Tüm alıştırma soruları

Soru 261Soru →

If $a$ , $b$ , and $c$ are non-zero real numbers such that $a^3 b^2 c < 0$ , $a b^3 > 0$ , and $\frac{a}{c} < 0$ , which of the following expressions MUST be positive?

\frac{b - c}{a}

\frac{a b}{c}

c(a + b)

\frac{a + c}{b}

b(c - a)

Cevabı ve açıklamayı göster

Cevap: $\frac{b - c}{a}$

Cevap

\frac{b - c}{a}

Deducing the signs of

a

,

b

, and

c

reveals two possible cases: either

a > 0, b > 0, c < 0

or

a < 0, b < 0, c > 0

. In Case 1,

b - c > 0

and

a > 0

, so

\frac{b - c}{a} > 0

. In Case 2,

b - c < 0

and

a < 0

, so

\frac{b - c}{a} > 0

. Therefore,

\frac{b - c}{a}

is strictly positive in all cases.

Adım Adım Çözüm

1

Analyze the sign conditions from the given inequalities.

From

a b^3 > 0

,

a

and

b

must have the same sign. From

\frac{a}{c} < 0

,

a

and

c

must have opposite signs. From

a^3 b^2 c < 0

, since

b^2 > 0

for non-zero

b

, we have

a^3 c < 0

, which confirms

a

and

c

have opposite signs.

Odd powers preserve the sign of a variable, whereas even powers are strictly positive for non-zero real numbers.

2

Determine the two possible sign scenarios for

(a, b, c)

.

Case 1:

a > 0, b > 0, c < 0

.
Case 2:

a < 0, b < 0, c > 0

.

Since

a

and

b

share the same sign and

c

has the opposite sign, these are the only two valid assignments.

3

Evaluate the sign of the numerator and denominator of

\frac{b - c}{a}

in Case 1.

In Case 1 (

a > 0, b > 0, c < 0

):

b - c = \text{positive} - \text{negative} = \text{positive}

. Denominator

a > 0

. Ratio

\frac{\text{positive}}{\text{positive}} > 0

.

Subtracting a negative number from a positive number yields a positive result.

4

Evaluate the sign of the numerator and denominator of

\frac{b - c}{a}

in Case 2.

In Case 2 (

a < 0, b < 0, c > 0

):

b - c = \text{negative} - \text{positive} = \text{negative}

. Denominator

a < 0

. Ratio

\frac{\text{negative}}{\text{negative}} > 0

.

Dividing two negative values produces a positive quotient.

Anahtar Kavram

Positive and Negative Number Properties in Inequalities

Tahmini Süre:2m 0s

Soru 262Soru →

A cosmetic chemist creates a skin care product by mixing two vitamin C solutions. Solution A consists of $200$ milliliters of a liquid that is $15\%$ vitamin C by volume. Solution B is a liquid that is $25\%$ vitamin C by volume. How many milliliters of Solution B must be added to Solution A so that the resulting mixture is $21\%$ vitamin C by volume?

Cevabı ve açıklamayı göster

Cevap: 300

Cevap

300

milliliters of Solution B must be added.

Adding

300

mL of Solution B contributes

300 \times 0.25 = 75

mL of pure vitamin C. Combined with Solution A's

30

mL of vitamin C, the total amount of vitamin C is

105

mL in a total mixture volume of

200 + 300 = 500

mL. The resulting concentration is

\frac{105}{500} = 21\%

.

Adım Adım Çözüm

1

Calculate the volume of pure solute (vitamin C) contained in Solution A.

200 \times 0.15 = 30

mL of pure vitamin C.

The solute volume is calculated by multiplying total volume by the percentage concentration.

2

Express the solute contribution of Solution B and set up the equation for the combined concentration.

\frac{30 + 0.25x}{200 + x} = 0.21

The final concentration is equal to total volume of pure solute divided by the total volume of the final mixture.

3

Solve the algebraic equation for

x

.

30 + 0.25x = 42 + 0.21x \implies 0.04x = 12 \implies x = 300

Cross-multiplying and isolating

x

yields the required volume of Solution B in milliliters.

Anahtar Kavram

Weighted average concentration equation for mixing two liquids.

Tahmini Süre:1m 30s

Soru 263Soru →

If $x$ is a real number satisfying the equation $|2x - 3| = 5x - 12$ , what is the sum of all valid real solutions for $x$ ?

3

\frac{15}{7}

\frac{36}{7}

-3

\frac{45}{7}

Cevabı ve açıklamayı göster

Cevap: $3$

Cevap

The sum of all valid real solutions is

3

.

The option specifying

3

correctly identifies

x = 3

as the only valid solution after discarding the extraneous root

x = \frac{15}{7}

, which produces a negative right-hand side in the original equation.

Adım Adım Çözüm

1

Set up the two algebraic cases for the absolute value equation

|2x - 3| = 5x - 12

.

Case 1:

2x - 3 = 5x - 12

Case 2:

2x - 3 = -(5x - 12) = -5x + 12

By definition,

|a| = b

implies

a = b

or

a = -b

, provided

b \ge 0

.

2

Solve Case 1 for

x

.

2x - 3 = 5x - 12 \implies 3x = 9 \implies x = 3

Isolate the variable

x

algebraically.

3

Solve Case 2 for

x

.

2x - 3 = -5x + 12 \implies 7x = 15 \implies x = \frac{15}{7}

Isolate the variable

x

algebraically.

4

Check candidate solutions for extraneous roots in the original equation.

For

x = 3

:

|2(3) - 3| = |3| = 3

and

5(3) - 12 = 3

. Valid.
For

x = \frac{15}{7}

:

|2(\frac{15}{7}) - 3| = |\frac{9}{7}| = \frac{9}{7}

, but

5(\frac{15}{7}) - 12 = -\frac{9}{7}

. Since

\frac{9}{7} \neq -\frac{9}{7}

,

x = \frac{15}{7}

is extraneous.

The output of an absolute value expression cannot be negative, so any candidate root making the right side negative must be discarded.

5

Calculate the sum of all valid real solutions.

Sum =

3

There is only one valid solution,

x = 3

.

Anahtar Kavram

Solving absolute value equations with a variable expression on the right-hand side requires checking candidate solutions to eliminate extraneous roots.

Soru 264Soru →

Sequence $A$ is an arithmetic sequence with first term $a_1 = 5$ and common difference $d = 3$ .
Sequence $B$ is a geometric sequence with first term $b_1 = 2$ and common ratio $r = \sqrt{3}$ .

Arrange the four quantities defined below in ascending order (from smallest to largest value).

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct ascending order is Quantity K, followed by Quantity N, Quantity M, and finally Quantity L.

Evaluating each expression gives Quantity K = 47, Quantity N ≈ 142.07, Quantity M = 162, and Quantity L = 185. Comparing these values from smallest to largest yields the order K, N, M, L.

Adım Adım Çözüm

1

Calculate Quantity K (

a_{15}

for Sequence A)

a_{15} = 5 + (15 - 1) \times 3 = 47

The

n

-th term of an arithmetic sequence is given by

a_n = a_1 + (n - 1)d

.

2

Calculate Quantity L (

S_{10}

for Sequence A)

S_{10} = \frac{10}{2} \times [2(5) + (10 - 1) \times 3] = 5 \times (10 + 27) = 185

The sum of the first

n

terms of an arithmetic sequence is given by

S_n = \frac{n}{2}[2a_1 + (n - 1)d]

.

3

Calculate Quantity M (

b_9

for Sequence B)

b_9 = 2 \times (\sqrt{3})^{9 - 1} = 2 \times (\sqrt{3})^8 = 2 \times 3^4 = 162

The

n

-th term of a geometric sequence is given by

b_n = b_1 r^{n-1}

.

4

Calculate and approximate Quantity N (

S_6

for Sequence B)

S_6 = \frac{2((\sqrt{3})^6 - 1)}{\sqrt{3} - 1} = \frac{2(27 - 1)}{\sqrt{3} - 1} = \frac{52}{\sqrt{3} - 1} = 52(\sqrt{3} + 1) \approx 52 \times 2.732 = 142.07

The sum of a geometric series is

S_n = \frac{b_1(r^n - 1)}{r - 1}

. Rationalizing the denominator yields

52(\sqrt{3} + 1)

.

5

Compare all four values to establish ascending order

47 < 142.07 < 162 < 185

, which corresponds to

K < N < M < L

Ordering the numerical outputs from least to greatest gives the required sequence.

Anahtar Kavram

Calculating specific terms and sums of arithmetic and geometric sequences using explicit formulas and ordering calculated quantities.

Soru 265Soru →

Tank P contains $80$ liters of a liquid fertilizer solution that is $30\%$ nitrogen by volume. Tank Q contains $120$ liters of a liquid fertilizer solution that is $15\%$ nitrogen by volume. If $x$ liters of solution are transferred from Tank P into Tank Q, causing the nitrogen concentration in Tank Q to become $20\%$ , what is the value of $x$ ?

20

30

40

60

80

Cevabı ve açıklamayı göster

Cevap: $60$

Cevap

The volume transferred,

x

, is

60

liters.

The initial amount of nitrogen in Tank Q is

0.15 \times 120 = 18

liters. Transferring

x

liters from Tank P adds

0.30x

liters of nitrogen and increases Tank Q's total volume to

120 + x

liters. Setting the new concentration to

20\%

, we get

\frac{18 + 0.30x}{120 + x} = 0.20

. Solving for

x

gives

18 + 0.30x = 24 + 0.20x \implies 0.10x = 6 \implies x = 60

. Thus,

60

liters must be transferred.

Adım Adım Çözüm

1

Calculate the initial volume of pure nitrogen in Tank Q.

Initial nitrogen in Tank Q =

15\% \text{ of } 120 = 0.15 \times 120 = 18

liters.

To determine the new concentration, we must know the starting amount of solute.

2

Express the added nitrogen and the new total volume of Tank Q in terms of

x

.

Nitrogen added from Tank P =

0.30x

liters; New total volume of Tank Q =

120 + x

liters.

The solution transferred carries

30\%

nitrogen per liter and increases both the nitrogen content and total volume of Tank Q.

3

Set up the concentration equation for Tank Q and solve for

x

.

\frac{18 + 0.30x}{120 + x} = 0.20 \implies 18 + 0.30x = 0.20(120 + x) \implies 18 + 0.30x = 24 + 0.20x \implies 0.10x = 6 \implies x = 60.

The concentration is defined as total solute divided by total solution volume.

Anahtar Kavram

Weighted Average and Mixture Equations

Soru 266Soru →

A university library surveyed $240$ graduate students regarding the research methodologies used in their dissertations: Quantitative Analysis ( $Q$ ), Qualitative Interviews ( $I$ ), and Archival Research ( $A$ ). Every student surveyed used at least one of these three methodologies. A total of $130$ students used Quantitative Analysis, $110$ used Qualitative Interviews, and $90$ used Archival Research. Additionally, $40$ students used both Quantitative Analysis and Qualitative Interviews, $30$ used both Qualitative Interviews and Archival Research, and $35$ used both Quantitative Analysis and Archival Research. How many students used all three research methodologies?

15

25

40

90

105

Cevabı ve açıklamayı göster

Cevap: $15$

Cevap

15 students used all three research methodologies.

According to the Principle of Inclusion-Exclusion for three sets,

\text{Total} = |Q| + |I| + |A| - (|Q \cap I| + |I \cap A| + |Q \cap A|) + |Q \cap I \cap A| + \text{Neither}

. Substituting the given values gives

240 = 130 + 110 + 90 - (40 + 30 + 35) + x + 0

, which simplifies to

240 = 225 + x

, yielding

x = 15

. Thus, the option stating 15 is correct.

Adım Adım Çözüm

1

State the three-set inclusion-exclusion principle formula for total population.

Total =

|Q| + |I| + |A| - (|Q \cap I| + |I \cap A| + |Q \cap A|) + |Q \cap I \cap A| + \text{Neither}

To set up an algebraic equation relating the total number of students to their set intersections.

2

Substitute the known numerical values into the formula.

240 = 130 + 110 + 90 - (40 + 30 + 35) + |Q \cap I \cap A| + 0

Every student used at least one methodology, so

\text{Neither} = 0

.

3

Simplify the equation and solve for the target variable

|Q \cap I \cap A|

.

240 = 330 - 105 + |Q \cap I \cap A| \implies 240 = 225 + |Q \cap I \cap A| \implies |Q \cap I \cap A| = 15

Subtracting 225 from 240 yields the number of students who used all three methodologies.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle

Tahmini Süre:1m 30s

Soru 267Soru →

An antique dealer purchased three items—an armchair, a desk, and a cabinet—for a total combined cost of $\$4,000$ . The cost of the desk was $20\%$ greater than the cost of the armchair, and the cost of the cabinet was $50\%$ greater than the cost of the desk. To determine the list price for each item, the dealer marked up the cost of the armchair by $40\%$ , the cost of the desk by $50\%$ , and the cost of the cabinet by $30\%$ . During a clearance event, the dealer sold the armchair at a $15\%$ discount off its list price, the desk at a $20\%$ discount off its list price, and the cabinet at its full list price with no discount. What was the dealer's overall profit as a percentage of the total combined purchase cost of the three items?

23.00\%

24.25\%

28.75\%

19.52\%

38.00\%

Cevabı ve açıklamayı göster

Cevap: $24.25\%$

Cevap

The dealer's overall profit was

24.25\%

of the total combined purchase cost.

The correct response of

24.25\%

accurately calculates the individual item costs, applies sequential markup and discount multipliers (

1.40 \times 0.85 = 1.19

for armchair,

1.50 \times 0.80 = 1.20

for desk,

1.30

for cabinet), sums the final sales revenue to

\$4,970

, and evaluates net profit (

\$970

) over total cost (

\$4,000

).

Adım Adım Çözüm

1

Determine the individual cost of each item

Armchair cost =

\$1,000

, Desk cost =

\$1,200

, Cabinet cost =

\$1,800

Let

A

be the cost of the armchair. The desk cost is

1.2A

, and the cabinet cost is

1.5 \times 1.2A = 1.8A

. The total cost is

A + 1.2A + 1.8A = 4.0A = 4,000 \implies A = 1,000

.

2

Calculate the list price and selling price for each item

Armchair selling price =

\$1,190

; Desk selling price =

\$1,440

; Cabinet selling price =

\$2,340

Armchair list price =

\$1,000 \times 1.40 = \$1,400

; selling price =

\$1,400 \times 0.85 = \$1,190

. Desk list price =

\$1,200 \times 1.50 = \$1,800

; selling price =

\$1,800 \times 0.80 = \$1,440

. Cabinet selling price = list price =

\$1,800 \times 1.30 = \$2,340

.

3

Find total revenue and total profit

Total revenue =

\$4,970

, Total profit =

\$970

Total revenue =

\$1,190 + \$1,440 + \$2,340 = \$4,970

. Total profit =

\$4,970 - \$4,000 = \$970

.

4

Compute overall profit percentage on total cost

24.25\%

Overall profit percentage =

(\frac{970}{4000}) \times 100\% = 24.25\%

.

Anahtar Kavram

Successive Percentage Changes and Weighted Profit Base Calculations

Tahmini Süre:2m 0s

Soru 268Soru →

An investor allocates a sum of money across three accounts— $A$ , $B$ , and $C$ —which earn simple annual interest rates of $5\%$ , $4\%$ , and $6\%$ , respectively. The total interest earned from all three accounts in one year is $\$1,060$ . The amount invested in Account $C$ equals the total amount invested in Accounts $A$ and $B$ combined. If the amount in Account $A$ were increased by $50\%$ and the amount in Account $B$ were decreased by $25\%$ while Account $C$ remained unchanged, the total annual interest earned would increase by $\$110$ . What is the total amount, in dollars, invested across all three accounts?

\$10,000

\$15,000

\$18,000

\$20,000

\$24,000

Cevabı ve açıklamayı göster

Cevap: $\$20,000$

Cevap

The total amount invested across all three accounts is

\$20,000

.

The system of linear equations derived from the problem constraints uniquely solves to

a = \$6,000

,

b = \$4,000

, and

c = \$10,000

. Summing these three amounts gives a total investment of

\$20,000

.

Adım Adım Çözüm

1

Define variables and translate the problem statements into algebraic equations.

Let

a

,

b

, and

c

be the amounts invested in Accounts

A

,

B

, and

C

.
Equation 1 (Total Interest):

0.05a + 0.04b + 0.06c = 1,060 \implies 5a + 4b + 6c = 106,000

.
Equation 2 (Relationship among principal amounts):

c = a + b \implies a + b - c = 0

.
Equation 3 (Change in interest): Increasing

a

by

50\%

adds

0.05(0.50a) = 0.025a

interest. Decreasing

b

by

25\%

reduces interest by

0.04(0.25b) = 0.01b

. The net change is

0.025a - 0.01b = 110 \implies 25a - 10b = 110,000 \implies 5a - 2b = 22,000

.

Establishing a complete 3-variable system of linear equations is necessary to determine the unknowns.

2

Substitute

c = a + b

into Equation 1 to reduce the system to two variables.

5a + 4b + 6(a + b) = 106,000 \implies 11a + 10b = 106,000

.

Using substitution eliminates variable

c

, leaving a linear system in

a

and

b

.

3

Solve the two-variable system using elimination.

We have:
(1)

11a + 10b = 106,000

(2)

5a - 2b = 22,000

Multiply (2) by 5:

25a - 10b = 110,000

.
Add this to (1):

(11a + 10b) + (25a - 10b) = 106,000 + 110,000 \implies 36a = 216,000 \implies a = 6,000

.
Substitute

a = 6,000

back into (2):

5(6,000) - 2b = 22,000 \implies 30,000 - 2b = 22,000 \implies 2b = 8,000 \implies b = 4,000

.

Elimination allows straightforward calculation of individual values for

a

and

b

.

4

Calculate

c

and find the total sum invested across all three accounts.

c = a + b = 6,000 + 4,000 = 10,000

.
Total investment

= a + b + c = 6,000 + 4,000 + 10,000 = 20,000

.

The question asks for the total amount invested in all three accounts combined.

Anahtar Kavram

Systems of Linear Equations in Three Variables

Tahmini Süre:2m 0s

Soru 269Soru →

A positive integer $n$ has exactly four distinct prime factors, the three smallest of which are $2$ , $3$ , and $5$ . If $n$ is divisible by $360$ and has exactly $48$ positive divisors, what is the minimum possible value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 2520

Cevap

The minimum possible value of

n

is 2520.

To minimize

n

, we analyze its prime factorization

n = 2^a \times 3^b \times 5^c \times p^d

, where

p

is the fourth distinct prime factor. Divisibility by

360 = 2^3 \times 3^2 \times 5^1

requires

a \ge 3

,

b \ge 2

, and

c \ge 1

. The number of positive divisors is given by

(a+1)(b+1)(c+1)(d+1) = 48

. To minimize

n

, we pick the smallest prime greater than 5, which is

p = 7

, and set

d = 1

. This simplifies the divisor equation to

(a+1)(b+1)(c+1) = 24

. Since

a+1 \ge 4

,

b+1 \ge 3

, and

c+1 \ge 2

, the minimal product of these terms is

4 \times 3 \times 2 = 24

. This uniquely determines

a = 3

,

b = 2

, and

c = 1

. Substituting these values gives

n = 2^3 \times 3^2 \times 5^1 \times 7^1 = 2520

.

Adım Adım Çözüm

1

Determine the prime factorization of the divisor requirement.

360 = 2^3 \times 3^2 \times 5^1

.

Divisibility by 360 imposes lower bounds on the exponents of the prime factors 2, 3, and 5 in

n

.

2

Formulate the general prime factorization for

n

and state exponent constraints.

n = 2^a \times 3^b \times 5^c \times p^d

with

a \ge 3

,

b \ge 2

,

c \ge 1

,

d \ge 1

, and prime

p > 5

.

n

has four distinct prime factors, three of which are 2, 3, and 5.

3

Apply the divisor counting formula to set up an algebraic equation.

(a+1)(b+1)(c+1)(d+1) = 48

.

The number of positive divisors of

n = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k}

is given by

(e_1+1)(e_2+1)\dots(e_k+1)

.

4

Minimize

n

by choosing optimal values for

p

and

d

.

p = 7

and

d = 1

, leading to

(a+1)(b+1)(c+1) = 24

.

To make

n

as small as possible, the fourth prime

p

should be the smallest available prime (

7

) and its exponent

d

should be minimized (

1

).

5

Solve for exponents

a

,

b

, and

c

under the given inequality constraints.

a = 3

,

b = 2

,

c = 1

.

Since

a+1 \ge 4

,

b+1 \ge 3

, and

c+1 \ge 2

, the minimum possible product

(a+1)(b+1)(c+1)

is

4 \times 3 \times 2 = 24

. Hence,

a=3, b=2, c=1

is the unique solution.

6

Calculate the value of

n

.

n = 2^3 \times 3^2 \times 5^1 \times 7^1 = 2520

.

Multiplying out the prime factors yields the smallest integer matching all conditions.

Anahtar Kavram

Prime Factorization and Divisor Counting Constraints

Tahmini Süre:2m 0s

Soru 270Soru →

A commercial bakery operates two automated production lines, Line A and Line B, to produce specialized pastry boxes. Line A requires $4$ minutes of mixing and $2$ minutes of baking per box. Line B requires $3$ minutes of mixing and $5$ minutes of baking per box. On a given shift, the bakery utilizes a total of $180$ minutes of mixing time and $160$ minutes of baking time, with both lines running continuously without downtime.

If $x$ represents the number of pastry boxes produced by Line A and $y$ represents the number of pastry boxes produced by Line B during the shift, which of the following statements regarding the production parameters must be true? Select all that apply.

Geçerli olan tümünü seçin

Line A produced

10

more pastry boxes than Line B during the shift.

The total number of pastry boxes produced by both lines combined is equal to

50

.

The linear equation modeling total mixing time across both lines is

4x + 2y = 180

.

Line B accounted for

60\%

of the total pastry boxes produced during the shift.

The system of linear equations modeling the resource constraints yields a unique solution of

x = 30

and

y = 20

.

Cevabı ve açıklamayı göster

Cevap: Line A produced $10$ more pastry boxes than Line B during the shift.; The total number of pastry boxes produced by both lines combined is equal to $50$ .; The system of linear equations modeling the resource constraints yields a unique solution of $x = 30$ and $y = 20$ .

Cevap

The statements that must be true are: Line A produced 10 more pastry boxes than Line B during the shift; the total number of pastry boxes produced by both lines combined is equal to 50; and the system of linear equations modeling the resource constraints yields a unique solution of

x = 30

and

y = 20

.

Solving the system of equations representing mixing time (

4x + 3y = 180

) and baking time (

2x + 5y = 160

) gives

x = 30

boxes for Line A and

y = 20

boxes for Line B. Consequently, Line A produced

30 - 20 = 10

more boxes than Line B, the total production is

30 + 20 = 50

boxes, and the system indeed yields the unique solution

(30, 20)

.

Adım Adım Çözüm

1

Set up the linear equations based on mixing time and baking time constraints.

Mixing constraint:

4x + 3y = 180

; Baking constraint:

2x + 5y = 160

.

Each unit of

x

requires

4

min mixing and

2

min baking; each unit of

y

requires

3

min mixing and

5

min baking.

2

Solve the system of equations using elimination.

Multiply the baking equation by

2

:

4x + 10y = 320

. Subtracting the mixing equation (

4x + 3y = 180

) yields

7y = 140 \implies y = 20

. Substituting

y = 20

into

4x + 3(20) = 180

gives

4x = 120 \implies x = 30

.

Eliminating

x

yields the exact value for

y

, which then gives

x

.

3

Evaluate each candidate statement against the solution

(x, y) = (30, 20)

.

Difference:

x - y = 30 - 20 = 10

(True). Total:

x + y = 30 + 20 = 50

(True). Mixing equation:

4x + 3y = 180

, not

4x + 2y = 180

(False). Line B share:

\frac{20}{50} = 40\%

, not

60\%

(False). Unique solution

(30, 20)

(True).

Checking each statement determines which options are valid.

Anahtar Kavram

Algebraic modeling of multi-variable resource constraint word problems using linear systems of equations

Tahmini Süre:2m 0s

Soru 271Soru →

If $x$ is a real number satisfying the inequality $\frac{|2x - 5|}{x + 1} \le 1$ , which of the following represents the complete set of all possible values of $x$ ?

x < -1

or

\frac{4}{3} \le x \le 6

\frac{4}{3} \le x \le 6

-1 < x \le 6

x < -1

or

x \ge \frac{4}{3}

x \le -1

or

\frac{4}{3} \le x \le 6

Cevabı ve açıklamayı göster

Cevap: $x < -1$ or $\frac{4}{3} \le x \le 6$

Cevap

x < -1

or

\frac{4}{3} \le x \le 6

The correct answer accounts for both possible sign states of the denominator

x + 1

. When

x > -1

, multiplying gives

|2x - 5| \le x + 1

, yielding

\frac{4}{3} \le x \le 6

. When

x < -1

, multiplying flips the inequality to

|2x - 5| \ge x + 1

, which is trivially satisfied by all

x < -1

because an absolute value is non-negative and

x + 1

is negative. Combining both cases yields

x < -1

or

\frac{4}{3} \le x \le 6

.

Adım Adım Çözüm

1

Determine domain restrictions and break the inequality into cases based on the denominator's sign.

The expression is undefined at

x = -1

, so

x \neq -1

. We evaluate Case 1 (

x > -1

) and Case 2 (

x < -1

).

Multiplying an inequality by an algebraic expression requires knowing whether that expression is positive or negative to maintain or flip the inequality sign.

2

Solve Case 1 where

x + 1 > 0

(

x > -1

).

|2x - 5| \le x + 1 \implies -(x + 1) \le 2x - 5 \le x + 1

. Solving

-x - 1 \le 2x - 5

yields

x \ge \frac{4}{3}

. Solving

2x - 5 \le x + 1

yields

x \le 6

. Combining gives

\frac{4}{3} \le x \le 6

.

For positive denominators, multiplying both sides by

x + 1

preserves the inequality direction.

3

Solve Case 2 where

x + 1 < 0

(

x < -1

).

|2x - 5| \ge x + 1

. Since

|2x - 5| \ge 0

for all real

x

and

x + 1 < 0

, the left-hand side is non-negative while the right-hand side is strictly negative, which is always true for all

x < -1

.

Multiplying by a negative expression flips the inequality sign, and any non-negative quantity is strictly greater than any negative quantity.

4

Combine solutions from both cases.

The final solution set is

x < -1

or

\frac{4}{3} \le x \le 6

.

Taking the union of valid solution regions from both disjoint cases gives the complete solution.

Anahtar Kavram

Solving Rational Inequalities with Absolute Values by Case Analysis

Soru 272Soru →

If $a$ and $b$ are positive integers such that $2^{a+3} \cdot 5^b - 2^a \cdot 5^{b+1} = 6,000$ , what is the value of $a \cdot b$ ?

7

10

12

15

16

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

Factoring out

2^a \cdot 5^b

yields

2^a \cdot 5^b (2^3 - 5^1) = 3 \cdot 2^a \cdot 5^b = 6,000

. Dividing by 3 yields

2^a \cdot 5^b = 2,000

. Expressing

2,000

in prime factored form gives

2^4 \cdot 5^3

. Matching exponents for the prime bases gives

a = 4

and

b = 3

. The product

a \cdot b

is

4 \times 3 = 12

.

Adım Adım Çözüm

1

Rewrite the terms in the expression using exponent rules to isolate common bases.

2^{a+3} \cdot 5^b - 2^a \cdot 5^{b+1} = (2^a \cdot 2^3) \cdot 5^b - 2^a \cdot (5^b \cdot 5^1)

Applying

x^{m+n} = x^m \cdot x^n

allows us to extract common powers of

2^a

and

5^b

.

2

Factor out the common term

2^a \cdot 5^b

from the left side of the equation.

2^a \cdot 5^b (2^3 - 5^1) = 2^a \cdot 5^b (8 - 5) = 3 \cdot 2^a \cdot 5^b

Simplifying the constant factor in parentheses simplifies the equation.

3

Divide both sides of the equation by 3 and perform prime factorization on the resulting integer.

3 \cdot 2^a \cdot 5^b = 6,000 \implies 2^a \cdot 5^b = 2,000 = 2^4 \cdot 5^3

Prime factorization of

2,000

determines the unique integer exponents for bases 2 and 5.

4

Equate the corresponding exponents and calculate the requested product

a \cdot b

.

a = 4

and

b = 3 \implies a \cdot b = 4 \times 3 = 12

Since 2 and 5 are prime numbers, the prime factorization representation is unique.

Anahtar Kavram

Factoring Exponents and Prime Factorization

Tahmini Süre:2m 0s

Soru 273Soru →

An automotive testing center evaluated $350$ electric vehicle models for three advanced driver-assistance features: Lane Keeping Assist ( $L$ ), Automatic Emergency Braking ( $A$ ), and Blind Spot Detection ( $B$ ). The evaluation showed that $180$ models had feature $L$ , $150$ had feature $A$ , and $160$ had feature $B$ . Additionally, $65$ models had both $L$ and $A$ , $50$ had both $A$ and $B$ , $60$ had both $L$ and $B$ , and $25$ models had all three features. How many of the evaluated electric vehicle models had none of these three features?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The number of electric vehicle models that had none of the three features is 10.

Using the formula for three overlapping sets,

|L \cup A \cup B| = |L| + |A| + |B| - (|L \cap A| + |A \cap B| + |L \cap B|) + |L \cap A \cap B|

, we get

180 + 150 + 160 - (65 + 50 + 60) + 25 = 340

vehicles with at least one feature. Subtracting this from the total evaluated group of

350

yields

350 - 340 = 10

vehicles with none of the features.

Adım Adım Çözüm

1

Use the Principle of Inclusion-Exclusion for three overlapping sets to find the total number of models with at least one feature.

|L \cup A \cup B| = 180 + 150 + 160 - (65 + 50 + 60) + 25 = 340

models.

Adding individual set counts overcounts pairwise overlap regions twice and the central triple overlap three times. Subtracting pairwise intersections corrects for double counting, and adding back the triple intersection accounts for its over-subtraction.

2

Subtract the number of models having at least one feature from the total number of tested models.

\text{None} = 350 - 340 = 10

models.

The entire group consists of models with at least one feature plus models with none of the features.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle and Venn Diagram Region Partitioning

Soru 274Soru →

If $x = \sqrt[3]{5 + 2\sqrt{13}} + \sqrt[3]{5 - 2\sqrt{13}}$ , what is the value of $x^3 + 9x$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

By defining

x = u + v

with

u = \sqrt[3]{5 + 2\sqrt{13}}

and

v = \sqrt[3]{5 - 2\sqrt{13}}

, cubing both sides gives

x^3 = u^3 + v^3 + 3uv(u + v)

. Evaluating the components yields

u^3 + v^3 = 10

and

uv = \sqrt[3]{25 - 52} = -3

. Substituting these back gives

x^3 = 10 - 9(x)

, which rearranges to

x^3 + 9x = 10

.

Adım Adım Çözüm

1

Express

x

as a sum of two variables

u

and

v

x = u + v

, where

u = \sqrt[3]{5 + 2\sqrt{13}}

and

v = \sqrt[3]{5 - 2\sqrt{13}}

Grouping the binomial terms simplifies algebraic expansion using standard polynomial identities.

2

Calculate the sum of the cubes

u^3 + v^3

u^3 + v^3 = (5 + 2\sqrt{13}) + (5 - 2\sqrt{13}) = 10

Eliminating the cube roots allows for simple additive cancellation of the radical terms.

3

Calculate the product

uv

uv = \sqrt[3]{(5 + 2\sqrt{13})(5 - 2\sqrt{13})} = \sqrt[3]{5^2 - (2\sqrt{13})^2} = \sqrt[3]{25 - 52} = \sqrt[3]{-27} = -3

Applying the difference of squares property inside the cube root simplifies the product of conjugate radicals to a single integer.

4

Cube both sides of

x = u + v

and substitute evaluated terms

x^3 = u^3 + v^3 + 3uv(u + v) = 10 + 3(-3)x = 10 - 9x

Using

(u+v)^3 = u^3 + v^3 + 3uv(u+v)

connects

x^3

directly to

x

without expanding long radical terms.

5

Rearrange the equation to isolate

x^3 + 9x

x^3 + 9x = 10

Adding

9x

to both sides gives the exact numerical value of the requested expression.

Anahtar Kavram

Algebraic Identities with Polynomials and Radicals

Soru 275Soru →

Set $A$ consists of $k$ consecutive odd integers, and Set $B$ consists of $k$ consecutive even integers, where $k > 1$ . The smallest integer in Set $B$ is $3$ greater than the median of Set $A$ . If all integers in Set $A$ are positive, the sum of all integers in Set $A$ is $145$ , and the median of Set $B$ is $36$ , what is the smallest integer in Set $A$ ?

21

23

25

27

29

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The smallest integer in Set A is 25.

For an arithmetic sequence of

k

consecutive odd integers, the mean and median are equal to

\frac{\text{Sum}}{k} = \frac{145}{k}

. The smallest integer in Set

B

is

b_1 = \frac{145}{k} + 3

. Since Set

B

consists of

k

consecutive even integers, its median is

b_1 + k - 1

. Equating this to

36

yields

\frac{145}{k} + k + 2 = 36

, leading to

k^2 - 34k + 145 = 0

, whose roots are

k=5

and

k=29

. If

k=29

, the median of Set

A

is

5

, which implies negative integers exist in Set

A

. Since all integers in Set

A

are positive,

k=5

. With

k=5

, the median of Set

A

is

29

, and the smallest integer is

29 - 2(2) = 25

.

Adım Adım Çözüm

1

Express the median of Set A in terms of k.

M_A = \frac{145}{k}

For any evenly spaced set with an odd number of terms or symmetry, the arithmetic mean equals the median. The mean is the total sum divided by the number of terms

k

.

2

Express the median of Set B in terms of k using the given relationship for the smallest element of Set B.

M_B = \left(\frac{145}{k} + 3\right) + (k - 1) = \frac{145}{k} + k + 2

The smallest element in Set

B

is

b_1 = M_A + 3 = \frac{145}{k} + 3

. Since Set

B

contains

k

consecutive even integers (spacing

d=2

), its median is

b_1 + \frac{2(k-1)}{2} = b_1 + k - 1

.

3

Set the median of Set B to 36 and solve the quadratic equation for k.

k = 5

or

k = 29

Setting

\frac{145}{k} + k + 2 = 36

gives

\frac{145}{k} + k = 34

, which rearranges to

k^2 - 34k + 145 = 0

. Factoring gives

(k-5)(k-29) = 0

.

4

Determine the valid value of k and find the smallest integer in Set A.

Smallest integer in Set A is 25.

If

k = 29

,

M_A = \frac{145}{29} = 5

, and the smallest integer in Set

A

would be

5 - 2(14) = -23

, violating the condition that all integers in Set

A

are positive. Thus

k = 5

, making

M_A = 29

. The 5 consecutive odd integers are

25, 27, 29, 31, 33

, so the smallest integer is

25

.

Anahtar Kavram

Properties of consecutive integer sets, median-mean equivalence in arithmetic sequences, and term indexing.

Soru 276Soru →

If $x$ is a positive integer such that $\frac{6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6}{3^6 + 3^6 + 3^6} = 2^x$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

x

is 7.

Repeated addition converts to multiplication: six terms of

6^6

yield

6 \cdot 6^6 = 6^7

, and three terms of

3^6

yield

3 \cdot 3^6 = 3^7

. Dividing gives

\frac{6^7}{3^7} = \left(\frac{6}{3}\right)^7 = 2^7

. Comparing

2^7

to

2^x

gives

x = 7

.

Adım Adım Çözüm

1

Simplify the numerator by expressing repeated addition as multiplication.

6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 = 6 \times 6^6 = 6^7

Adding six identical terms of

6^6

is equivalent to multiplying

6^6

by 6. Using the power rule

a^1 \cdot a^n = a^{n+1}

, we get

6^7

.

2

Simplify the denominator by expressing repeated addition as multiplication.

3^6 + 3^6 + 3^6 = 3 \times 3^6 = 3^7

Adding three identical terms of

3^6

is equivalent to multiplying

3^6

by 3, yielding

3^7

.

3

Apply the quotient property of exponents for identical powers.

\frac{6^7}{3^7} = \left(\frac{6}{3}\right)^7 = 2^7

According to exponent laws,

\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n

for any non-zero real numbers

a

and

b

.

4

Equate exponents of equal bases to solve for

x

.

2^x = 2^7 \implies x = 7

Since the bases on both sides of the equation are equal to 2, the exponents must be equal.

Anahtar Kavram

Combining repeated addition into exponential products and dividing powers with equal exponents.

Soru 277Soru →

A project committee of $4$ members is to be selected from a pool of $n$ senior engineers and $6$ junior engineers, where $n \ge 4$ . The committee must contain at least one senior engineer and at least one junior engineer. If there are exactly $310$ different possible ways to select the committee, what is the value of $n$ ?

4

5

6

7

8

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

n

is 5.

Using complementary counting, the total number of ways to pick any 4 members from the

(n+6)

total engineers is

\binom{n+6}{4}

. The condition requires at least one senior engineer and at least one junior engineer. The invalid cases are when all 4 are junior engineers (which can occur in

\binom{6}{4} = 15

ways) or all 4 are senior engineers (which can occur in

\binom{n}{4}

ways). Setting up the equation:

\binom{n+6}{4} - \binom{6}{4} - \binom{n}{4} = 310

, which simplifies to

\binom{n+6}{4} - \binom{n}{4} = 325

. Testing

n = 5

gives

\binom{11}{4} - \binom{5}{4} = 330 - 5 = 325

, which matches the given condition.

Adım Adım Çözüm

1

Set up the combination formula for total unrestricted selections

Total selections from

(n + 6)

engineers choosing 4 is

\binom{n+6}{4}

.

Order of selection does not matter when forming a committee.

2

Identify the restricted (invalid) committee configurations

Committees with 0 senior engineers:

\binom{6}{4} = 15

. Committees with 0 junior engineers:

\binom{n}{4}

.

The committee must contain at least one member from each category.

3

Formulate the equation using complementary counting

\binom{n+6}{4} - \binom{n}{4} - 15 = 310 \implies \binom{n+6}{4} - \binom{n}{4} = 325

.

Subtracting invalid configurations from total configurations yields the valid configurations.

4

Test values for

n

For

n = 5

:

\binom{11}{4} - \binom{5}{4} = 330 - 5 = 325

.

Since

325 = 325

,

n = 5

satisfies the given constraint exactly.

Anahtar Kavram

Group selections with constraints using complementary counting:

\text{Valid} = \text{Total} - \text{Restricted}

.

Soru 278Soru →

A logistics company offers two freight pricing models for oversized cargo. Model X charges a fixed monthly account fee of $\$8$ plus $\$0.60$ per kilometer traveled. Model Y charges a fixed monthly account fee of $\$54$ plus $\$0.20$ per kilometer traveled. For how many kilometers in a month will the total monthly charge under Model X be exactly $20\%$ less than the total monthly charge under Model Y?

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

80 kilometers

The total monthly cost under Model X is

C_X = 8 + 0.60k

and under Model Y is

C_Y = 54 + 0.20k

. The condition that Model X is 20% less than Model Y means

C_X = 0.80 C_Y

. Substituting the expressions gives

8 + 0.60k = 0.80(54 + 0.20k) = 43.2 + 0.16k

. Subtracting

0.16k

and

8

from both sides yields

0.44k = 35.2

, which simplifies to

k = 80

.

Adım Adım Çözüm

1

Define variables and establish linear cost equations for both models.

Model X cost:

C_X = 8 + 0.60k

; Model Y cost:

C_Y = 54 + 0.20k

, where

k

is kilometers traveled.

Linear modeling translates flat fees and variable rates into algebraic expressions.

2

Formulate the linear equation based on the condition that Model X is 20% less than Model Y.

C_X = 0.80 C_Y \implies 8 + 0.60k = 0.80(54 + 0.20k)

.

Being 20% less than a base value means taking 80% (or 0.80) of that value.

3

Expand and simplify the algebraic equation.

8 + 0.60k = 43.2 + 0.16k \implies 0.44k = 35.2

.

Distributing 0.80 across

(54 + 0.20k)

yields

43.2 + 0.16k

, and subtracting

0.16k

and

8

isolates

k

on one side.

4

Calculate the value of

k

.

k = \frac{35.2}{0.44} = 80

.

Dividing

35.2

by

0.44

gives the exact number of kilometers required.

Anahtar Kavram

Linear Equations in One and Two Variables

Soru 279Soru →

If $x = \sqrt{7 - 4\sqrt{3}}$ and $y = \sqrt{7 + 4\sqrt{3}}$ , what is the value of $\frac{x^{-2} + y^{-2}}{(x+y)^{-1}}$ ?

\frac{1}{4}

\frac{7}{2}

14

56

64

Cevabı ve açıklamayı göster

Cevap: $56$

Cevap

56

The expression

x = \sqrt{7 - 4\sqrt{3}}

can be unnested by recognizing

7 - 4\sqrt{3}

as the perfect square

(2 - \sqrt{3})^2

, so

x = 2 - \sqrt{3}

and

y = 2 + \sqrt{3}

. This gives

x+y = 4

and

xy = 1

. The numerator

x^{-2} + y^{-2} = \frac{x^2+y^2}{(xy)^2} = \frac{(x+y)^2 - 2xy}{1} = 14

. The denominator

(x+y)^{-1} = \frac{1}{4}

. Dividing

14

by

\frac{1}{4}

yields

56

.

Adım Adım Çözüm

1

Simplify the radical expressions for

x

and

y

.

x = 2 - \sqrt{3}

and

y = 2 + \sqrt{3}

.

Note that

7 - 4\sqrt{3} = 4 - 4\sqrt{3} + 3 = (2 - \sqrt{3})^2

, so

\sqrt{7 - 4\sqrt{3}} = 2 - \sqrt{3}

. Similarly,

7 + 4\sqrt{3} = (2 + \sqrt{3})^2

.

2

Find the sum

x+y

and product

xy

.

x+y = 4

and

xy = 1

.

(2 - \sqrt{3}) + (2 + \sqrt{3}) = 4

, and

(2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1

.

3

Simplify the numerator

x^{-2} + y^{-2}

.

x^{-2} + y^{-2} = 14

.

x^{-2} + y^{-2} = \frac{1}{x^2} + \frac{1}{y^2} = \frac{x^2 + y^2}{(xy)^2} = \frac{(x+y)^2 - 2xy}{(xy)^2} = \frac{4^2 - 2(1)}{1^2} = 14

.

4

Evaluate the full expression

\frac{x^{-2} + y^{-2}}{(x+y)^{-1}}

.

\frac{14}{1/4} = 56

.

The denominator is

(x+y)^{-1} = (4)^{-1} = \frac{1}{4}

. Dividing

14

by

\frac{1}{4}

equals

14 \cdot 4 = 56

.

Anahtar Kavram

Nested Radical Simplification & Algebraic Exponent Identities

Alternatif Yöntem

Instead of unnesting the radicals first, observe that

x^2 = 7 - 4\sqrt{3}

and

y^2 = 7 + 4\sqrt{3}

. Then

x^2 y^2 = (7-4\sqrt{3})(7+4\sqrt{3}) = 49 - 48 = 1

, and

x^2 + y^2 = 14

. Thus

x^{-2} + y^{-2} = \frac{x^2+y^2}{x^2 y^2} = 14

. Next, find

(x+y)^2 = x^2 + y^2 + 2xy = 14 + 2(1) = 16

, so

x+y = 4

. Then

(x+y)^{-1} = \frac{1}{4}

, leading directly to

\frac{14}{1/4} = 56

.

Tahmini Süre:2m 0s

Soru 280Soru →

Let $M = 2^a \times 3^b \times 5^c$ and $N = 2^c \times 3^a \times 5^b$ , where $a$ , $b$ , and $c$ are distinct positive integers. If the greatest common divisor of $M$ and $N$ is $90$ and the least common multiple of $M$ and $N$ is $32,400$ , what is the value of $a + b + c$ ?

6

7

8

9

14

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

a + b + c

is 7.

The correct answer is 7. By applying the fundamental property

M \times N = \gcd(M, N) \times \text{lcm}(M, N)

, we get

2^{a+c} \times 3^{a+b} \times 5^{b+c} = 2^5 \times 3^6 \times 5^3

. Matching exponents gives

a + c = 5

,

a + b = 6

, and

b + c = 3

. Summing these three equations yields

2(a + b + c) = 14

, so

a + b + c = 7

.

Adım Adım Çözüm

1

Express the GCD and LCM of

M

and

N

in terms of their prime factorizations.

\gcd(M,N) = 90 = 2^1 \times 3^2 \times 5^1

and

\text{lcm}(M,N) = 32,400 = 2^4 \times 3^4 \times 5^2

.

Prime factorization allows us to relate the exponents of

M

and

N

directly to their GCD and LCM.

2

Use the identity

M \times N = \gcd(M,N) \times \text{lcm}(M,N)

to multiply the two numbers.

(

2^a \times 3^b \times 5^c) \times (2^c \times 3^a \times 5^b) = (2^1 \times 3^2 \times 5^1) \times (2^4 \times 3^4 \times 5^2)

, which simplifies to

2^{a+c} \times 3^{a+b} \times 5^{b+c} = 2^{1+4} \times 3^{2+4} \times 5^{1+2} = 2^5 \times 3^6 \times 5^3

.

The product of two positive integers is equal to the product of their greatest common divisor and least common multiple.

3

Equate the exponents for each prime base

2

,

3

, and

5

.

a + c = 5

,

a + b = 6

, and

b + c = 3

.

Since prime bases are unique, exponents of corresponding prime factors on both sides of the equation must be equal.

4

Sum the three equations and solve for

a + b + c

.

(a + c) + (a + b) + (b + c) = 5 + 6 + 3 \implies 2(a + b + c) = 14 \implies a + b + c = 7

.

Adding the three system equations yields twice the desired sum.

Anahtar Kavram

Prime Factorization, GCD and LCM Product Relationship

Tahmini Süre:2m 0s

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm