Algebra Drill
If |x + 1| = |3x − 2|, what are the possible values for x? A) ¼ and ¾ B) ¼ and ³/₂ C) ²/₃ and ³/₂ D) ²/₃ and ⁴/₃ E) ¾ and ⁴/₃
(x+1) = (3x−2) −2x+1 = −2 −2x = −3 x = 3/2 −(x+1) = −(3x−2) −x−1 = −3x+2 2x−1 = 2 2x = 3 x = 3/2 −(x+1) = (3x−2) −x−1 = 3x−2 −4x−1 = −2 −4x = −1 x = 1/4 (x+1) = −(3x−2) x+1 = −3x+2 4x+1 = 2 4x = 1 x = 1/4 B→ ¼ and ³/₂
10√12 ÷ 2√3
10×√4×√3 20√3 −−−−−−−− = −−−−− = 10 2√3 2√3 Algebra book - Chapter 4
If f(x) = 2x² − 4 and g(x) = 2x, for what values of x will f(x) = g(x)?
2x² −4 = 2x 2x² − 2x −4 = 0 2(x² − x − 2) = 0 x² − x −2 = 0 (x−2) (x+1) = 0 x = 2 or x = −1
If f(x) = 2x⁴ − x², what is the value of f(2√3)?
2x⁴ − x² 2(2√3)⁴ − (2√3)² 2(16√81) − (4×3) 2(16×9) − (12) 2(144) −12 288−12 276
If g(x) = 3x + √x, what is the value of g(d² + 6d + 9)?
3x + √x 3(d² + 6d + 9) + √(d² + 6d +9) 3d² + 18d + 27 + √(d+3)² 3d² + 18d + 27 + |d+3| 3d² + 18d + 27 + (d + 3) OR 3d² + 18d + 27 −(d + 3) 3d² + 18d + 27 + d + 3 OR 3d² + 18d + 27 − d − 3 3d² + 19d + 30 OR 3d² + 17d + 24
If 4x − 12 ≥ x + 9, which of the following must be true? A) x > 6 B) x < 7 C) x > 7 D) x > 8 E) x < 8
4x − 12 ≥ x + 9 3x − 12 ≥ 9 3x ≥ 21 x ≥ 7 MUST be A where x > 6 because it could be equal to 7 making B-E invalid ALGEBRA BOOK - Chapter 7
If 4ⁿ + 4∧ (n+1) = 4∧ (n+2) − 176, what is the value of n?
4ⁿ + 4∧ (n+1) − 4∧ (n+2) = −176 4ⁿ (1 + 4¹ − 4²) = −176 4ⁿ (1 + 4 − 16) = −176 4ⁿ (−11) = −176 4ⁿ = 16 n=2 Algebra book - Chapter 3
Estimate √60
60 is between 7² = 49 and 8² = 64 so estimate is 7.7 Difference between 64 & 49 is 15 so 60 is more than ²/₃ to 8² so 7.7 is a little more than ²/₃ of the way to 8 Algebra book - Chapter 4
Given that A = a² + b² + 2ab, what is a + b if A = 9? B B
9 = a² + b² + 2ab NOTE this is a special product (a+b)² = 9 a + b = ±3
If AB > 1 & A = B, which must be greater than 1? −− −− 7 14 A) A+B B) 1 − A C) 2A² D) A² − ½ E) A
AB(14) >7 2AB > 1 Can substitute A for B per problem 2A² > 1 → C ALGEBRA BOOK - Chapter 7
Simplify (4ⁿ + 4ⁿ + 4ⁿ + 4ⁿ) (3ⁿ + 3ⁿ + 3ⁿ) A) 4⁴ⁿ × 3³ⁿ B) 12∧ (n+1) C) 16ⁿ × 9ⁿ D) 12ⁿ E) 4ⁿ × 12ⁿ
B) 12∧ (n+1) 4 × 4ⁿ × 3 × 3ⁿ 12¹ × 4ⁿ × 3ⁿ 12¹ (4 × 3)ⁿ 12¹ (12)ⁿ 12 ∧(n+1) Algebra book -Chapter 3
Every attendee at a monster truck rally paid the same admission fee. How many people attended the rally? 1) If the admission fee had been raised to $20 and twice as many people had attended, the total admission fees collected would have been three times greater. 2) If the admission fee had been raised to $30 and two-thirds as many people had attended, the total admission fees collected would have been 150% of the actual admission fees collected.
Can we find the # of people? T = A × P STATEMENT 1 3T = (20) × 2P 3T = 40P Can't find P with this STATEMENT 2 1.5T = (30) ײ/₃P 1.5T= 20P Can't find P with this TOGETHER - they're basically the same statement so it won't help for them to be combined
The amount of electrical current that flows through a wire is inversely proportional to the resistance in that wire. If a wire currently carries 4 amperes of electrical current, but the resistance is then cut to one-third of its original value, how many amperes of electrical current will flow through the wire?
C₁R₁ = C₂R₂ (4) (3) = C₂ (1) 12 = C₂
If a/b = 16 and a/b² = 8, what is ab?
Divide the equations! a/b × b²/a = 16/8 ab²/ab = 2 b = 2 (after crossing everything out)
If 4 < −1, what is the possible range of values for x? −− −− x 3
If x > 0 4 × 3 > −x 12 > −x −12 < x −12<0<x ∴ x > 0 not possible because doesn't fit the original problem If x < 0 4 < −1 Multiply both sides by x, but since a neg - sign flips − −− x 3 4 > −x Multiply both sides by 3 −− 3 12 > −x −12<x ∴−12 < x < 0
If 4 < 1, what is the possible range of values for x? − − x 3
If x is POSITIVE no need to flip the sign when multiplying x>0 4 × 3 < 1x 12<x 0<12<x ∴12 < x If x is NEGATIVE you need to flip the sign when multiplying x<0 4 × 3 > x 12 > x x<0<12 ∴x<0
If x = 9b−3ab, what is x? −−−−−− 3 − a − − a 3 1) 9ab = 18 −−− −− 3+a 5 2) b=1
Multiply by 3a to get rid of denominators in bottom 9b−3ab (9b − 3ab) × 3a 3a(3b)(3−a) 9ab −−−−−−− = −−−−− = −−−−−−−−−− = −−−− 9 − a² 9 − a² (3−a)(3+a) 3+a −−−− 3a Since this is exactly what A is, then we know Statement 1 is sufficient. Statement 2 is insufficient since it tells us nothing about a TOGETHER - won't work doesn't solve the combo
If xy² = −96 and 1/xy = ¹/₂₄, what is y?
Multiply the equations together! xy² (1/xy) = −96(¹/₂₄) xy²/xy = ⁻⁹⁶/₂₄ y = ⁻⁸/₂ y = −4
Given that d + 8 + 3 = 0, what is d? − − 4 d
Multiply through by 4d to get rid of the denominators d² + 32 + 12d = 0 d² + 12d + 32 = 0 (d + 8) (d + 4) = 0 d = −8 or d = −4 Algebra book - Chapter 5
The "competitive edge" of a baseball team is defined by the formula √(W/L), where W represents the number of the team's wins, and L represents the number of the team's losses. This year the GMAT All-Stars had 3 times as many wins and one-half as many losses as they had last year. By what factor did their "competitive edge"increase?
Pick numbers for this one - you know to do this because there are no numbers given in in the problem so you can just try some out. LOOKING FOR - Increase not new number ORIGINAL √(4/2) = √2 If W = 4 and L = 2 in the original, then when W triples it will be 12 and when L halves, it will be 1 NEW √(12/1) = √12 New √12 −−−− = −−−− = √6 Old √2 ALGEBRA BOOK - Chapter 6
Is mn > −12 1) m > −3 2) n > −4
STATEMENT 1 Won't work because doesn't tell us anything about n STATEMENT 2 Won't work because doesn't tell us anything about m TOGETHER m = −2 and n = −3 then (−2)(−3) = 12 so that's right m = 10 and n = −3 then (10)(−3) = −30 so doesn't work CORRECT ANSWER IS E ALGEBRA BOOK - Chapter 11
If x and y are nonnegative integers and x + y = 25, what is x? 1) 20x + 10y < 300 2) 20x + 10y > 280
Since x + y = 25, the smallest value for 20x + 10y is 250 when x = 0 and y = 25 Statement 1 doesn't tell you what x is and neither does statement 2 - too many possibilities Since y = 25−x, substitute that into 20x + 10y and make: 20x + 10 (25−x) 20x + 250 − 10x 10x + 250 Put this between statements 1 & 2 to read 280 < 10x + 250 < 300 30 < 10x < 50 (remember to subtract from all 3 parts) 3 < x < 5 Since x must be an integer, it must be 4 so the answer is C
Is d>0 1) bc < 0 2) cd > 0
Statement 1 is insufficient because tells you nothing about d Statement 2 insufficient because either c & d are both positive or both negative So only test cases that fit both 1 and 2 Make a table such as this: b c d bc < 0 cd > 0 + − − Y Y − + + Y Y Since d could be either positive or negative, the correct answer is E.
Given that (p−3)² −5 = 0, what is p?
Take the square root first since quadratic formula gives you a non-integer solution (p−3)² = 5 |p−3| = √5 (every time you take the square root you should leave abs val) p=3±√5 (because abs val, you have to ±)
Eco Wildlife Preserve contains 5x Zebras and 2x Lions, where x is a positive integer. If the lions succeed in killing z of the zebras, is the new ratio of zebras to lions less than 2 to 1? 1) z > x 2) z = 4
This is a YES/NO question 5x − z 2 −−−−− < −− 2x 1 5x − z < 4x −z < −x z > x Statement 1 - Correct because it matches what we solved for Statement 2 - knowing z doesn't tell us anything about x TOGETHER - combining still doesn't tell us anything about x ALGEBRA BOOK - Chapter 8
Jake was 4½ feet tall on his 12th birthday, when he began to have a growth spurt. Between his 12th and 15th birthdays, he grew at a constant rate. If Jake was 20% taller on his 15th birthday than on his 13th birthday, how many inches per year did Jake grow during his growth spurt?
This is a linear growth problem where y = mx + b Where x = 0 represents Jake's 12th birthday x = 1 is his 13th birthday x = 2 is his 14th birthday x = 3 is his 15th birthday 4½ feet = 54 inches and serves as the constant b because it is the quantity at time 0 13th birthday: f(1) = m(1) + 54 = 54 + m 15th birthday: f(3) = m(3) + 54 = 54 + 3m and is 20% higher than 13th 54 + 3m = 1.2(54 + m) 54 + 3m = 64.8 + 1.2m 1.8m = 10.8 m = 6 Jake grew 6 inches per year
What is x? 1) x = 4y − 4 2) xy = 8
This is a value question so you need to be able to get 1 value for x Statement 1 doesn't give you a value Statement 2 doesn't give you a value TOGETHER: y = 8 / x and subbed in x = 4(8/x) - 4 x = (32 / x) - 4 Multiply by x to get rid of denominator x² = 32 - 4x Non perfect quadratic equation so it'll give you 2 answers Therefore, E - none of the answers work ALGEBRA BOOK - Chapter 5
A quantity increases in a manner such that the ratio of its values in any two consecutive years is constant. If the quantity doubles every 6 years, by what factor does it increase in two years?
Trying to find k² y(t) = y₀k∧t At time t(0) the quantity is y₀ Since the problem says it doubles every 6 years, the quantity at t = 6 must then be 2y₀ 2y₀ = y₀(k∧6) y₀ cancels out leaving... 2=k∧6 k= ⁶√2 ∴k² = 2∧²/₆ or 2∧¹/₃ = ³√2
Is x < y? 1) 1 1 −− < −− x y 2) x −− < 0 y
YES/NO Question STATEMENT 1 If x and y are the same signs, then this is true, but if they are opposite signs, then it would flip so doesn't work STATEMENT 2 One of them has to be negative, but we don't know which one so can't work to define. Together, we know that they have opposite signs so when the reciprocal is taken, the sign flips and answers NO to the question so C is correct.
If f(x) = ax² − x, and f(4) = 28, what is f(−2)?
f(4) = a(4)² − 4 = 28 16a − 4 = 28 16a = 32 a = 2 f(−2) = 2(−2)² − (−2) =2(4) + 2 =8+2 =10
If f(x) = x³ + 1, and g(x) = 2x, for what value of x does f(g(x)) = g(f(x))?
f(x) = x³ + 1 g(x) = 2x f(g(x)) = g(f(x)) f(2x) = g(x³ + 1) (2x)³ + 1 = 2(x³+1) 8x³ +1 = 2x³ + 2 6x³ = 1 x³ = ¹/₆ x = ³√¹/₆
The maximum height reached by an object thrown directly upward is directly proportional to the square of the velocity with which the object is thrown. If an object thrown upward at 16 feet per second reaches a maximum height of 4 feet, with what speed must the object be thrown upward to reach a maximum height of 9 feet?
h₁ h₂ − = − v₁² v₂² 4 9 −− = −− 16² v₂² v₂² = 9(16² / 4) v₂² = 9(64) = 576 v₂ = √576 = 24
A strain of bacteria multiplies such that the ratio of its population in any two consecutive minutes is constant. If the bacteria grows from a population of 5 million to a population of 40 million in one hour, by what factor does the population increase every 10 minutes?
looking for k¹⁰ y(t) = y₀(k∧t) 40,000,000 = 5,000,000 (k∧60) 40,000,000 = k∧60 −−−−−−−− 5,000,000 40 = k∧60 −− 5 8 = k∧60 ⁶⁰√8 = k y(10) = 5,000,000(⁶⁰√(2³))¹⁰ =5,000,000 (2³⁰/₆₀) k¹⁰ = √2
If mn = 3(m+1)+n and m and n are integers, m could be any of the following values EXCEPT: A) 2 B) 3 C) 4 D) 5 E) 7
mn = 3(m+1) + n mn−n = 3(m+1) n(m−1) = 3(m+1) n = 3(m+1) −−−−− (m−1) n = 3(2+1) 9 −−−−− = −−− = 9 (2−1) 1 n= 3(3+1) 12 −−−−− = −−− = 6 3−1 2 n = 3(4+1) 15 −−−−− = −− = 5 4−1 3 n= 3(5+1) 18 −−−−− = −− = NON INTEGER 5−1 4 n= 3(7+1) 24 −−−−− = −− = 4 7−1 6
The first term in an arithmetic sequence is −5 and the second term is −3. What is the 50th term? (Recall that in an arithmetic sequence, the difference between successive terms is constant.)
n₁ = −5 n₂ = −3 x = −5 k = 2 From −5 to the 50th term, there are 49 jumps of 2 50 - 1 = 49 So 49 x 2 = 98 −5 + 98 = 93 ∴ 93 is the 50th term ALGEBRA BOOK - Chapter 6
If c+d = 11 and c and d are positive integers, which of the following is a possible value for 5c + 8d? A) 55 B) 61 C) 69 D) 83 E) 88
with c = 10 and d = 1 5c + 8d = 50 + 8 = 58 with c = 9 and d = 2 5c + 8d = 45 + 16 = 61 with c = 8 and d = 3 5c + 8d = 40 + 24 = 64 There's an increase of 3 every time so if you hadn't already found it, then could just keep adding 3 until you get to a number that's listed.
For which of the following functions does f(x) = f(2−x)? A) f(x) = x + 2 B) f(x) = 2x − x² C) f(x) = 2 −x D) f(x) = (2−x)² E) f(x) = x²
x + 2 (2−x) + 2 4−x 2x−x² 2(2−x)−(2−x)² 4 −2x−((2−x)(2−x)) 4−2x−(4−2x−2x+x²) 4 − 2x − 4 + 4x − x² −2x + 4x − x² 2x − x²
y= 2x + 9 and 7x + 3y = −51
x = −6 y = −3 Algebra book - Chapter 2
If p = x∧(a+b) , what is the value of positive integer p? −−−−−− x∧b 1) x = 5 2) a = 0
x ∧(a+b) × x ∧−b x ∧a+b−b p = x ∧a Statement 1: Knowing x doesn't mean anything because a could be anything INSUFFICIENT Statement 2: Anything to the 0 power is 1 so this is SUFFICIENT ALGEBRA BOOK - Chapter 8
If f(x) = (x+√3)⁴, what is the range of the function f(x)? A) √3 < f(x) <4 B) f(x) ≥0 C) f(x) <0 D) f(x) ≠ 0
x⁴ + √81 = x⁴ + 9 f(x) ≥ 0 → B
22− | y + 14 | = 20
−12 or −16 Algebra book - Chapter 2
Evaluate ((4+8) / (2−(−6))) − (4 + 8 ÷ 2 − (−6))
−25 / 2 Algebra book - Chapter 1
Which of the following is equivalent to −3x + 7 ≤ 2x + 32? A) x ≥ −5 B) x ≥ 5 C) x ≤ 5 D) x ≤ −5
−3x ≤ 2x + 25 −5x ≤ 25 x ≥ −5 → A ALGEBRA BOOK - Chapter 7
√(x²y³ + 3x²y³), assuming x and y are both positive
√(x²)(y³)(1+3) xy√(y(1+3)) xy√(y×4) xy√y√4 2xy√y Algebra book - Chapter 4