Worksheet Chapter 6: Linear Inequalities in Two Variables

1. A factory produces two types of chairs, standard (x) and deluxe (y). The production is subject to the following constraints:
- Machine process time: Each standard chair takes 1 hour and each deluxe chair takes 2 hours. Total available machine time is 40 hours per week.
- Assembly time: Each standard chair takes 2 hours and each deluxe chair takes 1 hour. Total available assembly time is 50 hours per week.
- Non-zero constraint: Production counts cannot be negative.
(a) Write a system of linear inequalities for the constraints.
(b) Determine whether producing 15 standard chairs and 10 deluxe chairs per week satisfies all constraints.

2. State the three linear inequalities that define the shaded triangular region bounded by the y-axis (x = 0), the horizontal line y = 2, and the line x + y = 6 in the first quadrant.

3. Fill in the blank: In graphing linear inequalities, a ________ line is used when the inequality symbol contains '<' or '>'.

4. Determine whether the point (-1, 4) satisfies the linear inequality 3x + 2y ≥ 5. Show your calculations.

5. True or False: The point (3, 2) is part of the solution region for the inequality 2x - y > 4.

6. True or False: The inequality 5x - y² ≤ 10 is a linear inequality in two variables.

7. Write a linear inequality for each of the following real-world situations:
(a) A delivery truck carries x small packages and y large packages. The maximum payload count is 120 packages.
(b) The price of a shirt is RM x and a pair of trousers is RM y. Encik Ahmad spends at least RM 250 buying 3 shirts and 2 pairs of trousers.

8. Which of the following phrases represents the inequality symbol '≥'?

9. Given the boundary line equation 2y = -3x + 12:
(a) Find the x-intercept and y-intercept of the boundary line.
(b) State the linear inequality if the shaded region lies strictly above this boundary line.

10. A school sports day committee sells T-shirts (x) and caps (y) to raise funds. The conditions for the sale are as follows:
I. The total number of items sold is at least 50.
II. The number of T-shirts sold is at most twice the number of caps sold.
III. The maximum budget allocated is RM 600, where a T-shirt costs RM 12 and a cap costs RM 6.
Write down the system of three linear inequalities representing all the conditions above.