Unit notes and practice questions
Everything here is for unit 4 of the Grade 12 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.
The cards filed under this chapter on the Grade 12 Mathematics page. Open one at a time.
| Inequality Type | Symbols | Boundary Line | Points on Line Included |
|---|---|---|---|
| Strict | Dashed or broken | No | |
| Slack | Solid | Yes |
When you use it
Use this table to choose the correct line style and determine whether the boundary line is included in the solution region.
Watch out
Drawing a solid line for a strict inequality or a dashed line for a slack inequality is penalized in coordinate graphs.
Drafted from Grade 12 Mathematics, pages 237-302, then checked twice before it went up
When you use it
Use this rule whenever multiplying or dividing both sides of an inequality by a negative number.
Watch out
Multiplying or dividing by a negative number reverses the inequality sign, but adding or subtracting any number leaves the sign unchanged.
Drafted from Grade 12 Mathematics, pages 237-302, then checked twice before it went up
When you use it
Use this technique to determine which side of the boundary line to shade for a linear inequality in two variables.
Watch out
Never select a test point that lies on the boundary line. If the line passes through the origin, test a point like (0, 1) or (1, 0) instead of (0, 0).
Drafted from Grade 12 Mathematics, pages 237-302, then checked twice before it went up
When you use it
Use when translating word problems and everyday constraints into linear inequalities.
Watch out
At most translates to less than or equal to, not greater than. At least translates to greater than or equal to, not less than.
Drafted from Grade 12 Mathematics, pages 237-302, then checked twice before it went up
10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.
Question 1
Solve the inequality 7x + 3 < 5x + 9.
Answer: B x < 3
Why
Subtract 5x from both sides to get 2x + 3 < 9. Then subtract 3 from both sides to get 2x < 6. Divide both sides by the positive number 2 to get x < 3. The sign stays the same because you did not divide by a negative number.
Question 2
Alem has Birr 500 to buy wheat flour. One packet costs Birr 130. If x is the number of packets, the inequality is 130x < 500. Which value of x is not a solution?
Answer: D 4
Why
When x = 4, 130 × 4 = 520. But 520 is not less than 500, so the statement is false. The values 0, 1, 2 and 3 all make the inequality true.
Question 3
In the inequality y < 5x + 7, the boundary line is drawn as a dashed (broken) line. Why?
Answer: C The points on the line are not included, because the sign is strict (<).
Why
When the sign is < or >, the points on the line are not part of the solution. So you draw a dashed line. When the sign is ≤ or ≥, the points on the line are included, and you draw a solid line.
Question 4
A taxi in Addis Ababa charges the total cost T = 0.75x + 1.75, where x is the distance in km and T is in Birr. Selam has at most Birr 10 to spend. What is the maximum distance she can travel?
Answer: C 11 km
Why
Write 0.75x + 1.75 < 10. Subtract 1.75 to get 0.75x < 8.25. Divide by 0.75 to get x < 11. So the maximum distance is 11 km.
Question 5
A furniture dealer wants to maximize profit. The objective function is Z = 300x + 100y, where x is the number of tables and y is the number of chairs. The corner points of the feasible region are (0, 0), (20, 0), (10, 50) and (0, 60). What is the maximum profit?
Answer: C Birr 8,000
Why
Substitute each corner point into Z. At (0, 0), Z = 0. At (20, 0), Z = 6000. At (10, 50), Z = 3000 + 5000 = 8000. At (0, 60), Z = 6000. The largest value is 8000, at (10, 50), which means 10 tables and 50 chairs.
Question 6
The dealer has Birr 50,000 to invest. A table costs Birr 2500 and a chair costs Birr 500. Which simplified inequality shows this investment limit? (x = number of tables, y = number of chairs)
Answer: B 5x + y ≤ 100
Why
The total spent is 2500x + 500y, and it must be at most 50000. Divide every term by 500 to get 5x + y ≤ 100.
Question 7
A linear programming problem has the objective function Z = x + y. The feasible region has corner points (0, 0), (1, 0), (0, 2) and (2, 1). At which point is Z largest?
Answer: D (2, 1)
Why
Calculate Z at each corner. At (0, 0), Z = 0. At (1, 0), Z = 1. At (0, 2), Z = 2. At (2, 1), Z = 3. The largest value is 3, at (2, 1).
Question 8
Ficho's budget constraints are: the number of answer sheets (x) is at least 40 more than the number of pencils (y), and the cost is x + 2y ≤ 400 (sheets cost Birr 1 each, pencils cost Birr 2 each). Can Ficho buy 150 answer sheets and 100 pencils?
Answer: C Yes, because both conditions are met.
Why
Check the first condition: 150 is at least 100 + 40 = 140, so it holds. Check the cost: 150 + 2 × 100 = 350, which is at most 400, so it holds. Both conditions are met, so the purchase is possible.
Question 9
A world feed programme needs at least 800 kg of special feed each day. The minimum cost objective is Z = 30x + 90y, where x is kg of corn and y is kg of soybean meal. The two corner points that are candidates for the optimum are (400, 400) and (480, 320). What is the minimum daily cost?
Answer: A Birr 43,200
Why
At (400, 400), Z = 30 × 400 + 90 × 400 = 48,000. At (480, 320), Z = 30 × 480 + 90 × 320 = 14,400 + 28,800 = 43,200. The smaller value is the minimum cost, which is Birr 43,200 per day.
Question 10
A problem asks to minimize Z = 2x + 4y. The corner points are (0, 10), (2, 4) and (10, 0), giving Z values of 40, 20 and 20. Which statement is correct?
Answer: B The minimum value is 20, and every point on the segment between (2, 4) and (10, 0) gives this value, so the solution is not unique.
Why
Both (2, 4) and (10, 0) give Z = 20, which is the smallest value at the corners. Every point on the line segment between them also gives Z = 20. So the optimal solution is not unique. The value 40 at (0, 10) is larger, not smaller.
The app holds 12 questions on this unit, marked as you go. These are the first 10.
Unit 4 runs from page 237 to page 302 of the Ministry of Education edition.
Ministry of Education edition, English, 426 pages
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