Competency Focused Questions
Q1: The least positive integer value of x for which 3 – x < 4 is:
Step 1: Start with the given inequation:
3 – x < 4
Step 2: Subtract 3 from both sides:
3 – x – 3 < 4 – 3
-x < 1
Step 3: Multiply both sides by -1 (reverse the inequality):
x > -1
Step 4: x must be a **positive integer**:
x > -1 ⇒ smallest positive integer = 1
Answer: a. 1
Q2: The two integer values of x which satisfy the three conditions 2x + 6 ≥ 0, x ≠ -2, x < 0 are:
Step 1: Solve the first inequality 2x + 6 ≥ 0:
2x + 6 ≥ 0
2x ≥ -6
x ≥ -3
Step 2: Combine with the condition x < 0:
x ≥ -3 and x < 0 ⇒ x = -3, -2, -1
Step 3: Apply the condition x ≠ -2:
x = -3, -1
Answer: d. x = -3, -1
Q3: The largest integer m such that 5m < 23 is:
Step 1: Start with the inequality:
5m < 23
Step 2: Divide both sides by 5:
m < 23 / 5
m < 4.6
Step 3: The largest integer less than 4.6 is 4
Answer: c. 4
Q4: x is an integer such that \(-8 \le x <4\), and y is an integer such that \(-3 \le y \le 2\). The greatest value of (x + y) is:
Step 1: Determine the possible integer values of x:
-8 ≤ x < 4 ⇒ x = -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3
Step 2: Determine the possible integer values of y:
-3 ≤ y ≤ 2 ⇒ y = -3, -2, -1, 0, 1, 2
Step 3: To get the greatest value of (x + y), take the largest x and largest y:
largest x = 3, largest y = 2
Step 4: Compute (x + y):
x + y = 3 + 2 = 5
Answer: b. 5
Q5: If 2 ≥ y ≥ -3, y ∈ I and 4 > z ≥ -8, z ∈ I, then the least value of yz is:
Step 1: Identify the integer values of y:
2 ≥ y ≥ -3 ⇒ y ∈ {-3, -2, -1, 0, 1, 2}
Step 2: Identify the integer values of z:
4 > z ≥ -8 ⇒ z ∈ {-8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3}
Step 3: To get the **least value of yz**, multiply the smallest y by the largest negative z (or largest negative product):
y = 2 (largest positive) × z = -8 (largest negative magnitude)
yz = 2 × (-8) = -16
Step 4: Verify other combinations:
– y negative × z positive → yz negative but smaller magnitude
– y negative × z negative → yz positive
– y or z = 0 → yz = 0
Hence, the minimum value is indeed -16
Answer: a. -16
Q6: If -8 < 2x + 3, x ∈ I and 2x + 3 < 11, x ∈ I, then the least integer satisfying both the inequations is:
Step 1: Solve the first inequality:
-8 < 2x + 3
Subtract 3 from both sides:
-8 – 3 < 2x
-11 < 2x
Divide by 2:
-11/2 < x
x > -5.5
Step 2: Solve the second inequality:
2x + 3 < 11
Subtract 3 from both sides:
2x < 8
Divide by 2:
x < 4
Step 3: Combine both inequalities:
-5.5 < x < 4
x ∈ I ⇒ x = -5, -4, -3, …, 3
Step 4: Least integer satisfying both inequations:
x = -5
Answer: c. -5
Q7: If 2x + 7 < 3 and x ≥ -4, x ∈ I, then the sum of all the integers satisfying both the inequations is:
Step 1: Solve the first inequality:
2x + 7 < 3
Subtract 7 from both sides:
2x < -4
Divide by 2:
x < -2
Step 2: Combine with the second condition x ≥ -4:
-4 ≤ x < -2
x ∈ I ⇒ x = -4, -3
Step 3: Compute the sum of integers satisfying both:
Sum = -4 + (-3) = -7
Answer: a. -7
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