Q. Determine the solution set for the inequality 2x + 3 > 5.
A.
x < 1
B.
x > 1
C.
x < 2
D.
x > 2
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Solution
2x + 3 > 5 => 2x > 2 => x > 1.
Correct Answer: B — x > 1
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Q. Determine the solution set for the inequality 2x + 3 > 7.
A.
x < 2
B.
x > 2
C.
x < 3
D.
x > 3
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Solution
2x + 3 > 7 => 2x > 4 => x > 2.
Correct Answer: B — x > 2
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Q. Determine the solution set for the inequality 2x + 3 ≤ 7.
A.
x ≤ 2
B.
x ≥ 2
C.
x < 2
D.
x > 2
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Solution
2x + 3 ≤ 7 => 2x ≤ 4 => x ≤ 2.
Correct Answer: A — x ≤ 2
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Q. Determine the solution set for the inequality 3x - 4 < 2x + 5.
A.
x < 9
B.
x > 9
C.
x ≤ 9
D.
x ≥ 9
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Solution
3x - 4 < 2x + 5 => x < 9.
Correct Answer: A — x < 9
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Q. Determine the solution set for the inequality 4x - 1 > 3.
A.
x < 1
B.
x > 1
C.
x ≤ 1
D.
x ≥ 1
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Solution
4x - 1 > 3 => 4x > 4 => x > 1.
Correct Answer: B — x > 1
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Q. Determine the solution set for the inequality 4x - 1 > 3x + 2.
A.
x < 3
B.
x > 3
C.
x < 1
D.
x > 1
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Solution
4x - 1 > 3x + 2 => x > 3.
Correct Answer: B — x > 3
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Q. Determine the solution set for the inequality 4x - 1 < 3.
A.
x < 1
B.
x > 1
C.
x ≤ 1
D.
x ≥ 1
Show solution
Solution
4x - 1 < 3 => 4x < 4 => x < 1.
Correct Answer: A — x < 1
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Q. Determine the solution set for the inequality 4x - 8 < 0.
A.
x < 2
B.
x > 2
C.
x ≤ 2
D.
x ≥ 2
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Solution
4x - 8 < 0 => 4x < 8 => x < 2.
Correct Answer: A — x < 2
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Q. Determine the solution set for the inequality 5 - 2x ≤ 3.
A.
x < 1
B.
x > 1
C.
x ≤ 1
D.
x ≥ 1
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Solution
5 - 2x ≤ 3 => -2x ≤ -2 => x ≥ 1.
Correct Answer: C — x ≤ 1
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Q. Determine the solution set for the inequality 5 - x ≥ 2.
A.
x ≤ 3
B.
x < 3
C.
x ≥ 3
D.
x > 3
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Solution
5 - x ≥ 2 => -x ≥ -3 => x ≤ 3.
Correct Answer: C — x ≥ 3
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Q. Determine the solution set for the inequality 5x - 1 > 4.
A.
x < 1
B.
x > 1
C.
x ≤ 1
D.
x ≥ 1
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Solution
5x - 1 > 4 => 5x > 5 => x > 1.
Correct Answer: B — x > 1
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Q. Determine the solution set for the inequality 5x - 7 < 3.
A.
x < 2
B.
x > 2
C.
x ≤ 2
D.
x ≥ 2
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Solution
5x - 7 < 3 => 5x < 10 => x < 2.
Correct Answer: A — x < 2
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Q. Determine the solution set for the inequality 5x - 7 ≥ 3.
A.
x ≥ 2
B.
x < 2
C.
x > 2
D.
x ≤ 2
Show solution
Solution
5x - 7 ≥ 3 => 5x ≥ 10 => x ≥ 2.
Correct Answer: A — x ≥ 2
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Q. Determine the solution set for the inequality 6x + 4 < 10.
A.
x < 1
B.
x > 1
C.
x < 2
D.
x > 2
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Solution
6x + 4 < 10 => 6x < 6 => x < 1.
Correct Answer: A — x < 1
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Q. Determine the solution set for the inequality 6x - 4 < 2x + 8.
A.
x < 3
B.
x > 3
C.
x < 2
D.
x > 2
Show solution
Solution
6x - 4 < 2x + 8 => 4x < 12 => x < 3.
Correct Answer: B — x > 3
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Q. Determine the solution set for the inequality 7 - 3x < 1.
A.
x > 2
B.
x < 2
C.
x > 3
D.
x < 3
Show solution
Solution
7 - 3x < 1 => -3x < -6 => x > 2.
Correct Answer: B — x < 2
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Q. Determine the solution set for the inequality 7x + 2 ≥ 4.
A.
x ≥ 0
B.
x ≤ 0
C.
x ≥ 1/7
D.
x ≤ 1/7
Show solution
Solution
Subtract 2 from both sides: 7x ≥ 2. Then divide by 7: x ≥ 2/7.
Correct Answer: C — x ≥ 1/7
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Q. Determine the solution set for the inequality 7x + 3 < 4x + 12.
A.
x < 3
B.
x > 3
C.
x ≤ 3
D.
x ≥ 3
Show solution
Solution
7x + 3 < 4x + 12 => 3x < 9 => x < 3.
Correct Answer: B — x > 3
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Q. Determine the solution set for the inequality 7x - 4 ≥ 10.
A.
x ≥ 2
B.
x < 2
C.
x > 2
D.
x ≤ 2
Show solution
Solution
Add 4 to both sides: 7x ≥ 14. Then divide by 7: x ≥ 2.
Correct Answer: A — x ≥ 2
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Q. Determine the value of a for which the function f(x) = { x^2 + a, x < 1; 2x + 3, x >= 1 } is differentiable at x = 1.
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Solution
Setting f(1-) = f(1+) and f'(1-) = f'(1+) leads to a = 1 for differentiability.
Correct Answer: B — 0
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Q. Determine the value of c for which the function f(x) = { 3x + c, x < 1; 2x^2 - 1, x >= 1 } is continuous at x = 1.
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Solution
Setting the two pieces equal at x = 1 gives us 3 + c = 1. Thus, c = -2.
Correct Answer: A — -1
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Q. Determine the value of k for which the function f(x) = { kx + 1, x < 1; 2x - 3, x >= 1 } is continuous at x = 1.
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Solution
To ensure continuity at x = 1, k(1) + 1 = 2(1) - 3, solving gives k = 2.
Correct Answer: B — 2
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Q. Determine the value of k for which the function f(x) = { x^2 + k, x < 1; 2x + 3, x >= 1 } is continuous at x = 1.
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Solution
To ensure continuity at x = 1, we need to set the two pieces equal: k + 1^2 = 2(1) + 3. This gives k + 1 = 5, so k = 4.
Correct Answer: B — 0
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Q. Determine the value of k for which the function f(x) = { x^2 + k, x < 1; 2x + 1, x >= 1 } is continuous at x = 1.
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Solution
To ensure continuity at x = 1, we need to set the two pieces equal: 1^2 + k = 2(1) + 1. This gives k = 2.
Correct Answer: B — 1
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Q. Determine the value of k for which the function f(x) = { x^2 - 4, x < 2; k, x = 2; 3x - 4, x > 2 is continuous at x = 2.
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Solution
For f(x) to be continuous at x = 2, we need limit as x approaches 2 from left to equal limit as x approaches 2 from right. Thus, k must equal 0.
Correct Answer: B — 2
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Q. Determine the value of k for which the function f(x) = { x^2 - 4, x < 2; k, x = 2; 3x - 2, x > 2 is continuous at x = 2.
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Solution
For f(x) to be continuous at x = 2, we need limit as x approaches 2 from left to equal limit as x approaches 2 from right and equal to f(2). Thus, k = 4.
Correct Answer: B — 4
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Q. Determine the value of m for which the function f(x) = { mx + 1, x < 2; x^2 - 4, x >= 2 } is differentiable at x = 2.
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Solution
Setting f(2-) = f(2+) and f'(2-) = f'(2+) leads to m = 1 for differentiability.
Correct Answer: B — 0
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Q. Determine the value of n for which the function f(x) = { n^2 - 1, x < 0; 2x + 3, x >= 0 } is continuous at x = 0.
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Solution
Setting the two pieces equal at x = 0: n^2 - 1 = 3. Solving gives n = ±2.
Correct Answer: A — 1
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Q. Determine the value of p for which the function f(x) = { 2x + 3, x < 2; px + 1, x = 2; x^2 - 1, x > 2 is continuous at x = 2.
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Solution
Setting 2(2) + 3 = p(2) + 1 gives p = 3 for continuity.
Correct Answer: C — 3
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Q. Determine the value of p for which the function f(x) = { 3x - 1, x < 2; px + 4, x = 2; x^2 - 2, x > 2 is continuous at x = 2.
Show solution
Solution
Setting 3(2) - 1 = p(2) + 4 gives p = 2.
Correct Answer: C — 3
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