Linear Inequalities & Interval Notation (Grade 10) — Worksheet & Memo

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Grade 10 · Equations & Inequalities

Linear Inequalities & Interval Notation (Grade 10)

MARKING GUIDELINE

Marks

Duration

50 minutes

Questions

Name:

Class:

Date:

Mark

/ 33

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
Solve for x: 3x − 5 > 7
(1)
A)x > 4✓
B)x < 4
C)x > 2
D)x ≥ 4
Answer: A — 3x > 12, so x > 4.
- B — the sign reverses only when you divide by a NEGATIVE, and 3 is positive
- C — subtracted the 5 from the 7 instead of adding it
- D — the sign is >, so 4 itself is NOT included
1.2
Solve for x: −2x > 6
(1)
A)x < −3✓
B)x > −3
C)x < 3
D)x > 3
Answer: A — Dividing both sides by −2 REVERSES the inequality sign.
- B — did not reverse the sign when dividing by a negative
- C — divided 6 by 2 and lost the negative
- D — lost the negative AND kept the sign the same way round
1.3
Write x ≥ −2 in interval notation.
(1)
A)[−2 ; ∞)✓
B)(−2 ; ∞)
C)[−2 ; ∞]
D)(−∞ ; −2]
Answer: A — A square bracket includes −2, and infinity always takes a round bracket.
- B — a round bracket EXCLUDES −2, but ≥ includes it
- C — infinity is never reached, so it never takes a square bracket
- D — that is x ≤ −2, the other direction
1.4
Solve for x: −1 ≤ 2x − 1 < 5
(1)
A)0 ≤ x < 3✓
B)0 < x ≤ 3
C)−1 ≤ x < 5
D)−2 ≤ x < 4
Answer: A — Add 1 throughout to get 0 ≤ 2x < 6, then divide throughout by 2.
- B — swapped the two inequality signs on the way through
- C — never divided by the 2 at all
- D — SUBTRACTED 1 throughout instead of adding it, and never divided by 2
1.5
List the integers satisfying −2 < x ≤ 3.
(1)
A)−1 ; 0 ; 1 ; 2 ; 3✓
B)−2 ; −1 ; 0 ; 1 ; 2 ; 3
C)−1 ; 0 ; 1 ; 2
D)−2 ; −1 ; 0 ; 1 ; 2
Answer: A — The < sign excludes −2, and the ≤ sign includes 3.
- B — included −2, but < excludes it
- C — left out 3, but ≤ includes it
- D — read BOTH signs the wrong way round
1.6
Solve for x: x3 + 1 < 4
(1)
A)x < 9✓
B)x < 12
C)x < 3
D)x < 11
Answer: A — Subtract 1 to get x/3 < 3, then multiply both sides by 3.
- B — forgot to subtract the 1 before multiplying by 3
- C — subtracted the 1 but never multiplied by 3
- D — multiplied through by 3 but tripled only the x, not the 1
1.7
A taxi charges R12 plus R7 per kilometre. For the fare to stay UNDER R75, the trip must be …
(1)
A)less than 9 km✓
B)less than 10,71 km
C)more than 9 km
D)less than 63 km
Answer: A — 12 + 7d < 75 gives 7d < 63, so d < 9.
- B — divided 75 by 7 without first subtracting the R12 flag fall
- C — reversed the inequality — a SHORTER trip is the cheaper one
- D — forgot to divide the 63 by the R7 per kilometre
1.8
Solve for x: 5 − x ≥ 2
(1)
A)x ≤ 3✓
B)x ≥ 3
C)x ≤ 7
D)x ≥ −3
Answer: A — −x ≥ −3, and multiplying by −1 reverses the sign to give x ≤ 3.
- B — did not reverse the sign when multiplying by −1
- C — added the 2 to the 5 instead of subtracting it
- D — kept the −3 and did not reverse the sign either
1.9
The interval (2 ; 7] means …
(1)
A)x > 2 and x ≤ 7✓
B)x ≥ 2 and x < 7
C)x > 2 and x < 7
D)x ≥ 2 and x ≤ 7
Answer: A — A round bracket excludes the 2; a square bracket includes the 7.
- B — the two brackets have been read the wrong way round
- C — the square bracket at 7 INCLUDES it
- D — the round bracket at 2 EXCLUDES it
1.10
Solve for x: 4(x − 1) ≤ 2x + 6
(1)
A)x ≤ 5✓
B)x ≥ 5
C)x ≤ 3,5
D)x ≤ 10
Answer: A — 4x − 4 ≤ 2x + 6 gives 2x ≤ 10, so x ≤ 5.
- B — the sign reverses only when dividing by a NEGATIVE, and 2 is positive
- C — multiplied only the x by 4, leaving the −1 inside untouched
- D — forgot to divide the 10 by the 2

Answer grid — marking guideline

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[12 MARKS]

Solve for x, where x ∈ R. Show your answers on a number line and in interval notation where asked.

2.1
−3 ≤ 2x − 13 < −1
(5)
Multiply throughout by 3: −9 ≤ 2x − 1 < −3 (2)
Add 1: −8 ≤ 2x < −2 (1)
Divide by 2: −4 ≤ x < −1 (1)
Interval notation: [−4 ; −1) (1)
2.2
−2x > 10
(3)
Divide by −2 and REVERSE the sign (2)
x < −5 (1)
2.3
3 − 2x ≥ 11
(4)
−2x ≥ 8 (2)
Dividing by −2 reverses the sign (1)
x ≤ −4 (1)

Question 3

[11 MARKS]

Answer the questions below.

3.1
Write −2 ≤ x < 7 in interval notation, and describe what the two different bracket types mean.
(3)
[−2 ; 7) (1)
The square bracket includes −2 (1); the round bracket excludes 7 (1)
3.2
Write (−∞ ; 3] as an inequality, and explain why ∞ never takes a square bracket.
(3)
x ≤ 3 (1)
∞ is not a number and can never be reached (2)
3.3
The perimeter of a rectangle with length (x + 4) cm and breadth 3 cm is at most 26 cm. Determine the largest possible value of x.
(5)
2(x + 4) + 2(3) ≤ 26 (2)
2x + 14 ≤ 26 (1)
x ≤ 6 (1)
The largest value is x = 6 (1)

TOTAL: 33 marks

This question paper consists of 3 questions.