Linear Inequalities

16 Grade 9 Mathematics questions on Linear Inequalities, with worked marking schemes. 16 questions are available on this topic in full.

1. Question 1

   3 marks

   Solve the inequalities $5-2x<11$ and $3x+1\le 13$ simultaneously, and list all the integer values of $x$ that satisfy both.

   (3 marks)

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2. Question 2

   1 mark

   Which integer satisfies both x > −2 and x < 3?

   A. −2

   B. −3

   C. 0

   D. 4

   (1 mark)

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3. Question 3

   1 mark

   State the integer values of x in: −1 ≤ x ≤ 4.

   A. −1, 0, 1, 2, 3

   B. −1, 0, 1, 2, 3, 4

   C. 0, 1, 2, 3

   D. 0, 1, 2, 3, 4

   (1 mark)

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4. Question 4

   1 mark

   Solve: x + 4 > 10

   A. x < 6

   B. x > 6

   C. x < 14

   D. x > 14

   (1 mark)

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5. Question 5

   3 marks

   A school tuck shop sells pens at Ksh 15 each. A student has Ksh 100. Write an inequality and find the maximum number of pens the student can buy.

   (3 marks)

   Marking scheme

   Let $n$ be the number of pens. Each costs Ksh $15$, so the bill is $15n$, and it cannot exceed Ksh $100$: $15n\le 100$. (1)

   Divide both sides by $15$: $n\le \frac{100}{15}=6.67$ to two decimal places. (1)

   Pens are whole items, so $n$ must be a whole number, and the largest whole number below $6.67$ is $6$. (1)

   Final answer: $15n\le 100$, so at most $6$ pens. Check: $6\times 15=90$, while $7\times 15=105$ is too much.

6. Question 6

   2 marks

   The area of a rectangle is at least 48 cm². Its length is 8 cm. Write an inequality for the width w and solve it.

   (2 marks)

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7. Question 7

   1 mark

   On a number line, the solution of x ≤ 5 uses a ___ circle at 5.

   A. Open

   B. Filled/closed

   C. No circle

   D. Dotted

   (1 mark)

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8. Question 8

   1 mark

   Solve: −2x > 8

   A. x < 4

   B. x < −4

   C. x > 4

   D. x > −4

   (1 mark)

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9. Question 9

   1 mark

   Solve: 5 − x < 2

   A. x < 3

   B. x > 7

   C. x < 7

   D. x > 3

   (1 mark)

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10. Question 10

    1 mark

    Which of the following is in the solution set of x < 3?

    A. 4

    B. 2

    C. 5

    D. 3

    (1 mark)

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11. Question 11

    4 marks

    Find the integer values of x satisfying: −5 < 2x + 1 ≤ 9

    (4 marks)

    Marking scheme

    Work on all three parts of the chain at once. Subtract $1$ throughout: $-6<2x\le 8$. (1)

    Divide throughout by $2$, which is positive, so both signs are unchanged: $-3<x\le 4$. (1)

    So $x$ is strictly greater than $-3$ and at most $4$. (1)

    The integers in that range are $-2$, $-1$, $0$, $1$, $2$, $3$, $4$ — note $-3$ is excluded but $4$ is included. (1)

    Final answer: $-2,-1,0,1,2,3,4$.

12. Question 12

    1 mark

    The solution to 3x + 1 ≥ 13 is ___

    A. x ≥ 4

    B. x ≥ 14

    C. x ≤ 4

    D. x ≤ 14

    (1 mark)

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13. Question 13

    1 mark

    Solve: 2x − 3 ≤ 7

    A. x ≥ 5

    B. x ≤ 5

    C. x ≥ 2

    D. x ≤ 2

    (1 mark)

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14. Question 14

    1 mark

    A truck can carry a maximum of 5,000 kg. A load of x kg plus a driver of 80 kg must fit. Which inequality applies?

    A. x + 80 ≤ 5,000

    B. x − 80 ≤ 5,000

    C. x × 80 ≤ 5,000

    D. x + 80 ≥ 5,000

    (1 mark)

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15. Question 15

    3 marks

    Solve and represent on a number line: 2x − 4 < 8

    (3 marks)

    Marking scheme

    Add $4$ to both sides: $2x<12$. (1)

    Divide both sides by $2$, a positive number, so the sign stays the same: $x<6$. (1)

    On the number line, draw an open circle at $6$ because $6$ itself is not included, and shade to the left. (1)

    Final answer: $x<6$.

16. Question 16

    3 marks

    Solve: 3(x − 2) ≥ x + 4, and list the first three integer values in the solution set.

    (3 marks)

    Marking scheme

    Expand the bracket: $3x-6\ge x+4$. (1)

    Collect the $x$ terms on the left and the numbers on the right: $3x-x\ge 4+6$, so $2x\ge 10$. (1)

    Divide by $2$, which is positive, so the sign is unchanged: $x\ge 5$. (1)

    Final answer: $x\ge 5$; the first three integers are $5$, $6$ and $7$.

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