Mathematics · Textbook solutions

Linear Inequalities

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 59 questions

5.3 Algebraic Solutions of Linear Inequalities in One Variable

40 q

Solved Examples

Worked · 10
  1. Solve 30x<20030x < 200 when
    5.1 Eg.1(i)
    xx is a natural number.
  2. 5.1 Eg.1(ii)
    xx is an integer.
  3. Solve 5x−3<3x+15x - 3 < 3x + 1 when
    5.1 Eg.2(i)
    xx is an integer.
  4. 5.1 Eg.2(ii)
    xx is a real number.
  5. 5.1 Eg.3
    Solve 4x+3<6x+74x + 3 < 6x + 7.
  6. 5.1 Eg.4
    Solve 5−2x3≤x6−5\dfrac{5 - 2x}{3} \le \dfrac{x}{6} - 5.
  7. 5.1 Eg.5
    Solve 7x+3<5x+97x + 3 < 5x + 9. Show the graph of the solutions on number line.
  8. 5.1 Eg.6
    Solve 3x−42≥x+14−1\dfrac{3x - 4}{2} \ge \dfrac{x + 1}{4} - 1. Show the graph of the solutions on number line.
  9. 5.1 Eg.7
    The marks obtained by a student of Class XI in first and second terminal examination are 62 and 48, respectively. Find the minimum marks he should get in the annual examination to have an average of at least 60 marks.
  10. 5.1 Eg.8
    Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40.

Exercise 5.1

Practice · 30
  1. Solve 24x<10024x < 100, when
    Ex 5.1 Q1(i)
    xx is a natural number.
  2. Ex 5.1 Q1(ii)
    xx is an integer.
  3. Solve −12x>30-12x > 30, when
    Ex 5.1 Q2(i)
    xx is a natural number.
  4. Ex 5.1 Q2(ii)
    xx is an integer.
  5. Solve 5x−3<75x - 3 < 7, when
    Ex 5.1 Q3(i)
    xx is an integer.
  6. Ex 5.1 Q3(ii)
    xx is a real number.
  7. Solve 3x+8>23x + 8 > 2, when
    Ex 5.1 Q4(i)
    xx is an integer.
  8. Ex 5.1 Q4(ii)
    xx is a real number.
  9. Ex 5.1 Q5
    Solve the inequality 4x+3<5x+74x + 3 < 5x + 7 for real xx.
  10. Ex 5.1 Q6
    Solve the inequality 3x−7>5x−13x - 7 > 5x - 1 for real xx.
  11. Ex 5.1 Q7
    Solve the inequality 3(x−1)≤2(x−3)3(x - 1) \le 2(x - 3) for real xx.
  12. Ex 5.1 Q8
    Solve the inequality 3(2−x)≥2(1−x)3(2 - x) \ge 2(1 - x) for real xx.
  13. Ex 5.1 Q9
    Solve the inequality x+x2+x3<11x + \dfrac{x}{2} + \dfrac{x}{3} < 11 for real xx.
  14. Ex 5.1 Q10
    Solve the inequality x3>x2+1\dfrac{x}{3} > \dfrac{x}{2} + 1 for real xx.
  15. Ex 5.1 Q11
    Solve the inequality 3(x−2)5≤5(2−x)3\dfrac{3(x - 2)}{5} \le \dfrac{5(2 - x)}{3} for real xx.
  16. Ex 5.1 Q12
    Solve the inequality 12(3x5+4)≥13(x−6)\dfrac{1}{2}\left(\dfrac{3x}{5} + 4\right) \ge \dfrac{1}{3}(x - 6) for real xx.
  17. Ex 5.1 Q13
    Solve the inequality 2(2x+3)−10<6(x−2)2(2x + 3) - 10 < 6(x - 2) for real xx.
  18. Ex 5.1 Q14
    Solve the inequality 37−(3x+5)≥9x−8(x−3)37 - (3x + 5) \ge 9x - 8(x - 3) for real xx.
  19. Ex 5.1 Q15
    Solve the inequality x4<(5x−2)3−(7x−3)5\dfrac{x}{4} < \dfrac{(5x - 2)}{3} - \dfrac{(7x - 3)}{5} for real xx.
  20. Ex 5.1 Q16
    Solve the inequality (2x−1)3≥(3x−2)4−(2−x)5\dfrac{(2x - 1)}{3} \ge \dfrac{(3x - 2)}{4} - \dfrac{(2 - x)}{5} for real xx.
  21. Ex 5.1 Q17
    Solve the inequality 3x−2<2x+13x - 2 < 2x + 1 and show the graph of the solution on number line.
  22. Ex 5.1 Q18
    Solve the inequality 5x−3≥3x−55x - 3 \ge 3x - 5 and show the graph of the solution on number line.
  23. Ex 5.1 Q19
    Solve the inequality 3(1−x)<2(x+4)3(1 - x) < 2(x + 4) and show the graph of the solution on number line.
  24. Ex 5.1 Q20
    Solve the inequality x2≥(5x−2)3−(7x−3)5\dfrac{x}{2} \ge \dfrac{(5x - 2)}{3} - \dfrac{(7x - 3)}{5} and show the graph of the solution on number line.
  25. Ex 5.1 Q21
    Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of at least 60 marks.
  26. Ex 5.1 Q22
    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 first four examinations are 87, 92, 94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade 'A' in the course.
  27. Ex 5.1 Q23
    Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.
  28. Ex 5.1 Q24
    Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.
  29. Ex 5.1 Q25
    The longest side of a triangle is 3 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.
  30. Ex 5.1 Q26
    A man wants to cut three lengths from a single piece of board of length 91cm. The second length is to be 3cm 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 5cm longer than the second? [Hint: If xx is the length of the shortest board, then xx, (x+3)(x + 3) and 2x2x are the lengths of the second and third piece, respectively. Thus, x+(x+3)+2x≤91x + (x + 3) + 2x \le 91 and 2x≥(x+3)+52x \ge (x + 3) + 5].

Miscellaneous Exercise on Chapter 5

19 q

Miscellaneous Examples

Worked · 5
  1. Misc Eg.9
    Solve −8≤5x−3<7-8 \le 5x - 3 < 7.
  2. Misc Eg.10
    Solve −5≤5−3x2≤8-5 \le \dfrac{5 - 3x}{2} \le 8.
  3. Misc Eg.11
    Solve the system of inequalities: 3x−7<5+x3x - 7 < 5 + x ... (1) 11−5x≤111 - 5x \le 1 ... (2) and represent the solutions on the number line.
  4. Misc Eg.12
    In an experiment, a solution of hydrochloric acid is to be kept between 30∘30^\circ and 35∘35^\circ Celsius. What is the range of temperature in degree Fahrenheit if conversion formula is given by C=59(F−32)\text{C} = \dfrac{5}{9}(\text{F} - 32), where C and F represent temperature in degree Celsius and degree Fahrenheit, respectively.
  5. Misc Eg.13
    A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 30% acid solution must be added to it so that acid content in the resulting mixture will be more than 15% but less than 18%?

Miscellaneous Exercise

Practice · 14
  1. Misc Q1
    Solve the inequality 2≤3x−4≤52 \le 3x - 4 \le 5.
  2. Misc Q2
    Solve the inequality 6≤−3(2x−4)<126 \le -3(2x - 4) < 12.
  3. Misc Q3
    Solve the inequality −3≤4−7x2≤18-3 \le 4 - \dfrac{7x}{2} \le 18.
  4. Misc Q4
    Solve the inequality −15<3(x−2)5≤0-15 < \dfrac{3(x - 2)}{5} \le 0.
  5. Misc Q5
    Solve the inequality −12<4−3x−5≤2-12 < 4 - \dfrac{3x}{-5} \le 2.
  6. Misc Q6
    Solve the inequality 7≤(3x+11)2≤117 \le \dfrac{(3x + 11)}{2} \le 11.
  7. Misc Q7
    Solve the inequalities 5x+1>−245x + 1 > -24, 5x−1<245x - 1 < 24 and represent the solution graphically on number line.
  8. Misc Q8
    Solve the inequalities 2(x−1)<x+52(x - 1) < x + 5, 3(x+2)>2−x3(x + 2) > 2 - x and represent the solution graphically on number line.
  9. Misc Q9
    Solve the inequalities 3x−7>2(x−6)3x - 7 > 2(x - 6), 6−x>11−2x6 - x > 11 - 2x and represent the solution graphically on number line.
  10. Misc Q10
    Solve the inequalities 5(2x−7)−3(2x+3)≤05(2x - 7) - 3(2x + 3) \le 0, 2x+19≤6x+472x + 19 \le 6x + 47 and represent the solution graphically on number line.
  11. Misc Q11
    A solution is to be kept between 68∘ F68^\circ\ \text{F} and 77∘ F77^\circ\ \text{F}. What is the range in temperature in degree Celsius (C) if the Celsius / Fahrenheit (F) conversion formula is given by F=95C+32\text{F} = \dfrac{9}{5}\text{C} + 32 ?
  12. Misc Q12
    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 we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added?
  13. Misc Q13
    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?
  14. Misc Q14
    IQ of a person is given by the formula IQ=MACA×100\text{IQ} = \dfrac{\text{MA}}{\text{CA}} \times 100, where MA is mental age and CA is chronological age. If 80≤IQ≤14080 \le \text{IQ} \le 140 for a group of 12 years old children, find the range of their mental age.