Linear Inequalities

Many real-life situations involve comparisons rather than exact equalities. A student needs at least 60 marks for a grade, and a budget allows spending of at most a fixed amount. Such statements are expressed as inequalities using the symbols <, >, <= and >=. This chapter explains strict and slack inequalities, linear inequalities in one variable such as ax + b < 0, and double inequalities such as 3 < x <= 5. The solution of an inequality is the set of all values of the variable that make it true. You will learn the rules for solving inequalities. Equal numbers may be added to or subtracted from both sides, and both sides may be multiplied or divided by the same positive number. The sign of the inequality is reversed when multiplying or dividing by a negative number. You will represent solutions on the number line, using a filled circle for an included end point and an open circle for an excluded one. Word problems complete the chapter.

Detailed Worksheet: Linear Inequalities Section A - Definitions (10 marks) 1. Distinguish between strict and slack inequalities with one example each. (2 marks) 2. Solve 24x < 100 when x is (i) a natural number (ii) an integer. (2 marks) 3. State the rule for multiplying both sides of an inequality by a negative number. Give an example. (2 marks) 4. Solve 3x + 8 > 2 when x is a real number. (2 marks) 5. Represent x >= -2 on the number line. (2 marks) Section B - Calculations and Applications (15 marks) 6. Solve 3(x - 1) <= 2(x - 3) and represent the solution on the number line. (3 marks) 7. Solve (3x - 4)/2 >= (x + 1)/4 - 1 and represent the solution on the number line. (3 marks) 8. Solve -8 <= 5x - 3 < 7 and represent the solution on the number line. (3 marks) 9. 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. (3 marks) 10. Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40. (3 marks) Section C - Diagrams (10 marks) 11. Represent the solution of 7 <= (3x + 11)/2 <= 11 on the number line. (4 marks) 12. Represent the solutions of x < 3 and x >= -1 on the same number line and show their common part. (3 marks) 13. Draw the number line for the solution of 2(2x + 3) - 10 < 6(x - 2) for real x. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A solution of 600 litres contains 12% acid. How many litres of a 30% acid solution must be added so that the acid content of the resulting mixture is more than 15% but less than 18%? (5 marks) 15. The IQ of a person is given by IQ = (MA / CA) x 100, where MA is mental age and CA is chronological age. If 80 <= IQ <= 140 for a group of 12-year-old children, find the range of their mental age. (5 marks) 16. A 91 cm long board is to be cut into three pieces. The second piece is 3 cm longer than the shortest, and the third is twice as long as the shortest. If the third piece must be at least 5 cm longer than the second, find the possible lengths of the shortest piece. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show all steps and represent solutions on the number line where asked.