Q. What is the solution set for the inequality 6x - 4 < 2x + 8?
A.
x < 3
B.
x > 3
C.
x < 2
D.
x > 2
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Solution
6x - 4 < 2x + 8 => 4x < 12 => x < 3.
Correct Answer: B — x > 3
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Q. What is the solution set for the inequality 7 - 4x ≤ 3?
A.
x ≥ 1
B.
x ≤ 1
C.
x ≥ -1
D.
x ≤ -1
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Solution
7 - 4x ≤ 3 => -4x ≤ -4 => x ≥ 1.
Correct Answer: B — x ≤ 1
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Q. What is the solution set for the inequality 7x - 5 > 2?
A.
x > 1
B.
x < 1
C.
x > 2
D.
x < 2
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Solution
7x - 5 > 2 => 7x > 7 => x > 1.
Correct Answer: A — x > 1
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Q. What is the solution set for the inequality x/3 - 1 ≤ 2?
A.
x ≤ 9
B.
x ≥ 9
C.
x < 9
D.
x > 9
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Solution
x/3 - 1 ≤ 2 => x/3 ≤ 3 => x ≤ 9
Correct Answer: A — x ≤ 9
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Q. What is the solution set of the equation x^2 + 1 = 0?
A.
{0}
B.
{±i}
C.
{1}
D.
{-1}
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Solution
The equation has no real solutions; the solutions are x = i and x = -i.
Correct Answer: B — {±i}
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Q. What is the solution set of the equation x^2 + 4x + 4 = 0?
A.
{-2}
B.
{-4, 0}
C.
{2, 2}
D.
{-2, -2}
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Solution
The equation factors to (x + 2)^2 = 0, giving a double root x = -2.
Correct Answer: D — {-2, -2}
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Q. What is the solution set of the equation x^2 - 5x + 6 = 0?
A.
{2, 3}
B.
{1, 6}
C.
{3, 4}
D.
{0, 5}
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Solution
The equation factors to (x-2)(x-3) = 0, giving solutions x = 2 and x = 3.
Correct Answer: A — {2, 3}
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Q. What is the solution to the compound inequality 1 < 2x + 3 < 7?
A.
-1 < x < 2
B.
0 < x < 2
C.
1 < x < 2
D.
1 < x < 3
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Solution
1 < 2x + 3 < 7 => -2 < 2x < 4 => -1 < x < 2.
Correct Answer: A — -1 < x < 2
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Q. What is the solution to the differential equation dy/dx = (x^2 + 1)y?
A.
y = Ce^(x^3/3 + x)
B.
y = Ce^(x^2 + 1)
C.
y = Ce^(x^2/2)
D.
y = Ce^(x^3)
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Solution
This is a separable equation. Integrating gives y = Ce^(x^3/3 + x).
Correct Answer: A — y = Ce^(x^3/3 + x)
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Q. What is the solution to the differential equation dy/dx = 2y + 3?
A.
y = Ce^(2x) - 3/2
B.
y = Ce^(2x) + 3/2
C.
y = 3e^(2x)
D.
y = 3/2e^(2x)
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Solution
Using the integrating factor method, we find y = Ce^(2x) + 3/2.
Correct Answer: B — y = Ce^(2x) + 3/2
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Q. What is the solution to the differential equation dy/dx = y^2?
A.
y = 1/(C - x)
B.
y = C/(x - 1)
C.
y = Cx
D.
y = e^(x + C)
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Solution
Separating variables and integrating gives y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
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Q. What is the solution to the differential equation y' = y^2 - 1?
A.
y = tan(x + C)
B.
y = 1/(C - x)
C.
y = C/(1 - Cx)
D.
y = C/(x - 1)
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Solution
Separating variables and integrating gives y = 1/(C - x).
Correct Answer: B — y = 1/(C - x)
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Q. What is the solution to the differential equation y'' + 4y = 0?
A.
y = C1 cos(2x) + C2 sin(2x)
B.
y = C1 e^(2x) + C2 e^(-2x)
C.
y = C1 e^(x) + C2 e^(-x)
D.
y = C1 sin(2x) + C2 cos(2x)
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Solution
This is a second-order linear homogeneous differential equation. The characteristic equation has roots ±2i, leading to the solution y = C1 cos(2x) + C2 sin(2x).
Correct Answer: A — y = C1 cos(2x) + C2 sin(2x)
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Q. What is the solution to the equation 2(x - 1) = 4?
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Solution
Dividing both sides by 2 gives x - 1 = 2, thus x = 3.
Correct Answer: C — 3
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Q. What is the solution to the equation 2x + 3 = 7?
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Solution
2x + 3 = 7 => 2x = 4 => x = 2.
Correct Answer: A — 1
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Q. What is the solution to the equation 2^x = 16?
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Solution
2^x = 2^4, so x = 4.
Correct Answer: B — 3
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Q. What is the solution to the equation 2^x = 32?
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Solution
Since 32 = 2^5, we have x = 5.
Correct Answer: B — 5
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Q. What is the solution to the equation 4x + 1 = 9?
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Solution
Subtract 1 from both sides: 4x = 8. Then divide by 4: x = 2.
Correct Answer: B — 2
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Q. What is the solution to the equation 5x - 7 = 3?
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Solution
Adding 7 to both sides gives 5x = 10, then dividing by 5 gives x = 2.
Correct Answer: A — 1
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Q. What is the solution to the equation x^2 - 4 = 0?
A.
-2, 2
B.
0, 4
C.
2, 4
D.
1, 3
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Solution
Factoring gives (x - 2)(x + 2) = 0, leading to solutions x = 2 and x = -2.
Correct Answer: A — -2, 2
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Q. What is the solution to the equation y' = y(1 - y)?
A.
y = 1/(C - x)
B.
y = C/(C + x)
C.
y = C/(1 + Cx)
D.
y = C/(1 - Cx)
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Solution
This is a separable equation. Integrating gives y = C/(C + x).
Correct Answer: B — y = C/(C + x)
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Q. What is the solution to the equation |x - 3| = 5?
A.
{8, -2}
B.
{3, 5}
C.
{5, 1}
D.
{0, 6}
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Solution
The absolute value equation gives two cases: x - 3 = 5 or x - 3 = -5, leading to x = 8 or x = -2.
Correct Answer: A — {8, -2}
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Q. What is the solution to the first-order linear differential equation dy/dx + y = e^x?
A.
y = e^x + Ce^(-x)
B.
y = e^x - Ce^(-x)
C.
y = Ce^x - e^x
D.
y = Ce^(-x) + e^x
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Solution
Using the integrating factor e^x, we solve to get y = e^x + Ce^(-x).
Correct Answer: A — y = e^x + Ce^(-x)
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Q. What is the solution to the inequality -2x + 4 > 0?
A.
x < 2
B.
x > 2
C.
x ≤ 2
D.
x ≥ 2
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Solution
-2x + 4 > 0 => -2x > -4 => x < 2.
Correct Answer: B — x > 2
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Q. What is the solution to the inequality -2x + 5 > 1?
A.
x < 2
B.
x > 2
C.
x < 3
D.
x > 3
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Solution
-2x + 5 > 1 => -2x > -4 => x < 2.
Correct Answer: B — x > 2
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Q. What is the solution to 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: A — x < 2
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Q. What is the solution to the inequality 2x + 5 > 3x - 1?
A.
x < 6
B.
x > 6
C.
x < 4
D.
x > 4
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Solution
2x + 5 > 3x - 1 => 6 > x => x < 6
Correct Answer: D — x > 4
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Q. What is the solution to the inequality 3(x - 1) < 2(x + 2)?
A.
x < 5
B.
x > 5
C.
x < 4
D.
x > 4
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Solution
3(x - 1) < 2(x + 2) => 3x - 3 < 2x + 4 => x < 7.
Correct Answer: C — x < 4
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Q. What is the solution to the inequality 3x - 5 ≤ 4?
A.
x ≤ 3
B.
x ≥ 3
C.
x < 3
D.
x > 3
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Solution
3x - 5 ≤ 4 => 3x ≤ 9 => x ≤ 3
Correct Answer: A — x ≤ 3
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Q. What is the solution to the inequality 4 - x < 2?
A.
x > 2
B.
x < 2
C.
x > 4
D.
x < 4
Show solution
Solution
4 - x < 2 => -x < -2 => x > 2.
Correct Answer: D — x < 4
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