Tüm alıştırma soruları

Soru 1381Soru →

Printer X prints at a constant rate of $60$ pages per minute, and Printer Y prints at a constant rate of $40$ pages per minute. If both printers operate simultaneously at their respective constant rates, which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

Working together, the two printers print

100

pages in

1

minute.

Printer X prints

60\%

of the total number of pages produced when both printers operate for the same duration.

The ratio of the time taken by Printer Y to print a job to the time taken by Printer X to print the same job is

2:3

.

Working together, the two printers require

5

minutes to print

500

pages.

The combined printing rate of the two printers working together is

50

pages per minute.

Cevabı ve açıklamayı göster

Cevap: Working together, the two printers print $100$ pages in $1$ minute.; Printer X prints $60\%$ of the total number of pages produced when both printers operate for the same duration.; Working together, the two printers require $5$ minutes to print $500$ pages.

Cevap

The correct statements are that working together the printers print 100 pages in 1 minute, Printer X prints 60% of the total pages, and working together they require 5 minutes to print 500 pages.

The combined rate is

60 + 40 = 100

pages per minute. In

1

minute,

100

pages are printed. In

5

minutes,

5 \times 100 = 500

pages are printed. Out of every

100

pages, Printer X produces

60

, which corresponds to

60\%

.

Adım Adım Çözüm

1

Calculate the combined rate of both printers

Combined rate =

60 + 40 = 100

pages per minute

When two entities work together simultaneously, their work rates add directly.

2

Evaluate the proportion of total output printed by Printer X

Proportion =

\frac{60}{60 + 40} = \frac{60}{100} = 60\%

The fraction of work done by one printer is its rate divided by the total combined rate.

3

Determine the time ratio between Printer Y and Printer X for a fixed job

Time ratio Y : X =

\frac{1/40}{1/60} = \frac{60}{40} = 3:2

Time required for a fixed task is inversely proportional to the work rate.

4

Calculate time needed for 500 pages working together

Time =

\frac{500\text{ pages}}{100\text{ pages/min}} = 5\text{ minutes}

Time equals total work divided by combined rate.

Anahtar Kavram

Additive work rates and inverse relationship between rate and time in ratio problems

Tahmini Süre:1m 0s

Soru 1382Soru →

Fill in the blank in the sentence below with the word that best completes the passage based on the structural contrast clues.

Aşağıdaki boşlukları doldurun

Paradoxically, despite the author's longstanding reputation for writing prose that alienated lay readers, her final manuscript was widely celebrated for its remarkable lucidity and broad accessibility.

Cevabı ve açıklamayı göster

Cevap

abstruse (or semantically equivalent contrast terms such as recondite, esoteric, arcane, or opaque)

The sentence relies on dual contrast signals ('Paradoxically' and 'despite') to establish a reversal between the author's previous writing style and her final manuscript. Since the final manuscript is praised for its 'lucidity and broad accessibility,' the correct word for the blank must describe prose that is obscure or difficult to comprehend. The word 'abstruse' (or synonyms like 'recondite' or 'esoteric') fits both the contrast requirement and the contextual clue that her earlier work 'alienated lay readers.'

Adım Adım Çözüm

1

Identify structural transition signals in the sentence

The words 'Paradoxically' and 'despite' act as reversal and contrast pivots signaling an opposing relationship between the target blank and the description in the second half of the sentence.

Contrast pivots dictate that the missing word must carry a meaning opposite to the subsequent outcome described.

2

Analyze contextual clues and target valence

The manuscript was celebrated for 'lucidity and broad accessibility' and previously 'alienated lay readers.'

The blank describes prose that is difficult for general readers to comprehend.

3

Determine the required meaning for the blank

The blank requires a term meaning obscure, dense, or hard to understand, such as 'abstruse'.

'Abstruse' directly contrasts with 'lucidity' and logically accounts for alienating lay readers.

Anahtar Kavram

Identifying structural reversal signals (such as 'paradoxically' and 'despite') to determine semantic opposition in Text Completion questions.

Soru 1383Soru →

Passage:

In late eighteenth-century France, agronomic advocate Antoine-Augustin Parmentier sought to elevate the potato from a despised livestock feed to a staple crop for human consumption. Following his imprisonment in Prussia during the Seven Years' War—where he observed prisoners surviving solely on potato tubers—Parmentier realized the crop's caloric efficiency and resilience against famine. Prior to his campaign, French peasant communities widely believed that consuming potatoes transmitted leprosy, a superstition reinforced by municipal edicts prohibiting their cultivation in several northern provinces. To dismantle this institutional resistance, Parmentier employed unconventional publicity tactics rather than relying exclusively on academic treatises. In 1787, he planted a high-yield potato plot on royal land at Sablons, surrounding the perimeter with heavily armed guards during the day to convey an aura of immense value. Crucially, Parmentier instructed the guards to accept bribes and abandon their posts at night, allowing local farmers to intentionally 'steal' the tubers and cultivate them in private plots. This psychological strategy successfully transformed public perception, accelerating widespread domestic cultivation prior to the harvest failures of 1789.

According to the passage, French peasant resistance to potato cultivation prior to Parmentier's campaign was partly attributable to which of the following?

Official opposition from royal authorities who refused to grant Parmentier land for agricultural experiments

A complete lack of interest among agronomic scholars in promoting alternative staple crops

A widespread superstitious belief that consuming the tubers transmitted leprosy

The high financial cost of importing Prussian tubers into French peasant communities

Municipal edicts that encouraged peasants to use potato crops exclusively for livestock feed

Cevabı ve açıklamayı göster

Cevap: A widespread superstitious belief that consuming the tubers transmitted leprosy

Cevap

A widespread superstitious belief that consuming the tubers transmitted leprosy

The passage explicitly states that prior to Parmentier's campaign, 'French peasant communities widely believed that consuming potatoes transmitted leprosy.' The correct choice accurately retrieves and paraphrases this explicitly stated fact.

Adım Adım Çözüm

1

Analyze the prompt to determine the target detail

The prompt asks for a reason cited in the passage for French peasant resistance to potato cultivation prior to Parmentier's campaign.

Explicit detail retrieval requires matching the question's target criteria directly to the text.

2

Scan the passage for keywords related to peasant belief and resistance prior to the campaign

Located the sentence: 'Prior to his campaign, French peasant communities widely believed that consuming potatoes transmitted leprosy, a superstition reinforced by municipal edicts prohibiting their cultivation in several northern provinces.'

This section specifically details the cause of peasant resistance.

3

Evaluate the options against the explicit text statement

The option identifying a widespread superstitious belief that consuming tubers transmitted leprosy directly matches the statement in the text.

The correct answer must accurately paraphrase facts explicitly stated in the passage without adding outside assumptions.

Anahtar Kavram

Explicit Detail Retrieval

Tahmini Süre:1m 30s

Soru 1384Soru →

A circle has a radius of $6$ . A central angle of $60^\circ$ intercepts an arc on the circle. What is the perimeter of the sector defined by this central angle?

2\pi

6\pi

2\pi + 6

2\pi + 12

12\pi

Cevabı ve açıklamayı göster

Cevap: $2\pi + 12$

Cevap

2\pi + 12

The sector's perimeter includes the curved arc length and the two straight radii bounding it. The central angle of

60^\circ

represents

\frac{60}{360} = \frac{1}{6}

of the full circle. The arc length is

\frac{1}{6} \times 2\pi(6) = 2\pi

. Adding the two radii of length

6

gives

2\pi + 6 + 6 = 2\pi + 12

.

Adım Adım Çözüm

1

Calculate the arc length of the sector

Arc length =

2\pi

The arc length formula is

\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r

. Substituting

\theta = 60^\circ

and

r = 6

yields

\frac{60}{360} \times 12\pi = \frac{1}{6} \times 12\pi = 2\pi

.

2

Calculate the perimeter of the sector by adding the arc length to the two radii

Perimeter =

2\pi + 12

The perimeter of a sector consists of the arc length plus two radii (

2r

). Thus,

\text{Perimeter} = 2\pi + 2(6) = 2\pi + 12

.

Anahtar Kavram

Perimeter of a sector equals arc length plus twice the radius (

L + 2r

).

Tahmini Süre:45s

Soru 1385Soru →

If $x > 0$ and $x^{\frac{1}{2}} + x^{-\frac{1}{2}} = 3$ , what is the value of $x^2 + x^{-2}$ ?

7

47

49

51

81

Cevabı ve açıklamayı göster

Cevap: 47

Cevap

47

Squaring both sides of

x^{\frac{1}{2}} + x^{-\frac{1}{2}} = 3

gives

x + 2 + x^{-1} = 9

, which simplifies to

x + x^{-1} = 7

. Squaring both sides of

x + x^{-1} = 7

gives

x^2 + 2 + x^{-2} = 49

, which yields

x^2 + x^{-2} = 47

.

Adım Adım Çözüm

1

Square both sides of the given equation

x^{\frac{1}{2}} + x^{-\frac{1}{2}} = 3

.

(x^{\frac{1}{2}} + x^{-\frac{1}{2}})^2 = 3^2 \implies x + 2(x^{\frac{1}{2}})(x^{-\frac{1}{2}}) + x^{-1} = 9

Applying the binomial expansion identity

(a+b)^2 = a^2 + 2ab + b^2

.

2

Simplify the middle term and solve for

x + x^{-1}

.

x + 2(1) + x^{-1} = 9 \implies x + x^{-1} = 7

Since

x^{\frac{1}{2}} \cdot x^{-\frac{1}{2}} = x^0 = 1

, subtracting 2 from both sides isolates

x + x^{-1}

.

3

Square both sides of

x + x^{-1} = 7

.

(x + x^{-1})^2 = 7^2 \implies x^2 + 2(x)(x^{-1}) + x^{-2} = 49

Squaring

x + x^{-1}

generates the terms

x^2

and

x^{-2}

.

4

Simplify the middle term and solve for

x^2 + x^{-2}

.

x^2 + 2 + x^{-2} = 49 \implies x^2 + x^{-2} = 47

Subtracting 2 from both sides isolates the desired expression

x^2 + x^{-2}

.

Anahtar Kavram

Algebraic Exponents and Binomial Expansion

Soru 1386Soru →

A dataset consists of 15 numerical values with a mean of 50 and a standard deviation of 8. If a constant value of 5 is added to every number in the dataset, what is the standard deviation of the resulting dataset?

8

13

40

55

64

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The standard deviation of the resulting dataset is 8.

Adding a constant to every data point in a set shifts the central location (the mean and median) by that constant, but leaves the distances between points unchanged. Because standard deviation depends solely on the distances between each point and the mean, the standard deviation remains 8.

Adım Adım Çözüm

1

Recall the effect of adding a constant to data points on measures of dispersion.

Measures of dispersion (range, IQR, standard deviation) describe the spread or scatter of data relative to its center.

When a constant

k

is added to every value

x_i

, the new mean becomes

\text{mean} + k

.

2

Evaluate the distance of each transformed data point from the new mean.

(x_i + k) - (\text{mean} + k) = x_i - \text{mean}

.

The deviation of each point from the mean remains identical to its original deviation.

3

Determine the standard deviation of the new dataset.

The standard deviation is unchanged and remains 8.

Because all individual deviations from the mean are preserved, the average distance from the mean does not change.

Anahtar Kavram

Invariance of standard deviation under addition of a constant

Soru 1387Soru →

If $x$ is an integer that satisfies both $|2x - 5| \le 9$ and $3 - x < 5$ , how many possible values of $x$ exist?

4

8

9

10

0

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

There are 9 possible integer values for x.

Solving

|2x - 5| \le 9

gives

-2 \le x \le 7

. Solving

3 - x < 5

gives

x > -2

. Taking the intersection yields

-2 < x \le 7

. The integers in this interval are

-1, 0, 1, 2, 3, 4, 5, 6, 7

, amounting to 9 values in total.

Adım Adım Çözüm

1

Solve the absolute value inequality

|2x - 5| \le 9

-2 \le x \le 7

Unfold

|2x - 5| \le 9

as

-9 \le 2x - 5 \le 9

, add 5 to obtain

-4 \le 2x \le 14

, and divide by 2.

2

Solve the linear inequality

3 - x < 5

x > -2

Subtract 3 to get

-x < 2

, then divide by

-1

and reverse the inequality symbol.

3

Determine the intersection of both inequalities

-2 < x \le 7

Combine

-2 \le x \le 7

and

x > -2

on the real number line.

4

Count the integer values within the intersection

-2 < x \le 7

9 integer values

The valid integers are

-1, 0, 1, 2, 3, 4, 5, 6, 7

, giving a total count of

7 - (-1) + 1 = 9

.

Anahtar Kavram

Linear Inequalities and Absolute Value Bounds

Tahmini Süre:1m 30s

Soru 1388Soru →

If $x$ is a real number that satisfies the equation $\sqrt{x + 6\sqrt{x - 9}} + \sqrt{x - 6\sqrt{x - 9}} = 10$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 34

Cevap

34

Using the substitution

u = \sqrt{x - 9} \ge 0

, we have

x = u^2 + 9

. The expressions under the square roots become

x + 6\sqrt{x - 9} = u^2 + 6u + 9 = (u + 3)^2

and

x - 6\sqrt{x - 9} = u^2 - 6u + 9 = (u - 3)^2

. Taking square roots gives

\sqrt{(u + 3)^2} + \sqrt{(u - 3)^2} = (u + 3) + |u - 3| = 10

. For

u \ge 3

, this simplifies to

(u + 3) + (u - 3) = 10 \implies 2u = 10 \implies u = 5

. Finally, substituting

u = 5

back yields

x = 5^2 + 9 = 34

.

Adım Adım Çözüm

1

Define a variable substitution to simplify the nested radical structure.

Let

u = \sqrt{x - 9}

where

u \ge 0

. Squaring both sides gives

u^2 = x - 9

, so

x = u^2 + 9

.

This substitution allows the expressions inside the outer square roots to be rewritten as polynomials in terms of

u

.

2

Rewrite the expressions under each square root as perfect square trinomials.

x + 6\sqrt{x - 9} = (u^2 + 9) + 6u = (u + 3)^2

and

x - 6\sqrt{x - 9} = (u^2 + 9) - 6u = (u - 3)^2

.

Expressing terms as perfect squares allows the outer radicals to be simplified.

3

Simplify the square root expressions using absolute values.

\sqrt{(u + 3)^2} + \sqrt{(u - 3)^2} = (u + 3) + |u - 3| = 10

.

For any real number

a

,

\sqrt{a^2} = |a|

. Since

u \ge 0

,

u + 3 > 0

, so

|u + 3| = u + 3

.

4

Solve the absolute value equation across valid domain intervals.

If

u \ge 3

,

|u - 3| = u - 3

, giving

(u + 3) + (u - 3) = 10 \implies 2u = 10 \implies u = 5

. If

0 \le u < 3

,

|u - 3| = 3 - u

, giving

(u + 3) + (3 - u) = 6 \neq 10

(no solution). Thus,

u = 5

.

Splitting into cases based on the definition of absolute value isolates the valid root.

5

Substitute

u = 5

back into the expression for

x

.

x = 5^2 + 9 = 25 + 9 = 34

.

Converting from

u

back to

x

provides the solution to the original equation.

Anahtar Kavram

Simplifying nested radicals by completing the square under the radical sign and applying the identity

\sqrt{a^2} = |a|

.

Soru 1389Soru →

A company allocated a total budget of $B$ dollars for a project. In the first phase of the project, $\frac{2}{5}$ of the total budget plus $\$3,000$ was spent. In the second phase, $\frac{1}{3}$ of the remaining budget after the first phase was spent. If the unspent amount after both phases is $\$14,000$ , what was the total initial budget $B$ ?

\$27,200

\$30,000

\$40,000

\$42,500

\$75,000

Cevabı ve açıklamayı göster

Cevap: $\$40,000$

Cevap

\$40,000

The correct answer of

\$40,000

is obtained by systematically tracking the remaining budget after each phase. After Phase 1, the remaining amount is

B - (\frac{2}{5}B + 3,000) = \frac{3}{5}B - 3,000

. Spending

\frac{1}{3}

of this balance in Phase 2 leaves

\frac{2}{3}

of it unspent:

\frac{2}{3}(\frac{3}{5}B - 3,000) = \frac{2}{5}B - 2,000

. Setting this equal to the final unspent amount of

\$14,000

gives

\frac{2}{5}B = 16,000

, which solves to

B = 40,000

.

Adım Adım Çözüm

1

Express the remaining budget after the first phase in terms of

B

.

Amount spent in Phase 1 =

\frac{2}{5}B + 3,000

. Remaining after Phase 1 =

B - \left(\frac{2}{5}B + 3,000\right) = \frac{3}{5}B - 3,000

.

Subtracting the first phase expenses from the initial total budget

B

determines the balance available for the second phase.

2

Express the unspent budget after the second phase.

Since

\frac{1}{3}

of the remaining budget was spent in Phase 2,

1 - \frac{1}{3} = \frac{2}{3}

of that remaining budget is left. Remaining after Phase 2 =

\frac{2}{3}\left(\frac{3}{5}B - 3,000\right)

.

Taking

\frac{2}{3}

of the Phase 1 remainder directly gives the final unspent amount.

3

Expand and simplify the algebraic equation setting the unspent amount equal to

\$14,000

.

\frac{2}{3}\left(\frac{3}{5}B - 3,000\right) = 14,000 \implies \frac{2}{5}B - 2,000 = 14,000.

Distributing

\frac{2}{3}

across both terms inside the parentheses clears the fraction product.

4

Solve the linear equation for

B

.

\frac{2}{5}B = 16,000 \implies B = 16,000 \times \frac{5}{2} = 40,000.

Adding

2,000

to both sides and multiplying by the reciprocal

\frac{5}{2}

yields the total budget

B

.

Anahtar Kavram

Formulating and solving multi-step linear equations in one variable with fractional quantities and consecutive remaining balances.

Tahmini Süre:2m 0s

Soru 1390Soru →

If $x$ and $y$ are positive integers satisfying $3^x - 3^y = 702$ and $\sqrt{x + y} = 3$ , what is the value of $x^2 - y^2$ ?

9

18

27

45

81

Cevabı ve açıklamayı göster

Cevap: $27$

Cevap

The correct value of

x^2 - y^2

is 27.

Squaring

\sqrt{x + y} = 3

gives

x + y = 9

. Factoring

3^x - 3^y = 702

gives

3^y(3^{x-y} - 1) = 702

. Since

702 = 3^3 \times 26

and

(3^{x-y} - 1)

is coprime to 3, we deduce

3^y = 3^3 \implies y = 3

, and

3^{x-3} - 1 = 26 \implies 3^{x-3} = 27 \implies x = 6

. Substituting

x = 6

and

y = 3

into

x^2 - y^2

gives

36 - 9 = 27

.

Adım Adım Çözüm

1

Eliminate the radical from the given linear equation.

Squaring both sides of

\sqrt{x + y} = 3

gives

x + y = 9

.

Squaring both sides removes the square root operator to establish a linear relationship between

x

and

y

.

2

Factor out the common exponential term

3^y

from

3^x - 3^y = 702

.

3^y(3^{x-y} - 1) = 702

.

Since

x

and

y

are positive integers and

702 > 0

, it must be true that

x > y

, allowing factoring by exponent rules

3^x = 3^y \cdot 3^{x-y}

.

3

Find the prime factorization of 702 and match the power of 3.

702 = 27 \times 26 = 3^3 \times 26

, so

3^y(3^{x-y} - 1) = 3^3 \times 26

.

The factor

(3^{x-y} - 1)

is not divisible by 3 because

3^{x-y}

is a multiple of 3 for

x > y

. Therefore, all powers of 3 in 702 must belong to

3^y

.

4

Solve for the values of

y

and

x

.

y = 3

and

3^{x-3} - 1 = 26 \implies 3^{x-3} = 27 = 3^3 \implies x - 3 = 3 \implies x = 6

.

Equating prime component bases yields

y = 3

and

x = 6

, which satisfies

x + y = 6 + 3 = 9

.

5

Calculate the target expression

x^2 - y^2

.

x^2 - y^2 = 6^2 - 3^2 = 36 - 9 = 27

.

Substituting

x = 6

and

y = 3

into

x^2 - y^2

yields 27 (or using

(x+y)(x-y) = 9 \times 3 = 27

).

Anahtar Kavram

Factoring exponential expressions using prime factorization, radical simplification, and difference of squares.

Soru 1391Soru →

A dataset of $20$ numerical measurements has a range of $30$ and a standard deviation of $6.4$ . If every measurement in the dataset is multiplied by $-2$ and then increased by $5$ to construct a new dataset, what are the range and standard deviation of the new dataset, respectively?

Range is

65

and standard deviation is

17.8

Range is

-60

and standard deviation is

-12.8

Range is

60

and standard deviation is

12.8

Range is

60

and standard deviation is

25.6

Range is

30

and standard deviation is

6.4

Cevabı ve açıklamayı göster

Cevap: Range is $60$ and standard deviation is $12.8$

Cevap

The range of the new dataset is

60

and the standard deviation is

12.8

.

Under a linear transformation

Y = aX + b

, all measures of dispersion (range, interquartile range, standard deviation) are multiplied by

|a|

and are unaffected by the additive constant

b

. Here,

a = -2

and

b = 5

. The absolute multiplier is

|-2| = 2

. Therefore, the new range is

2 \times 30 = 60

and the new standard deviation is

2 \times 6.4 = 12.8

.

Adım Adım Çözüm

1

Analyze the general transformation formula for measures of dispersion under linear transformations of the form

Y = aX + b

.

For any constant multiplier

a

and constant shift

b

, the new range is

|a| \times \text{Range}_X

and the new standard deviation is

|a| \times \sigma_X

.

Measures of dispersion reflect spread; multiplying each data value by

a

scales the spread by

|a|

, while adding a constant shift

b

translates all values equally without expanding or contracting their relative distance.

2

Calculate the new range using

a = -2

and

b = 5

.

\text{New Range} = |-2| \times 30 = 2 \times 30 = 60.

The range scales by the absolute value of the multiplier

|-2| = 2

, while the additive constant

5

has no impact on the spread.

3

Calculate the new standard deviation using

a = -2

and

b = 5

.

\text{New Standard Deviation} = |-2| \times 6.4 = 2 \times 6.4 = 12.8.

The standard deviation scales by

|-2| = 2

, and the shift of

+5

does not change the dispersion around the mean.

Anahtar Kavram

Effect of Linear Transformations on Measures of Dispersion

Tahmini Süre:1m 30s

Soru 1392Soru →

How many integer values of $x$ satisfy the compound absolute value inequality $1 \le ||x - 4| - 3| \le 5$ ?

10

12

15

17

18

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

There are 15 integer values of

x

that satisfy the given compound inequality.

Solving

1 \le ||x - 4| - 3| \le 5

requires breaking the nested absolute value into its boundary constraints. The upper bound

\|x - 4| - 3| \le 5

restricts

x

to

[-4, 12]

. The lower bound

||x - 4| - 3| \ge 1

requires either

|x - 4| \ge 4

(giving

x \le 0

or

x \ge 8

) or

|x - 4| \le 2

(giving

2 \le x \le 6

). Taking the intersection produces three distinct inclusive integer intervals:

[-4, 0]

,

[2, 6]

, and

[8, 12]

. Each interval contains 5 integers, yielding a total of 15 integer solutions.

Adım Adım Çözüm

1

Decompose the double inequality into two separate absolute value inequalities:

||x - 4| - 3| \le 5

and

||x - 4| - 3| \ge 1

.

Two simultaneous inequalities to solve for

x

.

A double inequality

a \le |u| \le b

requires satisfying both

|u| \le b

and

|u| \ge a

.

2

Solve the upper bound inequality

||x - 4| - 3| \le 5

.

-5 \le |x - 4| - 3 \le 5 \implies -2 \le |x - 4| \le 8

. Since

|x - 4| \ge 0 \ge -2

is always true, this simplifies to

|x - 4| \le 8 \implies -4 \le x \le 12

.

Absolute value is non-negative, so the lower bound of

-2

imposes no extra constraint.

3

Solve the lower bound inequality

||x - 4| - 3| \ge 1

.

This splits into two cases:

|x - 4| - 3 \ge 1

OR

|x - 4| - 3 \le -1

.
Case A:

|x - 4| \ge 4 \implies x - 4 \ge 4

or

x - 4 \le -4 \implies x \ge 8

or

x \le 0

.
Case B:

|x - 4| \le 2 \implies -2 \le x - 4 \le 2 \implies 2 \le x \le 6

.

The absolute value inequality

|u| \ge 1

holds when

u \ge 1

or

u \le -1

.

4

Intersect the solution set from the upper bound

[-4, 12]

with the solution set from the lower bound

(-\infty, 0] \cup [2, 6] \cup [8, \infty)

.

The valid solution set is

x \in [-4, 0] \cup [2, 6] \cup [8, 12]

.

Both conditions must hold simultaneously.

5

Count the integer values in each of the three valid intervals.

Interval

[-4, 0]

has 5 integers:

\{-4, -3, -2, -1, 0\}

.
Interval

[2, 6]

has 5 integers:

\{2, 3, 4, 5, 6\}

.
Interval

[8, 12]

has 5 integers:

\{8, 9, 10, 11, 12\}

.
Total integer solutions =

5 + 5 + 5 = 15

.

The number of integers in an inclusive integer range

[a, b]

is

b - a + 1

.

Anahtar Kavram

Linear Inequalities and Absolute Value

Soru 1393Soru →

A company's annual budget of $\$84,000$ is split among three departments: Research, Marketing, and Operations. The Marketing department receives $\frac{2}{3}$ as much funding as the Research department. The Operations department receives $\$6,000$ more than half of the combined funding of the Research and Marketing departments. What is the amount, in dollars, allocated to the Research department?

24,000

28,800

31,200

33,600

36,000

Cevabı ve açıklamayı göster

Cevap: $31,200$

Cevap

31,200

Defining Research funding as

x

, Marketing funding becomes

\frac{2}{3}x

, and Operations funding becomes

\frac{1}{2}(x + \frac{2}{3}x) + 6000 = \frac{5}{6}x + 6000

. Summing all three department allocations yields

x + \frac{2}{3}x + \frac{5}{6}x + 6000 = 84,000

. Combining the variable terms gives

\frac{5}{2}x + 6000 = 84,000

, which simplifies to

\frac{5}{2}x = 78,000

and yields

x = 31,200

.

Adım Adım Çözüm

1

Define the unknown variable for the target quantity.

Let

x

represent the dollar amount allocated to the Research department.

The problem asks specifically for the Research department allocation.

2

Express the allocations of Marketing and Operations in terms of

x

.

Marketing

= \frac{2}{3}x

. Combined Research and Marketing

= x + \frac{2}{3}x = \frac{5}{3}x

. Operations

= \frac{1}{2}\left(\frac{5}{3}x\right) + 6,000 = \frac{5}{6}x + 6,000

.

Translating word problem relationships into algebraic expressions.

3

Set up the single-variable linear equation for the total budget.

x + \frac{2}{3}x + \left(\frac{5}{6}x + 6,000\right) = 84,000

The sum of allocations across all three departments must equal the total budget of

\$84,000

.

4

Combine like terms using a common denominator.

\frac{6}{6}x + \frac{4}{6}x + \frac{5}{6}x + 6,000 = 84,000 \implies \frac{15}{6}x + 6,000 = 84,000 \implies \frac{5}{2}x + 6,000 = 84,000

Simplifying fractional coefficients by finding the common denominator 6.

5

Isolate

x

to solve the linear equation.

\frac{5}{2}x = 78,000 \implies 5x = 156,000 \implies x = 31,200

Subtracting

6,000

from both sides and multiplying by

\frac{2}{5}

.

Anahtar Kavram

Linear Equations in One Variable

Soru 1394Soru →

If $x$ is a real number such that $|4 - 3x| \leq 13$ , and $y$ is an integer such that $-5 < \frac{1 - 2y}{3} \leq 3$ , what is the least possible integer value of $x^2 - y$ ?

-8

-7

-4

2

9

Cevabı ve açıklamayı göster

Cevap: $-7$

Cevap

The least possible integer value of

x^2 - y

is

-7

.

To find the minimum value of

x^2 - y

, we must minimize

x^2

and maximize

y

. The absolute value inequality

|4 - 3x| \leq 13

simplifies to

-3 \leq x \leq \frac{17}{3}

, which contains

0

, so the minimum of

x^2

is

0

. The inequality

-5 < \frac{1 - 2y}{3} \leq 3

simplifies to

-4 \leq y < 8

. Since

y

is an integer, its maximum value is

7

. Therefore, the minimum possible value of

x^2 - y

is

0 - 7 = -7

.

Adım Adım Çözüm

1

Solve the absolute value inequality

|4 - 3x| \leq 13

for

x

.

-13 \leq 4 - 3x \leq 13 \implies -17 \leq -3x \leq 9 \implies \frac{17}{3} \geq x \geq -3 \implies -3 \leq x \leq \frac{17}{3}

.

An absolute value inequality

|u| \leq k

expands to

-k \leq u \leq k

. Dividing by a negative number flips the inequality signs.

2

Determine the minimum possible value of

x^2

.

Minimum

x^2 = 0

.

Since

x

can take any real value in the interval

[-3, \frac{17}{3}]

, which contains

0

, the minimum square of any real number in this interval is

0

at

x = 0

.

3

Solve the double inequality

-5 < \frac{1 - 2y}{3} \leq 3

for

y

.

-15 < 1 - 2y \leq 9 \implies -16 < -2y \leq 8 \implies 8 > y \geq -4 \implies -4 \leq y < 8

.

Multiplying by

3

preserves inequalities, subtracting

1

preserves inequalities, and dividing by

-2

reverses all inequality signs.

4

Find the maximum integer value of

y

.

Maximum integer

y = 7

.

The solution set for

y

is the half-open interval

[-4, 8)

. Since

y

is constrained to be an integer, the largest integer strictly less than

8

is

7

.

5

Minimize the expression

x^2 - y

.

\text{Minimum } (x^2 - y) = \text{Minimum } (x^2) - \text{Maximum } (y) = 0 - 7 = -7

.

To minimize a difference

A - B

, one must minimize the minuend

A

and maximize the subtrahend

B

.

Anahtar Kavram

Linear Inequalities and Absolute Value

Tahmini Süre:2m 0s

Soru 1395Soru →

A circle has a radius of $10$ units. A sector within this circle is defined by a central angle of $72^\circ$ . Which of the following statements about this sector are true? Select all such statements.

Geçerli olan tümünü seçin

The arc length of the sector is

4\pi

units.

The area of the sector is

20\pi

square units.

The area of the sector is

100\pi

square units.

The arc length of the sector is

20\pi

units.

The ratio of the sector's area to the circle's total area is

1

to

5

.

Cevabı ve açıklamayı göster

Cevap: The arc length of the sector is $4\pi$ units.; The area of the sector is $20\pi$ square units.; The ratio of the sector's area to the circle's total area is $1$ to $5$ .

Cevap

The statements confirming an arc length of

4\pi

units, a sector area of

20\pi

square units, and a sector-to-total area ratio of

1

to

5

are correct.

The central angle fraction is

\frac{72^\circ}{360^\circ} = \frac{1}{5}

. Multiplying the full circumference

20\pi

by

\frac{1}{5}

gives an arc length of

4\pi

. Multiplying the total area

100\pi

by

\frac{1}{5}

gives a sector area of

20\pi

. The ratio of sector area to total area is also equal to

\frac{1}{5}

.

Adım Adım Çözüm

1

Find the central angle fraction of the circle

The fraction is

\frac{72^\circ}{360^\circ} = \frac{1}{5}

Arc length and sector area are proportional to the central angle relative to a full

360^\circ

turn.

2

Calculate the arc length of the sector

Arc Length

= \frac{1}{5} \times 2\pi(10) = 4\pi

units

The arc length is the circle's circumference multiplied by the central angle fraction.

3

Calculate the area of the sector

Sector Area

= \frac{1}{5} \times \pi(10)^2 = 20\pi

square units

The sector area is the circle's total area multiplied by the central angle fraction.

4

Determine the area ratio

Ratio

= \frac{\text{Sector Area}}{\text{Total Area}} = \frac{20\pi}{100\pi} = \frac{1}{5}

The ratio of the sector area to the total area is identical to the central angle fraction.

Anahtar Kavram

Arc Length and Sector Area Formulas

Tahmini Süre:1m 0s

Soru 1396Soru →

For all real numbers $x > 0$ , which of the following expressions are equivalent to $\left(\frac{x^{3/2} \cdot \sqrt[3]{x}}{x^{1/6}}\right)^2$ ? Select all such expressions.

Geçerli olan tümünü seçin

\sqrt[3]{x^{10}}

x^3 \sqrt[3]{x}

x^3 \sqrt{x}

x^5

\left(\sqrt[3]{x^5}\right)^2

Cevabı ve açıklamayı göster

Cevap: $\sqrt[3]{x^{10}}$ ; $x^3 \sqrt[3]{x}$ ; $\left(\sqrt[3]{x^5}\right)^2$

Cevap

The expressions equivalent to the given expression are

\sqrt[3]{x^{10}}

,

x^3 \sqrt[3]{x}

, and

\left(\sqrt[3]{x^5}\right)^2

.

Simplifying the given expression by converting all radical forms to fractional exponents yields

x^{10/3}

. The three expressions

\sqrt[3]{x^{10}}

,

x^3 \sqrt[3]{x}

, and

\left(\sqrt[3]{x^5}\right)^2

each rewrite to

x^{10/3}

when evaluated using standard power laws.

Adım Adım Çözüm

1

Convert radical expressions to fractional exponent form inside the parentheses.

\sqrt[3]{x} = x^{1/3}

, so the numerator becomes

x^{3/2} \cdot x^{1/3}

.

Converting all terms to exponent notation allows applying standard exponent addition and subtraction rules.

2

Simplify the numerator by adding exponents.

x^{3/2 + 1/3} = x^{9/6 + 2/6} = x^{11/6}

.

When multiplying exponential terms with the same base, add their exponents using a common denominator.

3

Divide by the denominator by subtracting exponents.

\frac{x^{11/6}}{x^{1/6}} = x^{11/6 - 1/6} = x^{10/6} = x^{5/3}

.

When dividing exponential terms with the same base, subtract the denominator's exponent from the numerator's exponent.

4

Apply the outer exponent of 2.

\left(x^{5/3}\right)^2 = x^{(5/3) \cdot 2} = x^{10/3}

.

When raising a power to another power, multiply the inner and outer exponents.

5

Compare

x^{10/3}

to each option.

\sqrt[3]{x^{10}} = x^{10/3}

,

x^3 \sqrt[3]{x} = x^{3 + 1/3} = x^{10/3}

, and

\left(\sqrt[3]{x^5}\right)^2 = (x^{5/3})^2 = x^{10/3}

are all equivalent.

Matching each candidate expression in fractional exponent form identifies all equivalent choices.

Anahtar Kavram

Simplification of algebraic expressions using rules of fractional exponents and radicals

Tahmini Süre:1m 30s

Soru 1397Soru →

Three straight lines, $R$ , $S$ , and $T$ , all intersect at a single point $P$ . Line $R$ is perpendicular to line $S$ . Line $T$ intersects line $R$ such that one of the acute angles formed between line $R$ and line $T$ measures $35^\circ$ . Which of the following degree measures represent angles formed between any two of the intersecting lines at point $P$ ? Select all that apply.

Geçerli olan tümünü seçin

55^\circ

90^\circ

145^\circ

115^\circ

70^\circ

Cevabı ve açıklamayı göster

Cevap: $55^\circ$ ; $90^\circ$ ; $145^\circ$

Cevap

The degree measures

55^\circ

,

90^\circ

, and

145^\circ

are all valid angle measures formed by the intersecting lines at point

P

.

The intersecting lines form several distinct angle measures at point

P

. The right angle between lines

R

and

S

measures

90^\circ

. The acute angle between lines

T

and

S

is complementary to the

35^\circ

angle, giving

90^\circ - 35^\circ = 55^\circ

. The obtuse angle between lines

T

and

R

along straight line

R

is supplementary to

35^\circ

, giving

180^\circ - 35^\circ = 145^\circ

. Therefore, the options stating

55^\circ

,

90^\circ

, and

145^\circ

are all correct.

Adım Adım Çözüm

1

Identify the angle measure between perpendicular lines R and S.

Since line

R \perp S

, the angle between them is

90^\circ

.

Perpendicular lines intersect at right angles (

90^\circ

).

2

Calculate the acute angle between line T and line S.

Angle between

T

and

S

=

90^\circ - 35^\circ = 55^\circ

.

The given

35^\circ

angle between

R

and

T

and the adjacent angle between

T

and

S

form the

90^\circ

right angle between

R

and

S

.

3

Determine the supplementary obtuse angles formed by line T with line R and line S.

Supplementary angle to

35^\circ

is

180^\circ - 35^\circ = 145^\circ

. Supplementary angle to

55^\circ

is

180^\circ - 55^\circ = 125^\circ

.

Adjacent angles along a straight line sum to

180^\circ

.

4

Compare calculated angle measures with the options.

The valid angle measures are

35^\circ

,

55^\circ

,

90^\circ

,

125^\circ

, and

145^\circ

. Thus

55^\circ

,

90^\circ

, and

145^\circ

are correct.

Matching calculated angle measures with the choices provided.

Anahtar Kavram

Perpendicular line relationships and supplementary angle properties of intersecting lines.

Tahmini Süre:1m 0s

Soru 1398Soru →

If $-3|x - 2| + 4 > -11$ , which of the following inequalities represents all possible real values of $x$ ?

-3 < x < 7

x < -3 \text{ or } x > 7

x < 7

-7 < x < 3

-5 < x < 5

Cevabı ve açıklamayı göster

Cevap: $-3 < x < 7$

Cevap

-3 < x < 7

Subtracting

4

from both sides gives

-3|x - 2| > -15

. Dividing by

-3

and reversing the inequality sign results in

|x - 2| < 5

. Converting to the compound inequality

-5 < x - 2 < 5

and adding

2

to each part yields the correct interval

-3 < x < 7

.

Adım Adım Çözüm

1

Subtract 4 from both sides of the inequality

-3|x - 2| > -15

Isolate the absolute value term on the left side.

2

Divide both sides by

-3

and reverse the inequality sign

|x - 2| < 5

Dividing an inequality by a negative number reverses the direction of the inequality symbol.

3

Express the absolute value inequality as a compound inequality

-5 < x - 2 < 5

An inequality of the form

|u| < c

(where

c > 0

) is equivalent to

-c < u < c

.

4

Add 2 to all three parts of the compound inequality

-3 < x < 7

Isolate

x

to find the complete range of solution values.

Anahtar Kavram

Linear Inequalities and Absolute Value

Soru 1399Soru →

A circle has a circumference of $20\pi$ . A sector of this circle is defined by a central angle that intercepts an arc of length $5\pi$ . What is the area of this sector?

10\pi

25\pi

50\pi

100\pi

5\pi

Cevabı ve açıklamayı göster

Cevap: $25\pi$

Cevap

The area of the sector is

25\pi

.

First, the radius is found using

2\pi r = 20\pi

, which gives

r = 10

. The total area of the circle is

\pi (10)^2 = 100\pi

. The fraction of the circle defined by the sector is

\frac{5\pi}{20\pi} = \frac{1}{4}

. Multiplying the total area by this fraction gives

\frac{1}{4} \times 100\pi = 25\pi

.

Adım Adım Çözüm

1

Find the radius of the circle from the circumference.

Since

C = 2\pi r = 20\pi

, solving for

r

gives

r = 10

.

The radius is required to calculate the total circle area.

2

Determine the fraction of the circle that the sector represents.

\text{Fraction} = \frac{\text{Arc Length}}{\text{Circumference}} = \frac{5\pi}{20\pi} = \frac{1}{4}.

The ratio of arc length to circumference gives the proportion of the total circle occupied by the sector.

3

Calculate the total area of the circle.

A = \pi r^2 = \pi (10)^2 = 100\pi.

The area of a circle with radius 10 is

100\pi

.

4

Multiply the total area by the sector fraction.

\text{Sector Area} = \frac{1}{4} \times 100\pi = 25\pi.

Applying the proportional fraction yields the sector area.

Anahtar Kavram

The ratio of a sector's arc length to the full circumference is equal to the ratio of the sector's area to the full circle's area.

Soru 1400Soru →

An original dataset $X$ consists of $20$ distinct positive numerical values with standard deviation $\sigma$ , range $R$ , and interquartile range $IQR$ . A new dataset $Y$ is created by transforming each data value $x_i$ in $X$ according to the rule $y_i = -3x_i + 5$ . Which of the following statements about the dispersion metrics of dataset $Y$ compared to dataset $X$ must be true? Select all such statements.

Geçerli olan tümünü seçin

The standard deviation of dataset

Y

is

3\sigma

.

The interquartile range of dataset

Y

is

3 \times IQR

.

The standard deviation of dataset

Y

is

3\sigma + 5

.

The range of dataset

Y

is

-3R + 5

.

The interquartile range of dataset

Y

is equal to

-3 \times IQR

.

Cevabı ve açıklamayı göster

Cevap: The standard deviation of dataset $Y$ is $3\sigma$ .; The interquartile range of dataset $Y$ is $3 \times IQR$ .

Cevap

The statements asserting that the standard deviation of dataset Y is 3 times the standard deviation of dataset X, and that the interquartile range of dataset Y is 3 times the interquartile range of dataset X, are both correct.

For any data set undergoing a linear transformation

y = ax + b

, all measures of dispersion (including standard deviation, range, and interquartile range) are scaled by the absolute value of the multiplier

|a|

and remain completely unaffected by the additive constant

b

. Since

a = -3

,

|a| = |-3| = 3

. Therefore, both the standard deviation and the interquartile range scale by a factor of 3.

Adım Adım Çözüm

1

Recall the general rule for linear transformation of measures of dispersion.

For any linear transformation

y = ax + b

, measures of dispersion (Range, IQR, Standard Deviation) scale by

|a|

and are unaffected by

b

.

Measures of dispersion evaluate the spread/distance between points, which shifts unchanged when a constant is added, and scales non-negatively when multiplied by a constant.

2

Identify the values of the multiplicative constant

a

and additive constant

b

.

a = -3

and

b = 5

, giving

|a| = |-3| = 3

.

The transformation equation is

y_i = -3x_i + 5

.

3

Apply the scaling factor

|a| = 3

to each measure of dispersion.

Standard deviation of

Y

is

3\sigma

, IQR of

Y

is

3 \times IQR

, and Range of

Y

is

3R

.

All measures of dispersion scale by a factor of 3 regardless of the negative sign of the multiplier or the addition of 5.

Anahtar Kavram

Linear Transformations on Measures of Dispersion

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Anahtar Kavram

Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Anahtar Kavram

Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Anahtar Kavram