All practice questions

Question 1461Question →

For all real numbers $x$ satisfying the absolute value inequality $|4x - 12| \le 20$ , the maximum possible value of the expression $|2 - 3x|$ is $M$ . What is the value of $M$ ?

Show answer & explanation

Answer: 22

Answer

The maximum possible value

M

of the expression

|2 - 3x|

on the domain

-2 \le x \le 8

is 22.

Solving

|4x - 12| \le 20

yields

-20 \le 4x - 12 \le 20

, which simplifies to

-2 \le x \le 8

. Evaluating

|2 - 3x|

over this interval gives a minimum of

0

(at

x = 2/3

) and endpoint values of

|2 - 3(-2)| = 8

and

|2 - 3(8)| = |-22| = 22

. Thus, the maximum value

M

is 22.

Step-by-Step Solution

1

Unfold the given absolute value inequality into a compound linear inequality.

-20 \le 4x - 12 \le 20

The inequality

|u| \le k

for

k \ge 0

is equivalent to

-k \le u \le k

.

2

Isolate the variable

x

by adding 12 and dividing by 4.

-2 \le x \le 8

Adding 12 gives

-8 \le 4x \le 32

. Dividing by positive 4 preserves inequality signs, yielding

-2 \le x \le 8

.

3

Evaluate the target expression

|2 - 3x|

at the boundary points of the interval

[-2, 8]

.

For

x = -2

:

|2 - 3(-2)| = |8| = 8

. For

x = 8

:

|2 - 3(8)| = |-22| = 22

.

The expression

f(x) = |2 - 3x|

is convex and non-negative, reaching its local minimum of 0 at

x = \frac{2}{3}

. Its maximum over a closed interval must occur at one of the endpoints.

4

Compare the evaluated values to find the maximum

M

.

M = \max(8, 22) = 22

Comparing 8 and 22 shows that 22 is the absolute maximum value achievable within the domain.

Key Concept

Solving linear absolute value inequalities to determine variable bounds and evaluating extreme values of absolute value expressions.

Estimated Time:2m 0s

Question 1462Question →

A corporate enterprise allocates a total annual budget of $\$52,000$ across three departments: Marketing, Operations, and Development. The amount allocated to Operations is $\$5,000$ less than twice the amount allocated to Marketing. The amount allocated to Development is $\$1,000$ more than half of the combined allocations of Marketing and Operations. What is the budget allocation for Development?

\$13,000

\$17,000

\$18,000

\$21,000

\$24,000

Show answer & explanation

Answer: $\$18,000$

Answer

The budget allocation for Development is

\$18,000

.

By defining the Marketing budget as

x

, Operations becomes

2x - 5,000

and Development becomes

\frac{1}{2}(3x - 5,000) + 1,000 = 1.5x - 1,500

. Adding these three expressions equals the total budget of

\$52,000

, giving

4.5x - 6,500 = 52,000

. Solving for

x

yields

x = 13,000

. Substituting

x = 13,000

into the expression for Development gives

1.5(13,000) - 1,500 = \$18,000

.

Step-by-Step Solution

1

Define the variable and express each department's budget in terms of that variable.

Let

x

be the Marketing budget in dollars. Then Operations

= 2x - 5,000

. The combined Marketing and Operations budget

= x + (2x - 5,000) = 3x - 5,000

. Therefore, Development

= \frac{1}{2}(3x - 5,000) + 1,000 = 1.5x - 1,500

.

Establishing a single linear variable allows all three department budgets to be combined into one linear equation.

2

Set up the total budget equation and solve for

x

.

x + (2x - 5,000) + (1.5x - 1,500) = 52,000 \implies 4.5x - 6,500 = 52,000 \implies 4.5x = 58,500 \implies x = 13,000

.

Summing the allocations for Marketing, Operations, and Development yields the total corporate budget of

\$52,000

.

3

Calculate the specific budget allocation requested for Development.

Development allocation

= 1.5(13,000) - 1,500 = 19,500 - 1,500 = 18,000

.

Substituting

x = 13,000

back into the algebraic expression for Development gives the final required dollar amount.

Key Concept

Formulating and solving multi-step linear equations in one variable from word problems.

Estimated Time:2m 30s

Question 1463Question →

In a right triangle, the lengths of the two legs are in a ratio of $3:4$ . If the perimeter of the triangle is $36$ centimeters, what is the area of the triangle, in square centimeters?

Show answer & explanation

Answer: 54

Answer

The area of the triangle is 54 square centimeters.

Since the ratio of the legs of the right triangle is 3:4, the triangle forms a classic 3-4-5 right triangle proportion. The perimeter is

3x + 4x + 5x = 12x

. Setting

12x = 36

yields

x = 3

. The legs are therefore

9

cm and

12

cm. Calculating the area using

\frac{1}{2} \times 9 \times 12

gives

54

square centimeters.

Step-by-Step Solution

1

Express the side lengths in terms of a variable

x

Legs are

3x

and

4x

, and hypotenuse is

5x

By the Pythagorean theorem, a right triangle with legs in ratio 3:4 has hypotenuse ratio

\sqrt{3^2 + 4^2} = 5

.

2

Solve for

x

using the given perimeter

x = 3

The sum of all three sides is

3x + 4x + 5x = 12x = 36

, giving

x = 3

.

3

Calculate the actual leg lengths and area

Legs are

9

cm and

12

cm; Area is

54

\text{cm}^2

The area of a right triangle is half the product of its perpendicular legs:

\frac{1}{2} \times 9 \times 12 = 54

.

Key Concept

Perimeter and area of right triangles using standard side ratios

Question 1464Question →

In $\triangle ABC$ , point $D$ lies on segment $BC$ such that segment $AD$ is perpendicular to $BC$ . The ratio of the area of $\triangle ABD$ to the area of $\triangle ADC$ is $5 : 16$ . If $AB = 13$ and the perimeter of $\triangle ABC$ is $54$ , what is the area of $\triangle ABC$ ?

96

104

126

130

252

Show answer & explanation

Answer: 126

Answer

126

The correct answer is 126. Since triangles

\triangle ABD

and

\triangle ADC

share height

AD

, their areas are in proportion to their bases

BD:DC = 5:16

. Setting

BD = 5k

and

DC = 16k

, the Pythagorean theorem yields altitude

AD = \sqrt{169 - 25k^2}

and hypotenuse

AC = \sqrt{169 + 231k^2}

. Substituting these into the perimeter equation

13 + 21k + AC = 54

yields

k = 1

(after rejecting an extraneous root). Thus

BC = 21

and

AD = 12

, making the area

\frac{1}{2} \times 21 \times 12 = 126

.

Step-by-Step Solution

1

Relate the areas of the sub-triangles to their base lengths.

\frac{\text{Area}(\triangle ABD)}{\text{Area}(\triangle ADC)} = \frac{\frac{1}{2} \cdot BD \cdot AD}{\frac{1}{2} \cdot DC \cdot AD} = \frac{BD}{DC} = \frac{5}{16}

. Thus,

BD = 5k

and

DC = 16k

for some positive constant

k

, giving

BC = 21k

.

Triangles sharing the same altitude have areas proportional to their bases.

2

Express altitude

AD

and side

AC

in terms of

k

using the Pythagorean theorem.

In right

\triangle ABD

:

AD = \sqrt{AB^2 - BD^2} = \sqrt{13^2 - (5k)^2} = \sqrt{169 - 25k^2}

. In right

\triangle ADC

:

AC = \sqrt{AD^2 + DC^2} = \sqrt{(169 - 25k^2) + (16k)^2} = \sqrt{169 + 231k^2}

.

Since

AD \perp BC

, both

\triangle ABD

and

\triangle ADC

are right triangles.

3

Set up and solve the perimeter equation for

k

.

Perimeter

= AB + BC + AC = 13 + 21k + \sqrt{169 + 231k^2} = 54 \implies \sqrt{169 + 231k^2} = 41 - 21k

. Squaring both sides yields

169 + 231k^2 = 1681 - 1722k + 441k^2 \implies 210k^2 - 1722k + 1512 = 0 \implies 5k^2 - 41k + 36 = 0

. Factoring gives

(5k - 36)(k - 1) = 0

, so

k = 1

or

k = 7.2

.

The given perimeter allows over-constraining the side length expressions to a quadratic in

k

.

4

Test roots for validity and calculate final triangle area.

For

k = 7.2

,

41 - 21(7.2) = -110.2 < 0

, which is extraneous. For

k = 1

,

BD = 5

,

DC = 16

,

BC = 21

,

AD = 12

, and

AC = 20

. Area

(\triangle ABC) = \frac{1}{2} \cdot BC \cdot AD = \frac{1}{2} \cdot 21 \cdot 12 = 126

.

Extraneous roots introduced by squaring must be discarded, and the valid root gives the exact area.

Key Concept

Decomposing triangles into adjacent right triangles, leveraging shared altitudes for area ratios, and applying algebraic perimeter constraints with Pythagorean equations.

Question 1465Question →

If

x

is a real number greater than

1

satisfying the exponential equation

\left(x^x\right)^{\sqrt{x}} = x^{\left(x^{\sqrt{x}}\right)}

what is the value of

x

?

\frac{3}{2}

\frac{4}{9}

\frac{9}{4}

\frac{27}{8}

\frac{81}{16}

Show answer & explanation

Answer: $\frac{9}{4}$

Answer

The value of

x

is

\frac{9}{4}

.

Applying the exponent rule

(a^b)^c = a^{bc}

simplifies the left side to

x^{x\sqrt{x}}

. Since the base

x > 1

is identical on both sides, we set the exponents equal:

x\sqrt{x} = x^{\sqrt{x}}

. Expressing

x\sqrt{x}

as

x^{3/2}

yields

x^{3/2} = x^{\sqrt{x}}

, which implies

\sqrt{x} = \frac{3}{2}

. Squaring both sides gives

x = \frac{9}{4}

.

Step-by-Step Solution

1

Apply the power rule

(a^b)^c = a^{bc}

to the left-hand side of the equation.

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

Raising a power to another exponent requires multiplying the exponents:

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

.

2

Equate the exponents of the expressions on both sides, as the bases are equal and

x > 1

.

x \sqrt{x} = x^{\sqrt{x}}

, which means

x^{3/2} = x^{\sqrt{x}}

If

x^A = x^B

and

x > 1

, then

A = B

.

3

Equate exponents once more for the base

x

.

\frac{3}{2} = \sqrt{x}

Since the bases are identical (

x > 1

), their exponents must be equal.

4

Square both sides to solve for

x

.

x = \left(\frac{3}{2}\right)^2 = \frac{9}{4}

Squaring

\sqrt{x}

isolates

x

.

Key Concept

Properties of exponents and nested power rules with radicals

Estimated Time:2m 0s

Question 1466Question →

For the function $f(x) = x^2 - 5x$ , what is the value of $f(-3)$ ?

24

6

-6

-24

-16

Show answer & explanation

Answer: 24

Answer

24

Substituting

x = -3

into the function yields

(-3)^2 - 5(-3) = 9 + 15 = 24

, which correctly evaluates both the exponent and linear terms following standard sign rules.

Step-by-Step Solution

1

Substitute

x = -3

into the function expression

f(x) = x^2 - 5x

.

f(-3) = (-3)^2 - 5(-3)

Function notation

f(-3)

requires replacing every instance of variable

x

with

-3

.

2

Evaluate the exponent term

(-3)^2

.

9

Squaring any negative number produces a positive result:

(-3) \times (-3) = 9

.

3

Evaluate the product term

-5(-3)

.

15

Multiplying two negative numbers yields a positive product:

-5 \times (-3) = 15

.

4

Sum the evaluated components.

9 + 15 = 24

Combine the results to obtain the final value of

f(-3)

.

Key Concept

Evaluating functions with negative input values

Estimated Time:45s

Question 1467Question →

How many integer values of $x$ satisfy the inequality $|3 - 2x| + |x + 4| \le 9$ ?

3

4

5

6

7

Show answer & explanation

Answer: 5

Answer

5 integer values (specifically -2, -1, 0, 1, and 2)

Solving the absolute value inequality using piecewise intervals yields the continuous solution set

[-2, \frac{8}{3}]

. The integers contained in this range are

-2, -1, 0, 1,

and

2

, giving a total of 5 integer values.

Step-by-Step Solution

1

Identify the critical points of the absolute value expressions.

The critical points are

x = \frac{3}{2}

and

x = -4

. These split the real number line into three intervals:

x < -4

,

-4 \le x \le \frac{3}{2}

, and

x > \frac{3}{2}

.

Absolute value expressions change definition at their zeroes.

2

Analyze Case 1:

x < -4

.

Here

|3 - 2x| = 3 - 2x

and

|x + 4| = -(x + 4)

. The inequality becomes

(3 - 2x) - (x + 4) \le 9 \implies -3x - 1 \le 9 \implies -3x \le 10 \implies x \ge -\frac{10}{3} \approx -3.33

. Since there is no overlap between

x < -4

and

x \ge -3.33

, no solutions exist in this interval.

Evaluating expressions according to the sign of terms when

x < -4

.

3

Analyze Case 2:

-4 \le x \le \frac{3}{2}

.

Here

|3 - 2x| = 3 - 2x

and

|x + 4| = x + 4

. The inequality becomes

(3 - 2x) + (x + 4) \le 9 \implies 7 - x \le 9 \implies -x \le 2 \implies x \ge -2

. Combining with the case interval gives

[-2, \frac{3}{2}]

.

Determining valid values of

x

within the middle interval.

4

Analyze Case 3:

x > \frac{3}{2}

.

Here

|3 - 2x| = 2x - 3

and

|x + 4| = x + 4

. The inequality becomes

(2x - 3) + (x + 4) \le 9 \implies 3x + 1 \le 9 \implies 3x \le 8 \implies x \le \frac{8}{3} \approx 2.67

. Combining with the case interval gives

(\frac{3}{2}, \frac{8}{3}]

.

Determining valid values of

x

within the upper interval.

5

Combine solution intervals and count integer solutions.

The total solution set is

[-2, \frac{8}{3}]

. The integer values within this interval are

-2, -1, 0, 1,

and

2

. Total count = 5.

Identifying all integer values within the bounded set

[-2, 2.67]

.

Key Concept

Solving absolute value inequalities with multiple absolute value terms using critical points and case analysis.

Estimated Time:2m 0s

Question 1468Question →

What is the sum of all integer values of $x$ that satisfy both $|2x - 5| \le 9$ and $|x + 2| \ge 4$ ?

Show answer & explanation

Answer: 27

Answer

The sum of all integer values of

x

that satisfy both inequalities is 27.

First, solving

|2x - 5| \le 9

yields

-9 \le 2x - 5 \le 9 \implies -4 \le 2x \le 14 \implies -2 \le x \le 7

. Second, solving

|x + 2| \ge 4

yields

x + 2 \ge 4 \implies x \ge 2

or

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

. Taking the intersection of

-2 \le x \le 7

and

(x \ge 2 \text{ or } x \le -6)

gives the solution set

2 \le x \le 7

. The integer values satisfying this condition are 2, 3, 4, 5, 6, and 7. Summing these integers gives

2 + 3 + 4 + 5 + 6 + 7 = 27

.

Step-by-Step Solution

1

Solve the inequality

|2x - 5| \le 9

-2 \le x \le 7

An inequality of the form

|u| \le a

(where

a \ge 0

) expands to

-a \le u \le a

. Adding 5 gives

-4 \le 2x \le 14

, and dividing by 2 yields

-2 \le x \le 7

.

2

Solve the inequality

|x + 2| \ge 4

x \ge 2 \text{ or } x \le -6

An inequality of the form

|u| \ge a

(where

a > 0

) expands to

u \ge a

or

u \le -a

. Subtracting 2 from both inequalities yields

x \ge 2

or

x \le -6

.

3

Determine the overlapping interval for both inequalities

2 \le x \le 7

Combining

-2 \le x \le 7

with

x \ge 2 \text{ or } x \le -6

eliminates

x \le -6

. The intersection of

[-2, 7]

and

[2, \infty)

is

[2, 7]

.

4

Identify the integer values in the solution interval and calculate their sum

27

The integers in the closed interval

[2, 7]

are 2, 3, 4, 5, 6, and 7. Adding them together gives

2 + 3 + 4 + 5 + 6 + 7 = 27

.

Key Concept

System of Linear Absolute Value Inequalities

Question 1469Question →

In the $xy$ -plane, line $k$ is defined by the equation $2x + 4y = 9$ . Line $m$ is perpendicular to line $k$ . What is the slope of line $m$ ?

2

-\frac{1}{2}

-2

\frac{1}{2}

-\frac{9}{4}

Show answer & explanation

Answer: $2$

Answer

2

First, rewrite the line equation

2x + 4y = 9

in slope-intercept form

y = mx + b

: subtracting

2x

from both sides gives

4y = -2x + 9

, and dividing by

4

yields

y = -\frac{1}{2}x + \frac{9}{4}

. Thus, the slope of line

k

is

m_k = -\frac{1}{2}

. Since line

m

is perpendicular to line

k

, its slope is the negative reciprocal of

-\frac{1}{2}

, which is

2

.

Step-by-Step Solution

1

Convert the equation of line

k

to slope-intercept form (

y = mx + b

).

4y = -2x + 9 \implies y = -\frac{1}{2}x + \frac{9}{4}

.

The coefficient of

x

in slope-intercept form gives the slope of line

k

(

m_k = -\frac{1}{2}

).

2

Calculate the slope of perpendicular line

m

.

m_m = -\frac{1}{m_k} = -\frac{1}{-\frac{1}{2}} = 2

.

Perpendicular lines have slopes that are negative reciprocals of each other.

Key Concept

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

Estimated Time:45s

Question 1470Question →

For real constants

a

and

b

, consider the linear equation in one variable

x

:

\frac{a(x - 2)}{3} - \frac{2x + 1}{4} = \frac{(a - 3)x + b}{12}

If this equation has infinitely many solutions for

x

, which of the following statements must be true? Select all such statements.

Select all that apply

a + b = -10

ab < 0

2a - b = 13

a - b = -10

ab = 11

Show answer & explanation

Answer: $a + b = -10$ ; $ab < 0$ ; $2a - b = 13$

Answer

The statements

a + b = -10

,

ab < 0

, and

2a - b = 13

are all true.

Clearing denominators gives

4a(x - 2) - 3(2x + 1) = (a - 3)x + b

. Expanding both sides yields

(4a - 6)x - (8a + 3) = (a - 3)x + b

, which rearranges to

(3a - 3)x = 8a + b + 3

. For a linear equation in one variable to have infinitely many solutions, both the coefficient of

x

and the constant term must be zero (

0x = 0

). Setting

3a - 3 = 0

gives

a = 1

, and substituting

a = 1

into

8a + b + 3 = 0

yields

b = -11

. Evaluating the statements with

a = 1

and

b = -11

shows that

a + b = 1 + (-11) = -10

is true,

ab = -11 < 0

is true, and

2a - b = 2(1) - (-11) = 13

is true.

Step-by-Step Solution

1

Clear the denominators by multiplying both sides of the equation by 12.

4a(x - 2) - 3(2x + 1) = (a - 3)x + b

Eliminating fractions simplifies expanding terms and combining like variables.

2

Expand all terms and group terms containing

x

on the left-hand side.

4ax - 8a - 6x - 3 = (a - 3)x + b \implies (4a - 6)x - (8a + 3) = (a - 3)x + b

Preparing the linear equation to be expressed in standard form

Ax = B

.

3

Rearrange into standard form

(A)x = B

.

[(4a - 6) - (a - 3)]x = 8a + b + 3 \implies (3a - 3)x = 8a + b + 3

A linear equation has infinitely many solutions if and only if

A = 0

and

B = 0

simultaneously.

4

Set the coefficient of

x

and the constant term equal to zero to determine

a

and

b

.

3a - 3 = 0 \implies a = 1

; then

8(1) + b + 3 = 0 \implies b + 11 = 0 \implies b = -11

The equation reduces to

0 \cdot x = 0

, which is satisfied by every real number

x

.

5

Evaluate the given statements using

a = 1

and

b = -11

.

a + b = 1 + (-11) = -10

(True);

ab = (1)(-11) = -11 < 0

(True);

2a - b = 2(1) - (-11) = 13

(True).

Direct substitution confirms which statements hold true.

Key Concept

Conditions for a linear equation in one variable to have infinitely many solutions

Estimated Time:2m 30s

Question 1471Question →

The table below shows the frequency distribution of customer satisfaction ratings collected by a service center over a 50-day monitoring period.

Rating Interval	Frequency (Days)
1–5	8
6–10	12
11–15	20
16–20	10

Based on the table, what percentage of the days had a customer satisfaction rating of 11 or higher?

20%

40%

50%

60%

75%

Show answer & explanation

Answer: 60%

Answer

60%

To find the percentage of days with a customer satisfaction rating of 11 or higher, add the frequencies of the intervals '11–15' (20 days) and '16–20' (10 days) to get 30 days. The total number of days across all intervals is 8 + 12 + 20 + 10 = 50 days. The percentage is (30 / 50) × 100% = 60%.

Step-by-Step Solution

1

Identify the relevant rating intervals.

The intervals corresponding to a rating of 11 or higher are '11–15' and '16–20'.

Ratings of 11 or higher include all data points within these two top intervals.

2

Sum the frequencies for the relevant intervals.

Frequency = 20 + 10 = 30 days.

Combining the frequencies gives the total number of favorable outcomes.

3

Calculate the total number of days across all intervals.

Total days = 8 + 12 + 20 + 10 = 50 days.

The total frequency represents the denominator for the percentage calculation.

4

Compute the percentage.

(30 / 50) × 100% = 60%.

Dividing the target frequency by the total frequency and multiplying by 100 yields the required percentage.

Key Concept

Frequency Table Percentages and Grouped Data Cutoffs

Estimated Time:45s

Question 1472Question →

Two concentric circles are centered at point $O$ . The inner circle has a radius of $6$ units, and the outer circle has a radius of $6\sqrt{3}$ units. Radii $OA$ and $OB$ of the outer circle form a central angle $\angle AOB = 60^\circ$ and intersect the inner circle at points $C$ and $D$ , respectively. Which of the following statements must be true regarding the region and boundary lengths defined by these figures? Select all that apply.

Select all that apply

The area of the region bounded by arc

AB

, arc

CD

, segment

AC

, and segment

BD

is equal to

12\pi

square units.

The straight-line distance between points

A

and

B

is

6\sqrt{3}

units.

The area of the circular segment bounded by chord

AB

and minor arc

AB

is

18\pi - 27\sqrt{3}

square units.

The ratio of the length of arc

CD

to the length of arc

AB

is

1 : 3

.

The perimeter of the region bounded by arc

AB

, arc

CD

, segment

AC

, and segment

BD

is

2\pi(1 + \sqrt{3}) + 6\sqrt{3}

units.

Show answer & explanation

Answer: The area of the region bounded by arc $AB$ , arc $CD$ , segment $AC$ , and segment $BD$ is equal to $12\pi$ square units.; The straight-line distance between points $A$ and $B$ is $6\sqrt{3}$ units.; The area of the circular segment bounded by chord $AB$ and minor arc $AB$ is $18\pi - 27\sqrt{3}$ square units.

Answer

The true statements are: the area of the region bounded by arc AB, arc CD, segment AC, and segment BD is 12π square units; the straight-line distance between points A and B is 6√3 units; and the area of the circular segment bounded by chord AB and minor arc AB is 18π - 27√3 square units.

The area of the region between the two concentric arcs is obtained by subtracting the inner sector area (6π) from the outer sector area (18π), giving 12π square units. The triangle OAB is equilateral because it has two sides of length 6√3 and an included angle of 60°, making chord AB equal to 6√3. Subtracting the area of this equilateral triangle (27√3) from the outer sector area (18π) yields the area of the circular segment bounded by chord AB and arc AB, which is 18π - 27√3.

Step-by-Step Solution

1

Calculate the area of sector OAB and sector OCD to evaluate the area of the annular sector region.

Sector OAB area = (60/360) * π * (6√3)^2 = (1/6) * 108π = 18π. Sector OCD area = (60/360) * π * (6^2) = (1/6) * 36π = 6π. Region area = 18π - 6π = 12π.

The area between two concentric sector arcs bounded by the same radii is the difference in sector areas.

2

Determine the length of chord AB using triangle properties.

In triangle OAB, OA = OB = 6√3 and angle AOB = 60°. An isosceles triangle with a 60° vertex angle is equilateral, so AB = 6√3.

All internal angles of an isosceles triangle with a 60° angle must equal 60°.

3

Calculate the area of circular segment AB.

Area of equilateral triangle OAB = (√3 / 4) * (6√3)^2 = 27√3. Segment area = Sector OAB area - Triangle OAB area = 18π - 27√3.

A circular segment's area is found by subtracting the area of the subtended triangle from the area of the corresponding sector.

4

Evaluate arc length ratio and total perimeter of the annular sector.

Arc CD / Arc AB = 6 / (6√3) = 1 / √3 ≠ 1 / 3. Total perimeter = Arc AB + Arc CD + 2*(R - r) = 2√3π + 2π + 2*(6√3 - 6) = 2π(1 + √3) + 12√3 - 12.

Verifies that statements regarding ratio 1:3 and incomplete perimeter calculations are mathematically false.

Key Concept

Concentric circle geometry, arc length proportions, sector area calculations, and circular segment area formulation.

Question 1473Question →

Train $X$ departs from a station traveling due east at a constant speed of $50$ miles per hour. Exactly $1$ hour later, Express Train $Y$ departs from the same station along the same track, traveling due east at a constant speed of $75$ miles per hour. How many hours after Express Train $Y$ departs will it catch up to Train $X$ ?

1

1.5

2

2.5

3

Show answer & explanation

Answer: $2$

Answer

Express Train

Y

will catch up to Train

X

exactly

2

hours after Express Train

Y

departs.

Let

t

represent the number of hours Express Train

Y

travels. Because Train

X

departed

1

hour earlier, it has been traveling for

t + 1

hours. For Express Train

Y

to catch up to Train

X

, both trains must cover the exact same distance from the starting station. Setting up the distance equation

75t = 50(t + 1)

gives

75t = 50t + 50

, which simplifies to

25t = 50

, yielding

t = 2

hours.

Step-by-Step Solution

1

Define variables for the time traveled by each train.

Let

t

be the time in hours that Express Train

Y

travels. Since Train

X

departed

1

hour earlier, Train

X

travels for

t + 1

hours.

Train

X

has a

1

-hour head start.

2

Express the distance traveled by each train using

\text{Distance} = \text{Rate} \times \text{Time}

.

Distance of Train

X = 50(t + 1)

miles; Distance of Express Train

Y = 75t

miles.

Both trains travel at constant rates along the same path.

3

Equate the two distance expressions to solve for

t

.

75t = 50(t + 1) \implies 75t = 50t + 50 \implies 25t = 50 \implies t = 2

.

Express Train

Y

catches Train

X

when both have covered the exact same distance.

Key Concept

Distance, Rate, and Time Modeling for Catch-up Scenarios

Question 1474Question →

Dataset $X$ consists of 2 numbers, each equal to 10. Dataset $Y$ consists of 8 numbers, each equal to 25. If Dataset $X$ and Dataset $Y$ are combined to form a single dataset of 10 numbers, what is the standard deviation of the combined dataset?

Show answer & explanation

Answer: 6

Answer

The standard deviation of the combined dataset is 6.

To find the standard deviation of the combined dataset, first calculate the combined mean:

\frac{2(10) + 8(25)}{10} = 22

. Next, compute the variance by finding the average of the squared deviations from 22:

\frac{2(10 - 22)^2 + 8(25 - 22)^2}{10} = \frac{2(144) + 8(9)}{10} = \frac{360}{10} = 36

. Taking the square root of the variance yields a standard deviation of

\sqrt{36} = 6

.

Step-by-Step Solution

1

Calculate the mean of the combined 10-number dataset.

The combined mean is 22.

The mean of the combined dataset is needed to compute individual deviations.

2

Compute the squared deviation of each data point from the combined mean and average them to determine the variance.

The variance is 36.

Variance is defined as the arithmetic mean of the squared deviations from the mean.

3

Calculate the square root of the variance to find the standard deviation.

The standard deviation is 6.

Standard deviation is the non-negative square root of variance.

Key Concept

Standard deviation of a combined dataset

Question 1475Question →

For how many integer values of $k$ does the inequality $|2x - k| + |x + 3| \le 7$ have at least one real solution $x$ such that $x \ge 1$ ?

8

9

10

11

13

Show answer & explanation

Answer: 10

Answer

10 integer values of

k

satisfy the given condition.

The correct answer is 10 because analyzing the condition

x \ge 1

simplifies the inequality to

|2x - k| \le 4 - x

, requiring

x \in [1, 4]

. The double inequality

k - 4 \le x \le \frac{k + 4}{3}

yields solutions overlapping with

[1, 4]

if and only if

-1 \le k \le 8

, which contains exactly 10 integers.

Step-by-Step Solution

1

Simplify the absolute value term

|x + 3|

using the given condition

x \ge 1

.

Since

x \ge 1

,

x + 3 > 0

, so

|x + 3| = x + 3

. The inequality becomes

|2x - k| + x + 3 \le 7

, which simplifies to

|2x - k| \le 4 - x

.

Establishing the sign of

x + 3

allows eliminating one set of absolute value bars.

2

Determine the valid range for

x

.

Since

|2x - k| \ge 0

, it must hold that

4 - x \ge 0

, which implies

x \le 4

. Combined with

x \ge 1

, any solution

x

must lie in the interval

[1, 4]

.

An absolute value quantity cannot be less than a negative number.

3

Unwrap the absolute value inequality

|2x - k| \le 4 - x

.

-(4 - x) \le 2x - k \le 4 - x

. Splitting into two linear inequalities:
1)

2x - k \le 4 - x \implies 3x \le k + 4 \implies x \le \frac{k + 4}{3}

.
2)

2x - k \ge x - 4 \implies x \ge k - 4

.
Thus,

k - 4 \le x \le \frac{k + 4}{3}

.

Rewriting absolute value inequalities as compound inequalities defines explicit bounds on

x

in terms of

k

.

4

Find the range of

k

for which

[k - 4, \frac{k + 4}{3}]

overlaps with

[1, 4]

.

For an overlapping solution to exist in

[1, 4]

:
1) The upper bound

\frac{k + 4}{3}

must be at least 1:

\frac{k + 4}{3} \ge 1 \implies k \ge -1

.
2) The lower bound

k - 4

must be at most 4:

k - 4 \le 4 \implies k \le 8

.
Combining these gives

-1 \le k \le 8

.

The solution interval for

x

must have a non-empty intersection with the allowed domain

[1, 4]

.

5

Count the total number of integer values of

k

in the interval

[-1, 8]

.

The integers are

-1, 0, 1, 2, 3, 4, 5, 6, 7, 8

, giving a total of

8 - (-1) + 1 = 10

integer values.

Counting inclusive integer endpoints gives the total count.

Key Concept

Solving absolute value inequalities involving parameters and restricted variable domains.

Estimated Time:2m 0s

Question 1476Question →

An express train travels along a straight track between Station A and Station B. For the first $40\%$ of the total distance, the train travels at a constant speed of $v$ miles per hour. For the next $50\%$ of the remaining distance, due to track maintenance, it travels at a constant speed that is $25\%$ slower than $v$ . For the final leg of the journey, the train increases its speed to a constant rate that is $20\%$ faster than $v$ . If the average speed for the entire journey from Station A to Station B is $60$ miles per hour, what is the value of $v$ , in miles per hour?

57.5

60.0

63.0

68.0

69.0

Show answer & explanation

Answer: $63.0$

Answer

The base speed

v

is

63.0

miles per hour.

The correct answer is derived by setting up a model for distance, speed, and time across all three segments. The first leg covers

0.4D

at speed

v

(time

= 0.4D/v

). The second leg covers

50\%

of the remaining

0.6D

, which is

0.3D

, at speed

0.75v

(time

= 0.3D/(0.75v) = 0.4D/v

). The third leg covers the remaining

0.3D

at speed

1.2v

(time

= 0.3D/(1.2v) = 0.25D/v

). Summing these gives total time

T = 1.05D/v

. The average speed is

D / (1.05D/v) = v / 1.05 = 20v/21

. Setting

20v/21 = 60

yields

v = 63

.

Step-by-Step Solution

1

Define distances for each leg of the trip in terms of total distance

D

.

Leg 1 distance

d_1 = 0.40D

. Remaining distance is

D - 0.40D = 0.60D

. Leg 2 distance

d_2 = 0.50 \times 0.60D = 0.30D

. Leg 3 distance

d_3 = 0.60D - 0.30D = 0.30D

.

The problem specifies percentages of remaining distance, requiring step-by-step subtraction of completed distance.

2

Determine the speed for each leg in terms of

v

.

Leg 1 speed

v_1 = v

. Leg 2 speed

v_2 = v(1 - 0.25) = 0.75v

. Leg 3 speed

v_3 = v(1 + 0.20) = 1.20v

.

Percentage increases and decreases are applied to the base rate

v

.

3

Calculate time spent on each leg (

t = \frac{d}{v}

) and sum for total time

T

.

t_1 = \frac{0.40D}{v}

,

t_2 = \frac{0.30D}{0.75v} = \frac{0.40D}{v}

,

t_3 = \frac{0.30D}{1.20v} = \frac{0.25D}{v}

. Total time

T = \frac{0.40D + 0.40D + 0.25D}{v} = \frac{1.05D}{v} = \frac{21D}{20v}

.

Average speed requires total distance divided by total time.

4

Set up the average speed equation and solve for

v

.

Average speed

= \frac{D}{T} = \frac{D}{\frac{21D}{20v}} = \frac{20v}{21} = 60 \implies 20v = 1260 \implies v = 63

.

Equating the algebraic average speed expression to the given value of 60 mph yields the value of

v

.

Key Concept

Weighted Average Speed in Multi-Phase Motion Problems

Question 1477Question →

In right triangle $ABC$ , the lengths of the two legs perpendicular to each other are $9$ and $12$ . What is the perimeter of triangle $ABC$ ?

21

30

36

42

54

Show answer & explanation

Answer: $36$

Answer

The perimeter of triangle

ABC

is

36

.

Using the Pythagorean theorem, the hypotenuse length is

\sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15

. Summing the two legs (

9

and

12

) with the hypotenuse (

15

) gives a total perimeter of

36

.

Step-by-Step Solution

1

Calculate the length of the hypotenuse using the Pythagorean theorem.

Hypotenuse

c = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15

.

For any right triangle with legs

a

and

b

and hypotenuse

c

,

a^2 + b^2 = c^2

holds.

2

Sum the lengths of all three sides to find the perimeter.

Perimeter

= 9 + 12 + 15 = 36

.

The perimeter of a polygon is the total distance around its outer boundary.

Key Concept

Pythagorean Theorem (

a^2 + b^2 = c^2

) and Perimeter of Triangles

Estimated Time:45s

Question 1478Question →

For all non-zero real numbers $x$ and $y$ , the custom operation $\diamond$ is defined by $x \diamond y = \frac{x^2 + y^2}{xy}$ . Which of the following statements must be true for all non-zero real numbers $a$ and $b$ ? Select all such statements.

Select all that apply

a \diamond b = b \diamond a

a \diamond (-a) = -2

(2a) \diamond (2b) = 2(a \diamond b)

a \diamond a = 1

Show answer & explanation

Answer: $a \diamond b = b \diamond a$ ; $a \diamond (-a) = -2$

Answer

The correct statements are that

a \diamond b = b \diamond a

and that

a \diamond (-a) = -2

.

The operation is symmetric with respect to

a

and

b

, making

a \diamond b = b \diamond a

true. Substituting

-a

into the operation yields

\frac{2a^2}{-a^2} = -2

, which makes

a \diamond (-a) = -2

true as well.

Step-by-Step Solution

1

Evaluate the commutative statement

a \diamond b

a \diamond b = \frac{a^2 + b^2}{ab}

and

b \diamond a = \frac{b^2 + a^2}{ba} = \frac{a^2 + b^2}{ab}

.

Addition and multiplication of real numbers are commutative, so the expressions are identical.

2

Evaluate the statement

a \diamond (-a)

a \diamond (-a) = \frac{a^2 + (-a)^2}{a(-a)} = \frac{a^2 + a^2}{-a^2} = \frac{2a^2}{-a^2} = -2

.

Squaring a negative quantity

(-a)^2

yields positive

a^2

, while the denominator evaluates to

-a^2

.

3

Evaluate the scaling statement

(2a) \diamond (2b)

(2a) \diamond (2b) = \frac{4a^2 + 4b^2}{4ab} = \frac{a^2 + b^2}{ab} = a \diamond b

.

The factor of 4 in the numerator and denominator cancels out, showing

(2a) \diamond (2b) = a \diamond b

, which is not equal to

2(a \diamond b)

.

4

Evaluate the self-operation statement

a \diamond a

a \diamond a = \frac{a^2 + a^2}{a^2} = \frac{2a^2}{a^2} = 2

.

Summing identical squared terms in the numerator produces

2a^2

, which divides by

a^2

to give 2.

Key Concept

Evaluating custom binary operations by algebraic substitution and simplifying terms.

Question 1479Question →

If $x$ is a real number that satisfies the inequality $|3 - 2x| \le 7$ , what is the maximum possible value of $5 - 3x$ ?

-10

-1

5

11

20

Show answer & explanation

Answer: 11

Answer

The maximum possible value of

5 - 3x

is

11

.

Solving

|3 - 2x| \le 7

gives

-7 \le 3 - 2x \le 7

. Subtracting

3

yields

-10 \le -2x \le 4

. Dividing by

-2

and flipping the inequality signs gives

-2 \le x \le 5

. Because

5 - 3x

has a negative coefficient for

x

, it decreases as

x

increases. Therefore, the maximum value of

5 - 3x

occurs at the smallest value in the domain,

x = -2

. Substituting

x = -2

into

5 - 3x

gives

5 - 3(-2) = 11

.

Step-by-Step Solution

1

Rewrite the absolute value inequality

|3 - 2x| \le 7

as a compound inequality.

-7 \le 3 - 2x \le 7

An absolute value inequality of the form

|u| \le k

(where

k \ge 0

) is equivalent to

-k \le u \le k

.

2

Isolate

x

by subtracting

3

from all parts of the inequality and then dividing by

-2

.

-10 \le -2x \le 4 \implies 5 \ge x \ge -2

, which is equivalent to

-2 \le x \le 5

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

3

Determine which value of

x

in the interval

-2 \le x \le 5

maximizes the linear expression

5 - 3x

.

The expression reaches its maximum at the minimum bound

x = -2

.

Since the coefficient of

x

is negative (

-3

), smaller values of

x

result in larger values for

5 - 3x

.

4

Substitute

x = -2

into

5 - 3x

.

5 - 3(-2) = 5 + 6 = 11

Evaluating the linear expression at the lower bound yields its maximum value.

Key Concept

Solving linear absolute value inequalities and optimizing linear expressions over a closed interval.

Estimated Time:1m 30s

Question 1480Question →

In triangle $ABC$ , point $D$ lies on side $AC$ such that segment $BD$ is perpendicular to side $AC$ . The length of segment $AD$ is $9$ and the length of segment $DC$ is $16$ . If the ratio of the length of side $AB$ to the length of side $BC$ is $3 : 4$ , what is the area of triangle $ABC$ ?

72

96

150

300

1,800

Show answer & explanation

Answer: 150

Answer

150

Applying the Pythagorean theorem to right triangles

ABD

and

CBD

gives

AB^2 = 81 + BD^2

and

BC^2 = 256 + BD^2

. Squaring the ratio

\frac{AB}{BC} = \frac{3}{4}

yields

\frac{81 + BD^2}{256 + BD^2} = \frac{9}{16}

. Cross-multiplying gives

1296 + 16 BD^2 = 2304 + 9 BD^2

, so

7 BD^2 = 1008 \implies BD = 12

. The base

AC = 9 + 16 = 25

, so the area of triangle

ABC

is

\frac{1}{2} \times 25 \times 12 = 150

.

Step-by-Step Solution

1

Set up expressions for side lengths

AB

and

BC

using the Pythagorean theorem on right triangles

ABD

and

CBD

.

AB^2 = 9^2 + BD^2 = 81 + BD^2

and

BC^2 = 16^2 + BD^2 = 256 + BD^2

.

Segment

BD

is an altitude perpendicular to

AC

, dividing triangle

ABC

into two right triangles.

2

Use the given side ratio

\frac{AB}{BC} = \frac{3}{4}

to solve for the height

BD

.

\frac{81 + BD^2}{256 + BD^2} = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \implies 16(81 + BD^2) = 9(256 + BD^2) \implies 7 BD^2 = 1008 \implies BD^2 = 144 \implies BD = 12

.

Squaring both sides of the ratio allows substitution of the expressions for

AB^2

and

BC^2

.

3

Determine the total length of base

AC

and compute the area of triangle

ABC

.

AC = AD + DC = 9 + 16 = 25

. Area

= \frac{1}{2} \times AC \times BD = \frac{1}{2} \times 25 \times 12 = 150

.

The area of a triangle is given by half the product of its base and corresponding altitude.

Key Concept

Properties of altitudes in triangles, Pythagorean theorem, and ratio setup for area determination.

Estimated Time:2m 0s

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