each of the following system of equations in R.

1. x + 3 > 0, 2x < 14 

Solve each of the following system of equations in R.

2x − 7 > 5 − x, 11 − 5x ≤ 1

Solve each of the following system of equations in R.

x − 2 > 0, 3x < 18 

2x + 6 ≥ 0, 4x − 7 < 0 

Solve each of the following system of equations in R. 

3x − 6 > 0, 2x − 5 > 0 

Solve each of the following system of equations in R. 

2x − 3 < 7, 2x > −4 

Solve each of the following system of equations in R. 

2x + 5 ≤ 0, x − 3 ≤ 0 

Solve each of the following system of equations in R.

5x − 1 < 24, 5x + 1 > −24 

Solve each of the following system of equations in R. 

3x − 1 ≥ 5, x + 2 > −1 

Solve each of the following system of equations in R.

11 − 5x > −4, 4x + 13 ≤ −11 

Solve each of the following system of equations in R. 

 4x − 1 ≤ 0, 3 − 4x < 0 

Solve the following system of equation in R. 

 x + 5 > 2(x + 1), 2 − x < 3 (x + 2)

Solve each of the following system of equations in R. 

2 (x − 6) < 3x − 7, 11 − 2x < 6 − x 

Solve each of the following system of equations in R.

\[5x - 7 < 3\left( x + 3 \right), 1 - \frac{3x}{2} \geq x - 4\]

Solve each of the following system of equations in R. 

\[\frac{2x - 3}{4} - 2 \geq \frac{4x}{3} - 6, 2\left( 2x + 3 \right) < 6\left( x - 2 \right) + 10\]

Solve each of the following system of equations in R. 

\[\frac{7x - 1}{2} < - 3, \frac{3x + 8}{5} + 11 < 0\]

Solve the following system of equation in R. 

\[\frac{2x + 1}{7x - 1} > 5, \frac{x + 7}{x - 8} > 2\] 

Solve each of the following system of equations in R. 

\[0 < \frac{- x}{2} < 3\] 

Solve each of the following system of equations in R.

 10 ≤ −5 (x − 2) < 20 

Solve each of the following system of equations in R. 

20. −5 < 2x − 3 < 5

Solve each of the following system of equations in R. \[\frac{4}{x + 1} \leq 3 \leq \frac{6}{x + 1}, x > 0\]

Solve  

\[\left| x + \frac{1}{3} \right| > \frac{8}{3}\] 

Solve  

\[\left| 4 - x \right| + 1 < 3\] 

Solve  

\[\left| \frac{3x - 4}{2} \right| \leq \frac{5}{12}\] 

Solve  \[\frac{\left| x - 2 \right|}{x - 2} > 0\] 

Solve  \[\frac{1}{\left| x \right| - 3} < \frac{1}{2}\]

Solve  \[\frac{\left| x + 2 \right| - x}{x} < 2\] 

Solve 

\[\left| \frac{2x - 1}{x - 1} \right| > 2\] 

Solve  \[\left| x - 1 \right| + \left| x - 2 \right| + \left| x - 3 \right| \geq 6\]

Solve  \[\frac{\left| x - 2 \right| - 1}{\left| x - 2 \right| - 2} \leq 0\] 

Solve  \[\frac{1}{\left| x \right| - 3} \leq \frac{1}{2}\]

Solve \[\left| x + 1 \right| + \left| x \right| > 3\] 

Solve \[1 \leq \left| x - 2 \right| \leq 3\] 

Solve  \[\left| 3 - 4x \right| \geq 9\]

Find all pairs of consecutive odd positive integers, both of which are smaller than 10, such that their sum is more than 11. 

Find all pairs of consecutive odd natural number, both of which are larger than 10, such that their sum is less than 40. 

Find all pairs of consecutive even positive integers, both of which are larger than 5, such that their sum is less than 23. 

The marks scored by Rohit in two tests were 65 and 70. Find the minimum marks he should score in the third test to have an average of at least 65 marks. 

A solution is to be kept between 86° and 95°F. What is the range of temperature in degree Celsius, if the Celsius (C)/ Fahrenheit (F) conversion formula is given by\[F = \frac{9}{5}C + 32\]

A solution is to be kept between 30°C and 35°C. What is the range of temperature in degree Fahrenheit? 

To receive grade 'A' in a course, one must obtain an average of 90 marks or more in five papers each of 100 marks. If Shikha scored 87, 95, 92 and 94 marks in first four paper, find the minimum marks that she must score in the last paper to get grade 'A' in the course. 

A company manufactures cassettes and its cost and revenue functions for a week are \[C = 300 + \frac{3}{2}x \text{ and } R = 2x\] respectively, where x is the number of cassettes produced and sold in a week. How many cassettes must be sold for the company to realize a profit? 

The longest side of a triangle is three times the shortest side and third side is 2 cm shorter than the longest side if the perimeter of the triangles at least 61 cm, find the minimum length of the shortest-side.

How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content? 

A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If there are 640 litres of the 8% solution, how many litres of 2% solution will have to be added?

The water acidity in a pool is considered normal when the average pH reading of three daily measurements is between 7.2 and 7.8. If the first two pH reading are 7.48 and 7.85, find the range of pH value for the third reading that will result in the acidity level being normal.

Represent to solution set of each of the following inequations graphically in two dimensional plane: 

x + 2y − y ≤ 0 

Represent to solution set of each of the following in equations graphically in two dimensional plane:

2. x + 2y ≥ 6 

Represent to solution set of each of the following inequations graphically in two dimensional plane: 

 x + 2 ≥ 0 

Represent to solution set of each of the following inequations graphically in two dimensional plane:

4. x − 2y < 0 

Represent to solution set of each of the following inequations graphically in two dimensional plane:

5. −3x + 2y ≤ 6 

Represent to solution set of each of the following inequations graphically in two dimensional plane:

6. x ≤ 8 − 4y

Represent to solution set of each of the following inequations graphically in two dimensional plane: 

0 ≤ 2x − 5y + 10 

Represent to solution set of each of the following inequations graphically in two dimensional plane: 

3y ≥ 6 − 2x 

Represent to solution set of each of the following inequations graphically in two dimensional plane: 

y ≥ 2x − 8 

Represent to solution set of each of the following inequations graphically in two dimensional plane: 

3x − 2y ≤ x + y − 8

Solve the following systems of linear inequation graphically:

 2x + 3y ≤ 6, 3x + 2y ≤ 6, x ≥ 0, y ≥ 0 

Solve the following systems of linear inequation graphically:

2x + 3y ≤ 6, x + 4y ≤ 4, x ≥ 0, y ≥ 0 

Solve the following systems of linear inequations graphically: 

x − y ≤ 1, x + 2y ≤ 8, 2x + y ≥ 2, x ≥ 0, y ≥ 0

Solve the following systems of linear inequations graphically: 

 x + y ≥ 1, 7x + 9y ≤ 63, x ≤ 6, y ≤ 5, x ≥ 0, y ≥ 0 

Solve the following systems of linear inequations graphically:

2x + 3y ≤ 35, y ≥ 3, x ≥ 2, x ≥ 0, y ≥ 0 

Show that the solution set of the following linear inequations is empty set: 

 x − 2y ≥ 0, 2x − y ≤ −2, x ≥ 0, y ≥ 0 

Show that the solution set of the following linear inequations is empty set: 

x + 2y ≤ 3, 3x + 4y ≥ 12, y ≥ 1, x ≥ 0, y ≥ 0 

Find the linear inequations for which the shaded area in Fig. 15.41 is the solution set. Draw the diagram of the solution set of the linear inequations: 

Find the linear inequations for which the solution set is the shaded region given in Fig. 15.42 

Show that the solution set of the following linear in equations is an unbounded set:
x + y ≥ 9
3x + y ≥ 12
x ≥ 0, y ≥ 0

Solve the following systems of inequations graphically:

2x + y ≥ 8, x + 2y ≥ 8, x + y ≤ 6 

Solve the following systems of inequations graphically: 

12x + 12y ≤ 840, 3x + 6y ≤ 300, 8x + 4y ≤ 480, x ≥ 0, y ≥ 0

Solve the following systems of inequations graphically: 

x + 2y ≤ 40, 3x + y ≥ 30, 4x + 3y ≥ 60, x ≥ 0, y ≥ 0 

Solve the following systems of inequations graphically: 

 5x + y ≥ 10, 2x + 2y ≥ 12, x + 4y ≥ 12, x ≥ 0, y ≥ 0

Show that the following system of linear equations has no solution:  

\[x + 2y \leq 3, 3x + 4y \geq 12, x \geq 0, y \geq 1\]

Show that the solution set of the following system of linear inequalities is an unbounded region:  

\[2x + y \geq 8, x + 2y \geq 10, x \geq 0, y \geq 0\] 

Mark the correct alternative in each of the following:

If x\[<\]7, then

Mark the correct alternative in each of the following: 

If − 3x\[+\]17\[< -\]13, then

Write the solution set of the inequation 

\[x + \frac{1}{x} \geq 2\] 

Write the set of values of x satisfying the inequation (x² − 2x + 1) (x − 4) < 0. 

Write the solution set of the equation |2 − x| = x − 2.

Write the set of values of x satisfying |x − 1| ≤ 3 and |x − 1| ≥ 1.

Write the solution set of the inequation \[\left| \frac{1}{x} - 2 \right| > 4\] 

Write the number of integral solutions of \[\frac{x + 2}{x^2 + 1} > \frac{1}{2}\]

Write the set of values of x satisfying the inequations 5x + 2 < 3x + 8 and \[\frac{x + 2}{x - 1} < 4\] 

Write the solution of set of\[\left| x + \frac{1}{x} \right| > 2\]

Write the solution set of the inequation |x − 1| ≥ |x − 3|.

Write the solution of the inequation\[\frac{x^2}{x - 2} > 0\]

Mark the correct alternative in each of the following:
Given that x, y and b are real numbers and x\[<\]y, b\[>\]0, then

Mark the correct alternative in each of the following:
If x is a real number and  \[\left| x \right|\]\[<\]5, then

Mark the correct alternative in each of the following:
If x and a are real numbers such that a\[>\]0 and \\left| x \right|\]\[>\]a, then

Mark the correct alternative in each of the following:

\[\left| x - 1 \right|\]\[>\]5, then 

Mark the correct alternative in each of the following:
If \[\left| x + 2 \right|\]\[\leq\]9, then

Mark the correct alternative in each of the following:
The inequality representing the following graph is 

Mark the correct alternative in each of the following:
The linear inequality representing the solution set given in

Mark the correct alternative in each of the following:
The solution set of the inequation \[\left| x + 2 \right|\]\[\leq\]5 is 

Mark the correct alternative in each of the following:
If  \[\frac{\left| x - 2 \right|}{x - 2}\]\[\geq\] then

Mark the correct alternative in each of the following:
If \[\left| x + 3 \right|\]\[\geq\]10, then