Nipun Maths

Grade 9 · Lesson 21

Linear inequalities

Solve simple inequalities, then mark the solution set on a number line — open circle for < and >, filled for ≤ and ≥.

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Adding and subtracting

Inequalities like x + a > b or x − a ≤ b. Add or subtract the same number on both sides — the inequality sign stays the same.

Number-line legend

All solutions · x > 2

Open circle at the boundary — that value is not included.

-4-3-2-1012345678x > 2

Open circle — does not include 2. Shade to the right (all solutions).

All solutions · x ≥ 2

Filled circle — the boundary value is included.

-4-3-2-1012345678x ≥ 2

Filled circle — includes 2. Shade to the right (all solutions).

Integer solutions · x > 2

Circle 3, 4, 5, … — not 2, because 2 is not greater than 2.

-4-3-2-1012345678x > 2

Orange circles mark each integer solution. No circle at 2 if it is not included.

Integer solutions · x ≥ 2

Circle 2, 3, 4, … — 2 is included, so it gets an orange circle.

-4-3-2-1012345678x ≥ 2

Orange circles mark each integer solution. No circle at 2 if it is not included.

Worked examples

Mark all solutions

Question: solve the inequality, then shade the number line to show all solutions (every real number that works).

  1. Subtract 3 from both sides
  2. x + 3 − 3 > 7 − 3
  3. x > 4

Solution: x > 4

-2-1012345678910x > 4

Open circle — does not include 4. Shade to the right (all solutions).

  1. Add 5 to both sides
  2. x − 5 + 5 ≤ 2 + 5
  3. x ≤ 7

Solution: x ≤ 7

12345678910x ≤ 7

Filled circle — includes 7. Shade to the left (all solutions).

  1. Subtract 1 from both sides
  2. x + 1 − 1 < 0 − 1
  3. x < −1

Solution: x < −1

-7-6-5-4-3-2-1012345x < −1

Open circle — does not include -1. Shade to the left (all solutions).

  1. Add 4 to both sides
  2. x − 4 + 4 ≥ −1 + 4
  3. x ≥ 3

Solution: x ≥ 3

-3-2-10123456789x ≥ 3

Filled circle — includes 3. Shade to the right (all solutions).

  1. Subtract 8 from both sides
  2. x + 8 − 8 ≤ 8 − 8
  3. x ≤ 0

Solution: x ≤ 0

-6-5-4-3-2-10123456x ≤ 0

Filled circle — includes 0. Shade to the left (all solutions).

  1. Add 2 to both sides
  2. x − 2 + 2 > −6 + 2
  3. x > −4

Solution: x > −4

-10-9-8-7-6-5-4-3-2-1012x > −4

Open circle — does not include -4. Shade to the right (all solutions).

  1. Subtract 5 from both sides
  2. x + 5 − 5 ≥ 12 − 5
  3. x ≥ 7

Solution: x ≥ 7

12345678910x ≥ 7

Filled circle — includes 7. Shade to the right (all solutions).

  1. Add 9 to both sides
  2. x − 9 + 9 < 1 + 9
  3. x < 10

Solution: x < 10

456789101112x < 10

Open circle — does not include 10. Shade to the left (all solutions).

  1. Subtract 7 from both sides
  2. x + 7 − 7 > −2 − 7
  3. x > −9

Solution: x > −9

-11-10-9-8-7-6-5-4-3x > −9

Open circle — does not include -9. Shade to the right (all solutions).

  1. Add 3 to both sides
  2. x − 3 + 3 ≤ −3 + 3
  3. x ≤ 0

Solution: x ≤ 0

-6-5-4-3-2-10123456x ≤ 0

Filled circle — includes 0. Shade to the left (all solutions).

Mark only integer solutions

Question: solve the inequality, then mark only the integer solutions. Each integer that works is shown in a coloured circle. Do not shade the whole ray.

  1. Subtract 2 from both sides
  2. x + 2 − 2 > 5 − 2
  3. x > 3

3 is not an integer solution because x must be greater than 3.

Solution: x > 3

Integer solutions: 4, 5, 6, 7, 8, 9, …

-3-2-10123456789x > 3

Orange circles mark each integer solution. No circle at 3 if it is not included.

  1. Add 4 to both sides
  2. x − 4 + 4 ≤ 1 + 4
  3. x ≤ 5

5 is included, so it gets a coloured circle.

Solution: x ≤ 5

Integer solutions: …, -1, 0, 1, 2, 3, 4, 5

-1012345678910x ≤ 5

Orange circles mark each integer solution. No circle at 5 if it is not included.

  1. Subtract 6 from both sides
  2. x + 6 − 6 < 8 − 6
  3. x < 2

2 is not circled — x must be strictly less than 2.

Solution: x < 2

Integer solutions: …, -4, -3, -2, -1, 0, 1

-4-3-2-1012345678x < 2

Orange circles mark each integer solution. No circle at 2 if it is not included.

  1. Add 1 to both sides
  2. x − 1 + 1 ≥ −2 + 1
  3. x ≥ −1

−1 is included, so it is circled.

Solution: x ≥ −1

Integer solutions: -1, 0, 1, 2, 3, 4, 5, …

-7-6-5-4-3-2-1012345x ≥ −1

Orange circles mark each integer solution. No circle at -1 if it is not included.

  1. Subtract 3 from both sides
  2. x + 3 − 3 ≤ −1 − 3
  3. x ≤ −4

Solution: x ≤ −4

Integer solutions: …, -10, -9, -8, -7, -6, -5, -4

-10-9-8-7-6-5-4-3-2-1012x ≤ −4

Orange circles mark each integer solution. No circle at -4 if it is not included.

  1. Add 5 to both sides
  2. x − 5 + 5 > 0 + 5
  3. x > 5

Circle 6, 7, 8, … — not 5.

Solution: x > 5

Integer solutions: 6, 7, 8, 9, 10, …

-1012345678910x > 5

Orange circles mark each integer solution. No circle at 5 if it is not included.

  1. Subtract 9 from both sides
  2. x + 9 − 9 ≥ 4 − 9
  3. x ≥ −5

Solution: x ≥ −5

Integer solutions: -5, -4, -3, -2, -1, 0, 1, …

-10-9-8-7-6-5-4-3-2-101x ≥ −5

Orange circles mark each integer solution. No circle at -5 if it is not included.

  1. Add 6 to both sides
  2. x − 6 + 6 < −2 + 6
  3. x < 4

Solution: x < 4

Integer solutions: …, -2, -1, 0, 1, 2, 3

-2-1012345678910x < 4

Orange circles mark each integer solution. No circle at 4 if it is not included.

  1. Subtract 4 from both sides
  2. x + 4 − 4 > −3 − 4
  3. x > −7

Solution: x > −7

Integer solutions: -6, -5, -4, -3, -2, -1, …

-10-9-8-7-6-5-4-3-2-1x > −7

Orange circles mark each integer solution. No circle at -7 if it is not included.

  1. Add 8 to both sides
  2. x − 8 + 8 ≤ −5 + 8
  3. x ≤ 3

Every integer up to and including 3 is circled.

Solution: x ≤ 3

Integer solutions: …, -3, -2, -1, 0, 1, 2, 3

-3-2-10123456789x ≤ 3

Orange circles mark each integer solution. No circle at 3 if it is not included.

Practice questions

8 questions for this part. After you check, mark the answer on the number line.

Question 1 of 8

x + 4 > 9

Write the solution as an inequality in x, e.g. x > 3 or x ≤ −2.

Lesson 21 · Linear inequalities · Grade 9

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