Explanationx^2 + 12x − 64 = (x + 16)(x - 4), with roots -16 and 4. Since the parabola opens upward, the expression is positive outside the roots, giving x < -16 or x > 4.
AlgebraGMATGREMathematicsPractice Set
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21
If f(x) = -2x + 18, what is f(6) − f(0)?
Algebra - Functions and Sequences (including word…
Medium
Explanationf(6) = 6 and f(0) = 18, so f(6) − f(0) = -12.
AlgebraGMATGREMathematicsPractice Set
22
What is the axis of symmetry of the parabola y = 3x^2 − 6x − 16?
A delivery service charges a fixed fee of Rs. 25 plus Rs. 25 per item. If a customer has a budget of Rs. 727, what is the maximum number of items they can order?
Algebra - Linear equations (1 and 2 variables) an…
Medium
Explanation25 + 25n ≤ 727, so n ≤ 702/25 ≈ 28.08, meaning at most 28 items.
ExplanationSince a = 6 is positive, the parabola opens upward.
AlgebraGMATGREMathematicsPractice Set
43
A geometric sequence has first term 4 and common ratio -3. What is the 4th term?
Algebra - Functions and Sequences (including word…
Medium
Explanationaₙ = a₁ × r^(n−1) = 4 × (-3)^3 = -108.
AlgebraGMATGREMathematicsPractice Set
44
Is f(x) = x^2 + x + 1 an even function, an odd function, or neither?
Algebra - Functions and Sequences (including word…
Medium
Explanationf(−x) is neither equal to f(x) nor to −f(x) for this function, so it is neither even nor odd.
AlgebraGMATGREMathematicsPractice Set
45
A cinema sells adult tickets for Rs. 21 and child tickets for Rs. 14. If 91 tickets were sold for a total of Rs. 1589, how many adult tickets were sold?
Algebra - Linear equations (1 and 2 variables) an…
Medium
ExplanationLet a = adult tickets, c = child tickets. a + c = 91 and 21a + 14c = 1589. Solving gives a = 45, c = 46.
AlgebraGMATGREMathematicsPractice Set
46
If f(x) = 12x − 9, what is f(-7)?
Algebra - Functions and Sequences (including word…
Medium
Explanationf(-7) = (12)(-7) − 9 = -93.
AlgebraGMATGREMathematicsPractice Set
47
An arithmetic sequence has first term -42 and common difference 4. What is the 28th term?
Algebra - Functions and Sequences (including word…
ExplanationA trinomial x^2 + bx + c is a perfect square exactly when c = (b/2)^2. Here (b/2)^2 = (-4)^2 = 16, which does not match c = 20, so it is not a perfect square trinomial.
AlgebraGMATGREMathematicsPractice Set
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51
Does the parabola y = 7x^2 + 24 open upward or downward?
Explanationx^2 − 45x + 416 = (x - 13)(x - 32), with roots 13 and 32. Since the parabola opens upward, the expression is positive outside the roots, giving x < 13 or x > 32.
AlgebraGMATGREMathematicsPractice Set
58
A cinema sells adult tickets for Rs. 14 and child tickets for Rs. 9. If 27 tickets were sold for a total of Rs. 273, how many adult tickets were sold?
Algebra - Linear equations (1 and 2 variables) an…
Medium
ExplanationLet a = adult tickets, c = child tickets. a + c = 27 and 14a + 9c = 273. Solving gives a = 6, c = 21.
AlgebraGMATGREMathematicsPractice Set
59
For the function f(x) = 1/(x − (-33)), which value of x must be excluded from the domain?
Algebra - Functions and Sequences (including word…
Medium
ExplanationThe denominator x − (-33) cannot equal 0, so x = -33 must be excluded from the domain.
A delivery service charges a fixed fee of Rs. 15 plus Rs. 23 per item. If a customer has a budget of Rs. 419, what is the maximum number of items they can order?
Algebra - Linear equations (1 and 2 variables) an…
Medium
Explanation15 + 23n ≤ 419, so n ≤ 404/23 ≈ 17.57, meaning at most 17 items.
AlgebraGMATGREMathematicsPractice Set
65
In an arithmetic sequence, the 6th term is 5 and the 7th term is 11. What is the common difference?
Algebra - Functions and Sequences (including word…
ExplanationSum of roots = 9, product of roots = 20, giving x^2 − (9)x + (20) = 0, i.e. x^2 − 9x + 20 = 0.
AlgebraGMATGREMathematicsPractice Set
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71
A rectangle has length (x + 13) and width (x + 7). If the area of the rectangle is 187 square units, what is the value of x (where x is a positive number)?
Explanationx^2 + 9x + 18 factors as the product of (x − r) for each root: (x + 3)(x + 6).
AlgebraGMATGREMathematicsPractice Set
76
A cinema sells adult tickets for Rs. 20 and child tickets for Rs. 10. If 37 tickets were sold for a total of Rs. 480, how many adult tickets were sold?
Algebra - Linear equations (1 and 2 variables) an…
Medium
ExplanationLet a = adult tickets, c = child tickets. a + c = 37 and 20a + 10c = 480. Solving gives a = 11, c = 26.
AlgebraGMATGREMathematicsPractice Set
77
What is the axis of symmetry of the parabola y = 4x^2 + 8x − 18?
ExplanationThe GCF of 16 and 6 (besides x) is 2, and factoring it out (along with one x) gives 2x(8x + 3).
AlgebraGMATGREMathematicsPractice Set
82
If f(x) = x − 9, for what value of x does f(x) = -3?
Algebra - Functions and Sequences (including word…
Medium
Explanationx − 9 = -3, so x = 6.
AlgebraGMATGREMathematicsPractice Set
83
A rectangle has length (x + 8) and width (x + 15). If the area of the rectangle is 1178 square units, what is the value of x (where x is a positive number)?
Explanation(x + 8)(x + 15) = 1178. Testing x = 23: (31)(38) = 1178, which checks out (the other algebraic root is negative and not valid here).
AlgebraGMATGREMathematicsPractice Set
84
What is the value of 3x − 28 when x = 7?
Algebra - Linear equations (1 and 2 variables) an…
Medium
Explanation3(7) − 28 = 21 − 28 = -7.
AlgebraGMATGREMathematicsPractice Set
85
If -9x + 3y = -147 and 5x − 6y = 112, what is x + y?
Algebra - Linear equations (1 and 2 variables) an…
Medium
ExplanationSolving the system gives x = 14 and y = -7, so x + y = 7.
AlgebraGMATGREMathematicsPractice Set
86
f(x) = x + 6 if x < -1, and f(x) = 3x + 13 if x ≥ -1. What is f(-11)?
Algebra - Functions and Sequences (including word…
Medium
ExplanationSince -11 < -1, use f(x) = x + 6: f(-11) = -5.
AlgebraGMATGREMathematicsPractice Set
87
A rectangle has length (x + 8) and width (x + 3). If the area of the rectangle is 456 square units, what is the value of x (where x is a positive number)?
Explanation(x + 8)(x + 3) = 456. Testing x = 16: (24)(19) = 456, which checks out (the other algebraic root is negative and not valid here).
AlgebraGMATGREMathematicsPractice Set
88
A rectangle has length (x + 6) and width (x + 8). If the area of the rectangle is 783 square units, what is the value of x (where x is a positive number)?
Explanationx^2 − 10x − 24 = (x + 2)(x - 12), with roots -2 and 12. Since the parabola opens upward, the expression is positive outside the roots, giving x < -2 or x > 12.
AlgebraGMATGREMathematicsPractice Set
98
Solve for x: -11x + 15 > 202
Algebra - Linear equations (1 and 2 variables) an…
Medium
ExplanationDividing both sides by the negative coefficient -11 reverses the inequality sign, giving x < -17.