Test: Solving inequalities - Year 9 MCQ

The "<" symbol represents "less than" in mathematics. It indicates that the value on the left is smaller than the value on the right. For example, in 2 < 5, 2 is less than 5. A helpful mnemonic is that the symbol points to the left, resembling an "L" for "less."

The ">" symbol represents "greater than" in mathematics. It indicates that the value on the left is larger than the value on the right. For example, in 5 > 2, 5 is greater than 2. The small end of the symbol points to the smaller number, and the wide end opens toward the larger number.

The "≤" symbol represents "less than or equal to" in mathematics. It indicates that the value on the left is either less than or equal to the value on the right. For example, if x ≤ 6, then x can be 6, 5, 4, etc., including 6 itself, unlike "<" which excludes 6.

The inequality x > 3 means x must be greater than 3. Since we are looking for the smallest integer, we consider integers greater than 3. The integers are ..., 2, 3, 4, 5, .... The smallest integer greater than 3 is 4, as 3 itself does not satisfy x > 3.

The inequality x ≥ 19 means x must be greater than or equal to 19.
We evaluate the options:

• A: 18 (18 < 19, so not possible)
• B: 19 (19 = 19, so possible)
• C: 20 (20 > 19, so possible)
• D: 21 (21 > 19, so possible)

Thus, 18 is not possible.

Solve the inequality 4x + 3 < 15:

1. Subtract 3 from both sides: 4x + 3 - 3 < 15 - 3 → 4x < 12

2. Divide both sides by 4: 4x / 4 < 12 / 4 → x < 3

Now, evaluate the options:

• A: 4x < 15 - 3 (Correct, as it simplifies to 4x < 12)

• B: 4x < 12 (Correct, as shown in the solution)

• C: x < 3 (Correct, as shown in the solution)

• D: x > 3 (Incorrect, as the solution is x < 3, not x > 3)

Solve the inequality 4x - 7 < 17:

1. Add 7 to both sides: 4x - 7 + 7 < 17 + 7 → 4x < 24

2. Divide both sides by 4: 4x / 4 < 24 / 4 → x < 6

Now, evaluate the options:

• A: 4x < 17 + 7 (Correct, as it simplifies to 4x < 24)

• B: 4x < 24 (Correct, as shown in the solution)

• C: x > 6 (Incorrect, as the solution is x < 6, not x > 6)

• D: x < 6 (Correct, as shown in the solution)

Solve the inequality 16x - 5 > 59:

1. Add 5 to both sides: 16x - 5 + 5 > 59 + 5 → 16x > 64

2. Divide both sides by 16: 16x / 16 > 64 / 16 → x > 4

Now, evaluate the options:

• A: 16x > 64 (Correct, as shown in the solution)

• B: 16x > 59 (Incorrect, as we added 5 to get 16x > 64)

• C: 16x < 59 (Incorrect, as the inequality direction is wrong)

• D: 16x < 64 (Incorrect, as the inequality direction is wrong)

Solve the inequality 3x - 2 > 10:

1. Add 2 to both sides: 3x - 2 + 2 > 10 + 2 → 3x > 12

2. Divide both sides by 3: 3x / 3 > 12 / 3 → x > 4

Now, evaluate the options:

• A: x > 4 (Correct, as shown in the solution)

• B: x < 4 (Incorrect)

• C: x > 12 (Incorrect)

• D: x < 12 (Incorrect)

Solve the inequality 2y - 5 ≤ 11:

1. Add 5 to both sides: 2y - 5 + 5 ≤ 11 + 5 → 2y ≤ 16

2. Divide both sides by 2: 2y / 2 ≤ 16 / 2 → y ≤ 8

Now, evaluate the options:

• A: y ≤ 8 (Correct, as shown in the solution)

• B: y ≥ 8 (Incorrect)

• C: y ≤ 3 (Incorrect)

• D: y ≥ 3 (Incorrect)