All practice questions

For all non-zero real numbers $x$ and $y$ , the expression

\frac{(x^2 y^{-3})^{-2} (x^{-1} y^4)^3}{(x^3 y^{-2})^d}

can be written in the form $x^p y^q$ , where $p$ and $q$ are integers. If $q = 2p$ , what is the value of $d$ ?

Show answer & explanation

Answer: -4

Answer

-4

Applying the rules of exponents yields the simplified expression

x^{-7-3d} y^{18+2d}

. Setting the exponent of

y

equal to twice the exponent of

x

gives the equation

18+2d = 2(-7-3d)

, which solves to

d = -4

.

Step-by-Step Solution

1

Apply the power of a power rule to the terms in the numerator.

(x^2 y^{-3})^{-2} = x^{-4} y^6

and

(x^{-1} y^4)^3 = x^{-3} y^{12}

To raise a power to another power, multiply the exponents:

(u^m)^n = u^{mn}

.

2

Multiply the simplified terms in the numerator together.

x^{-4} y^6 \cdot x^{-3} y^{12} = x^{-7} y^{18}

To multiply powers with the same base, add the exponents:

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

.

3

Apply the power of a power rule to the denominator.

(x^3 y^{-2})^d = x^{3d} y^{-2d}

Distribute the exponent

d

to both variables inside the parentheses by multiplying the exponents.

4

Divide the numerator by the denominator.

\frac{x^{-7} y^{18}}{x^{3d} y^{-2d}} = x^{-7-3d} y^{18-(-2d)} = x^{-7-3d} y^{18+2d}

To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator:

\frac{u^m}{u^n} = u^{m-n}

.

5

Set up the linear equation for

d

using

q = 2p

and solve.

18+2d = 2(-7-3d) \implies 18+2d = -14-6d \implies 8d = -32 \implies d = -4

The problem states the relationship between the final exponents is

q = 2p

, where

p = -7-3d

and

q = 18+2d

.

Key Concept

Properties of exponents (product, quotient, and power rules) combined with solving a linear equation.

Question 1862Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined as $z = \frac{a + bi}{1 - 2i}$ , where $a$ and $b$ are real numbers. If $z = 4 + 3i$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a + b

is 5.

To find the value of

a + b

, we start with the equation

\frac{a + bi}{1 - 2i} = 4 + 3i

. Multiplying both sides by the denominator gives

a + bi = (4 + 3i)(1 - 2i)

. Expanding the right side using the distributive property, we get

a + bi = 4(1) + 4(-2i) + 3i(1) + 3i(-2i) = 4 - 8i + 3i - 6i^2

. Substituting

i^2 = -1

simplifies the expression to

4 - 5i - 6(-1) = 4 - 5i + 6 = 10 - 5i

. By comparing the real and imaginary parts of both sides, we find that

a = 10

and

b = -5

. The sum of these two values is

a + b = 10 + (-5) = 5

.

Step-by-Step Solution

1

Isolate the numerator by multiplying both sides by the denominator.

a + bi = (4 + 3i)(1 - 2i)

To solve for the variables

a

and

b

in the numerator, we clear the fraction by multiplying by the denominator.

2

Expand the product of the two complex numbers.

a + bi = 4 - 8i + 3i - 6i^2

Distribute each term of the first binomial to each term of the second binomial.

3

Simplify the expression using the definition of

i^2

.

a + bi = 10 - 5i

Since

i^2 = -1

, the term

-6i^2

becomes

+6

. Combine the real parts (

4 + 6 = 10

) and imaginary parts (

-8i + 3i = -5i

).

4

Equate the components and calculate

a + b

.

a = 10

,

b = -5

, and

a + b = 5

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Therefore,

a = 10

and

b = -5

. Summing these yields

10 + (-5) = 5

.

Key Concept

Equality and multiplication of complex numbers

Question 1863Question →

A weather balloon's altitude in meters after $t$ minutes is given by the function $A(t) = 100 + 50t$ . The air temperature in degrees Celsius at an altitude of $a$ meters is modeled by the function $T(a) = 25 - 0.02a$ . What is the temperature of the air surrounding the balloon, in degrees Celsius, after $2$ minutes?

Show answer & explanation

Answer: 21

Answer

The temperature of the air surrounding the balloon after 2 minutes is 21 degrees Celsius.

The temperature of the air surrounding the balloon after 2 minutes is found by first calculating the balloon's altitude,

A(2) = 100 + 50(2) = 200

meters, and then using this altitude to evaluate the temperature function,

T(200) = 25 - 0.02(200) = 21

.

Step-by-Step Solution

1

Calculate the altitude of the weather balloon at

t = 2

minutes using the function

A(t)

.

A(2) = 200

meters

To find the temperature surrounding the balloon, we must first determine its altitude at the given time of 2 minutes.

2

Substitute the altitude of 200 meters into the temperature function

T(a)

.

T(200) = 21

degrees Celsius

The temperature function depends on the altitude, so evaluating

T(200)

yields the temperature at that height.

Key Concept

Evaluating composite functions in a real-world context.

Question 1864Question →

Which of the following inequality expressions represents the complete set of real values of $p$ that satisfy the inequality $|7 - 2p| < 13$ ?

p < -3

or

p > 10

p > -3

-3 < p < 10

-\frac{13}{5} < p < \frac{13}{5}

-10 < p < 3

Show answer & explanation

Answer: $-3 < p < 10$

Answer

-3 < p < 10

To solve

|7 - 2p| < 13

, write it as the compound inequality

-13 < 7 - 2p < 13

. Subtracting

7

from all parts gives

-20 < -2p < 6

. Dividing by

-2

and reversing the inequality signs yields

-3 < p < 10

.

Step-by-Step Solution

1

Write the absolute value inequality as a compound inequality.

-13 < 7 - 2p < 13

An absolute value inequality of the form

|x| < c

is equivalent to

-c < x < c

.

2

Subtract

7

from all three parts of the compound inequality.

-20 < -2p < 6

To isolate the variable term, we perform the inverse operation of adding

7

by subtracting

7

from each part.

3

Divide all parts by

-2

and reverse the inequality signs.

10 > p > -3

, which simplifies to

-3 < p < 10

Dividing an inequality by a negative number requires reversing the direction of the inequality signs to preserve the truth of the statement.

Key Concept

Solving absolute value inequalities of the form

|ax + b| < c

Question 1865Question →

A company allocates its monthly advertising budget between online advertisements and print media. Last month, the company spent $300$ more than half of its total budget on online advertisements, and $\frac{1}{6}$ of its total budget on print media. The remaining $700$ of the budget was spent on administrative fees. If $B$ represents the total budget in dollars, what is the value of $\frac{B}{10} + 150$ ?

270

300

450

405

315

Show answer & explanation

Answer: 450

Answer

The correct value is 450.

Evaluating the total budget equation yields

B = 3000

. Substituting this value into the expression

\frac{B}{10} + 150

gives

450

. This is correct because the individual allocations (online ads, print media, and administrative fees) sum to the total budget, and the fraction arithmetic is performed correctly.

Step-by-Step Solution

1

Define the variables and write the expressions for each category of expenses.

Let

B

be the total budget. Online advertisements budget is

\frac{1}{2}B + 300

, print media budget is

\frac{1}{6}B

, and administrative fees are

700

.

Translating verbal descriptions into algebraic expressions is necessary to build the equation.

2

Set up the linear equation representing the sum of all expenses equaling the total budget

B

.

B = \left(\frac{1}{2}B + 300\right) + \frac{1}{6}B + 700

The total budget is the sum of its individual components.

3

Group like terms and solve for the total budget

B

.

Combine constant terms:

300 + 700 = 1000

. Combine fraction terms:

\frac{1}{2}B + \frac{1}{6}B = \frac{3}{6}B + \frac{1}{6}B = \frac{4}{6}B = \frac{2}{3}B

. The equation becomes

B = \frac{2}{3}B + 1000

. Subtracting

\frac{2}{3}B

from both sides gives

\frac{1}{3}B = 1000

, which yields

B = 3000

.

This isolates the variable

B

to find the total budget value.

4

Evaluate the required expression

\frac{B}{10} + 150

using the solved value of

B

.

\frac{3000}{10} + 150 = 300 + 150 = 450

The question asks for the value of this specific expression rather than

B

itself.

Key Concept

Solving linear equations in one variable, including translating word problems with fractional terms and evaluating algebraic expressions.

Alternative Method

Instead of solving the equation algebraically, we could test the options to find the total budget

B

. Since each option represents the value of

V = \frac{B}{10} + 150

, we can express

B

as

B = 10(V - 150)

. For the correct option of

450

, we get

B = 10(450 - 150) = 3000

. Substituting

B = 3000

back into the original word problem description: half the budget plus

300

is

1500 + 300 = 1800

; one-sixth of the budget is

500

; the remaining is

3000 - 1800 - 500 = 700

, which matches the given administrative fees.

Estimated Time:2m 0s

Question 1866Question →

An online retailer determines that the cost to ship a package of weight $w$ pounds is given by the linear expression $C(w) = kw + b$ , where $k$ and $b$ are constants. Shipping a $3$ -pound package costs $11.50$ dollars, and shipping an $8$ -pound package costs $24.00$ dollars. If the total shipping cost for two packages is $47.00$ dollars, and one of the packages weighs $5$ pounds, what is the weight, in pounds, of the other package?

Show answer & explanation

Answer: 10.6

Answer

The weight of the other package is 10.6 pounds.

The correct weight of 10.6 pounds is found by setting up a linear cost function

C(w) = 2.5w + 4

using the two data points, calculating the cost of the 5-pound package as

16.50

dollars, subtracting this from the total cost of

47.00

dollars to get

30.50

dollars, and solving

2.5w + 4 = 30.50

for the weight.

Step-by-Step Solution

1

Set up the linear system from the given costs

3k + b = 11.50

and

8k + b = 24.00

To determine the relationship between weight and shipping cost.

2

Solve for the slope

k

k = 2.50

Subtracting the first equation from the second eliminates

b

.

3

Solve for the intercept

b

b = 4.00

Substitute

k = 2.50

back into the first equation.

4

Determine the cost of the 5-pound package

C(5) = 16.50

dollars

Evaluate the linear expression

2.50(5) + 4.00

.

5

Determine the remaining cost for the second package

C(w_2) = 30.50

dollars

Subtract the cost of the first package from the total cost (

47.00 - 16.50

).

6

Solve the linear equation for the second package's weight

w_2 = 10.6

Solve

2.50w_2 + 4.00 = 30.50

for

w_2

.

Key Concept

Solving Linear Equations

Question 1867Question →

Jordan starts a walk with $2{}500$ steps already recorded on a fitness tracker. Jordan then walks at a constant rate of $120$ steps per minute. If the fitness tracker shows a total of $7{}900$ steps at the end of the walk, for how many minutes did Jordan walk?

Show answer & explanation

Answer: 45

Answer

Jordan walked for

45

minutes.

The correct answer is

45

. The scenario describes a linear relationship with a constant rate of

120

steps per minute and a starting baseline of

2{}500

steps. Let

m

represent the number of minutes Jordan walked. The equation is

2{}500 + 120m = 7{}900

. Subtracting

2{}500

from both sides gives

120m = 5{}400

, and dividing by

120

yields

m = 45

.

Step-by-Step Solution

1

Set up the algebraic equation based on the word problem details.

2{}500 + 120m = 7{}900

The total steps are the sum of the starting steps (

2{}500

) and the product of the rate (

120

steps/min) and time (

m

minutes).

2

Isolate the variable term by subtracting

2{}500

from both sides of the equation.

120m = 5{}400

Subtracting the initial steps gives the steps accumulated solely during the walk.

3

Solve for

m

by dividing both sides of the equation by

120

.

m = 45

Dividing the total steps walked by the rate per minute yields the duration of the walk in minutes.

Key Concept

Translating verbal descriptions of constant rates and starting values into linear equations

Alternative Method

Solve arithmetically by subtracting the baseline steps from the final count (

7{}900 - 2{}500 = 5{}400

steps) and dividing the remaining steps by the walking rate (

5{}400 \div 120 = 45

minutes).

Estimated Time:45s

Question 1868Question →

Match each unsimplified algebraic expression on the left with its equivalent simplified form on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings match

3x(x - 2y) - 2x(x - 3y)

to

x^2

;

2(x^2 - xy) - 3(xy - y^2)

to

2x^2 - 5xy + 3y^2

;

x^2(x - 2y) - x(x^2 - xy)

to

-x^2y

; and

(x + y)^2 - (x - y)^2

to

4xy

.

Each unsimplified expression is correctly matched to its simplified equivalent by expanding parenthetical terms (taking care to distribute negative signs) and combining like terms.

Step-by-Step Solution

1

Distribute and combine like terms for the first expression

3x(x - 2y) - 2x(x - 3y)

.

The expression simplifies to

x^2

.

First distribute the coefficients to get

3x^2 - 6xy - 2x^2 + 6xy

. Then combine

3x^2 - 2x^2 = x^2

and

-6xy + 6xy = 0

.

2

Distribute and combine like terms for the second expression

2(x^2 - xy) - 3(xy - y^2)

.

The expression simplifies to

2x^2 - 5xy + 3y^2

.

Distribute the coefficients to get

2x^2 - 2xy - 3xy + 3y^2

. Note that distributing the negative sign of

-3

to

-y^2

results in

+3y^2

. Then combine

-2xy - 3xy = -5xy

.

3

Distribute and combine like terms for the third expression

x^2(x - 2y) - x(x^2 - xy)

.

The expression simplifies to

-x^2y

.

Distribute to get

x^3 - 2x^2y - x^3 + x^2y

. Note that distributing

-x

to

-xy

gives

+x^2y

. The

x^3 - x^3

terms cancel, leaving

-2x^2y + x^2y = -x^2y

.

4

Expand and simplify the fourth expression

(x + y)^2 - (x - y)^2

.

The expression simplifies to

4xy

.

Expand both squared binomials:

(x^2 + 2xy + y^2) - (x^2 - 2xy + y^2)

. Distribute the negative sign to get

x^2 + 2xy + y^2 - x^2 + 2xy - y^2

. Combine terms to cancel

x^2

and

y^2

, leaving

4xy

.

Key Concept

Simplifying Expressions and Combining Like Terms

Estimated Time:1m 30s

Question 1869Question →

For any positive real number $y$ , the expression $\frac{(y^3)^{1/2}}{y^{2/3}}$ is equivalent to which of the following?

y^{7/6}

y^{1/6}

y^{5/6}

y^{17/6}

y^{13/6}

Show answer & explanation

Answer: $y^{5/6}$

Answer

The simplified expression is

y^{5/6}

Applying the power of a power rule to the numerator gives

(y^3)^{1/2} = y^{3/2}

. Then, applying the quotient rule to divide by

y^{2/3}

requires subtracting the exponents:

\frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}

. This results in the equivalent expression

y^{5/6}

.

Step-by-Step Solution

1

Apply the power of a power rule to the numerator

(y^3)^{1/2}

.

y^{3 \cdot 1/2} = y^{3/2}

When raising a power to a power, multiply the exponents.

2

Apply the quotient of powers rule to divide

y^{3/2}

by

y^{2/3}

.

y^{3/2 - 2/3}

When dividing powers with the same base, subtract the exponent in the denominator from the exponent in the numerator.

3

Subtract the fractions in the exponent by finding a common denominator.

y^{9/6 - 4/6} = y^{5/6}

A common denominator for 2 and 3 is 6. Rewrite the fractions and subtract their numerators.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 1870Question →

If $x$ satisfies the equation $\frac{3(x - 2)}{5} - \frac{2x - 1}{3} = \frac{1}{15}$ , what is the value of the expression $2x + 7$ ?

Show answer & explanation

Answer: -21

Answer

The final value of the expression is

-21

.

Solving the equation by first multiplying both sides by the least common denominator of 15 yields the simplified equation

9(x - 2) - 5(2x - 1) = 1

. Distributing and combining terms yields

-x - 13 = 1

, which gives

x = -14

. Substituting this value into the expression

2x + 7

yields

-21

.

Step-by-Step Solution

1

Multiply the entire equation by the least common multiple of the denominators (

15

) to eliminate all fractions.

9(x - 2) - 5(2x - 1) = 1

Multiplying by the least common multiple of 5 and 3 eliminates the fractions and simplifies the equation.

2

Distribute the coefficients (

9

and

-5

) to their respective terms inside the parentheses.

9x - 18 - 10x + 5 = 1

Distributive property allows us to remove parentheses. Note that distributing

-5

to

-1

results in

+5

.

3

Combine the variable terms (

9x

and

-10x

) and the constant terms (

-18

and

5

) on the left side of the equation.

-x - 13 = 1

Combining like terms simplifies the expression to prepare for isolating the variable.

4

Isolate the variable term

-x

by adding

13

to both sides, then solve for

x

by multiplying by

-1

.

x = -14

Adding

13

yields

-x = 14

, and multiplying by

-1

isolates

x

to find its value.

5

Substitute

x = -14

into the given expression

2x + 7

.

2(-14) + 7 = -21

The question asks for the value of the expression

2x + 7

, not just

x

itself.

Key Concept

Solving multi-step linear equations involving fractions, distributive property with negative signs, and evaluating algebraic expressions.

Alternative Method

Instead of multiplying by the least common multiple first, you can separate the fractions:

\frac{3}{5}x - \frac{6}{5} - \frac{2}{3}x + \frac{1}{3} = \frac{1}{15}

. Combining the

x

terms gives

(\frac{9}{15} - \frac{10}{15})x = -\frac{1}{15}x

. Combining the constant terms gives

-\frac{18}{15} + \frac{5}{15} = -\frac{13}{15}

. The equation becomes

-\frac{1}{15}x - \frac{13}{15} = \frac{1}{15}

. Multiplying the entire equation by

15

yields

-x - 13 = 1

, which gives

x = -14

, and substituting into

2x + 7

yields

-21

.

Estimated Time:2m 0s

Question 1871Question →

A digital marketing firm runs advertisements on two platforms: SocialMedia and SearchEngine. The cost to run an advertisement on SocialMedia is $15$ dollars per day, and the cost to run an advertisement on SearchEngine is $25$ dollars per day. Last month, the firm ran advertisements on both platforms for a combined total of $60$ days. The total amount spent on SocialMedia advertisements was $350$ dollars more than half the total amount spent on SearchEngine advertisements. For how many days last month did the firm run advertisements on SocialMedia?

Show answer & explanation

Answer: 40

Answer

The firm ran advertisements on SocialMedia for 40 days.

Setting up the system of equations based on the problem description gives

S + E = 60

and

15S = 0.5(25E) + 350

. Substituting the first equation into the second yields

15S = 12.5(60 - S) + 350

. Solving this linear equation results in

S = 40

days.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

S

be the number of days the firm ran advertisements on SocialMedia, and

E

be the number of days they ran advertisements on SearchEngine.

Establishing variables is the first step in translating a word problem into algebraic equations.

2

Express the relationship for the total number of days.

S + E = 60

, which simplifies to

E = 60 - S

.

This allows us to express one variable in terms of the other, making it easier to solve the system by substitution.

3

Translate the cost relationship statement into an algebraic equation.

15S = 12.5E + 350

The cost of running advertisements on SocialMedia is

15S

. The cost on SearchEngine is

25E

. Half of the SearchEngine cost is

12.5E

. Adding

350

to half of the SearchEngine cost gives the SocialMedia cost.

4

Substitute the expression for

E

into the cost equation and solve for

S

.

15S = 12.5(60 - S) + 350 \Rightarrow 15S = 750 - 12.5S + 350 \Rightarrow 27.5S = 1100 \Rightarrow S = 40

.

Solving the resulting linear equation yields the number of days spent on SocialMedia ads.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:2m 0s

Question 1872Question →

A rectangular region has a width of $2x - 3$ meters and a length of $3x + 1$ meters. A square piece with a side length of $x - 2$ meters is removed from the region. The area, in square meters, of the remaining region can be expressed in the standard polynomial form $Ax^2 + Bx + C$ , where $A$ , $B$ , and $C$ are integers. What is the value of the coefficient $B$ ?

Show answer & explanation

Answer: -3

Answer

The coefficient of the linear term,

B

, is

-3

.

Subtracting the area of the square,

x^2 - 4x + 4

, from the area of the rectangle,

6x^2 - 7x - 3

, yields

5x^2 - 3x - 7

. Thus, the coefficient

B

of the

x

term is

-3

.

Step-by-Step Solution

1

Calculate the area of the rectangle.

Area =

6x^2 - 7x - 3

The area of a rectangle is found by multiplying its length and width:

(2x - 3)(3x + 1) = 6x^2 + 2x - 9x - 3 = 6x^2 - 7x - 3

.

2

Calculate the area of the square.

Area =

x^2 - 4x + 4

The area of a square is the square of its side length:

(x - 2)^2 = (x - 2)(x - 2) = x^2 - 4x + 4

.

3

Subtract the square's area from the rectangle's area.

Remaining Area =

5x^2 - 3x - 7

Subtracting the area of the removed square from the total area requires distributing the negative sign to each term of the square's polynomial:

(6x^2 - 7x - 3) - (x^2 - 4x + 4) = 6x^2 - 7x - 3 - x^2 + 4x - 4 = 5x^2 - 3x - 7

.

4

Identify the coefficient

B

.

B = -3

In the standard quadratic form

Ax^2 + Bx + C

, the coefficient of the linear term

x

is

B

, which corresponds to

-3

in the polynomial

5x^2 - 3x - 7

.

Key Concept

Operations on Polynomials (multiplication, squaring binomials, and subtraction with negative sign distribution)

Question 1873Question →

For what value of $y$ is the equation $3(y - 4) = 5y + 2$ true?

Show answer & explanation

Answer: -7

Answer

The value of

y

that satisfies the equation is

-7

.

Distributing the 3 yields

3y - 12 = 5y + 2

. Subtracting

3y

from both sides gives

-12 = 2y + 2

. Subtracting 2 from both sides gives

-14 = 2y

. Dividing by 2 results in

y = -7

.

Step-by-Step Solution

1

Distribute the 3 on the left side of the equation.

3y - 12 = 5y + 2

To simplify the expression by expanding the parentheses.

2

Subtract

3y

from both sides of the equation.

-12 = 2y + 2

To collect the variable terms on the right side of the equation.

3

Subtract 2 from both sides of the equation.

-14 = 2y

To isolate the variable term.

4

Divide both sides by 2.

y = -7

To solve for

y

.

Key Concept

Solving linear equations by distributing and isolating the variable.

Estimated Time:45s

Question 1874Question →

A commercial bakery has two bread-kneading machines, Machine X and Machine Y. Machine X can knead dough at a constant rate of $p$ pounds per hour. Machine Y's kneading rate is $14$ pounds per hour more than half the kneading rate of Machine X. On a busy morning, Machine X starts kneading and operates for exactly $5$ hours. Machine Y starts operating $1$ hour after Machine X starts and operates for the next $4$ hours. Together, the two machines knead a total of $336$ pounds of dough. What is the total number of pounds of dough kneaded by Machine Y?

34

116

136

148

200

Show answer & explanation

Answer: 136

Answer

The total number of pounds of dough kneaded by Machine Y is 136.

To find the total dough kneaded by Machine Y, we first define the rates of both machines. Let Machine X's rate be

p

pounds per hour. Machine Y's rate is

14

more than half of Machine X's rate, which is written as

0.5p + 14

pounds per hour. Machine X runs for

5

hours, producing

5p

pounds of dough. Machine Y runs for

4

hours, producing

4(0.5p + 14) = 2p + 56

pounds of dough. Setting their sum equal to the total of

336

gives the equation

7p + 56 = 336

, which simplifies to

7p = 280

and yields

p = 40

. Machine Y's total work is then

4(0.5(40) + 14) = 136

pounds.

Step-by-Step Solution

1

Define the variable for Machine X's rate and write the expression for Machine Y's rate based on the text.

Machine X's rate is

p

pounds per hour. Machine Y's rate is

0.5p + 14

pounds per hour.

We need to express both rates in terms of a single variable to set up the equation.

2

Determine the operating duration for each machine.

Machine X operates for

5

hours. Machine Y operates for

4

hours.

Machine X starts first and runs for

5

hours. Machine Y starts

1

hour later and operates for the remaining

4

hours.

3

Write the equation for the total pounds of dough kneaded by both machines combined.

5p + 4(0.5p + 14) = 336

Total work is the sum of the work done by Machine X (rate times time) and Machine Y (rate times time).

4

Solve the equation for the variable

p

.

5p + 2p + 56 = 336 \implies 7p + 56 = 336 \implies 7p = 280 \implies p = 40

.

Distribute the

4

and combine like terms to isolate

p

.

5

Calculate the total work done specifically by Machine Y.

Machine Y's total work =

4(0.5(40) + 14) = 4(20 + 14) = 4(34) = 136

pounds.

The question asks for the total pounds of dough kneaded by Machine Y, which is its rate multiplied by its operating time.

Key Concept

Translating verbal descriptions of rates and times into algebraic equations and solving them.

Alternative Method

Instead of solving for

p

first, we can write the equation directly in terms of Machine Y's work. Let

y

be the work done by Machine Y. Since Machine Y worked for

4

hours, its rate is

\frac{y}{4}

. This rate is

14

more than half of Machine X's rate, so

\frac{y}{4} = 0.5R_X + 14 \implies R_X = 2(\frac{y}{4} - 14) = \frac{y}{2} - 28

. Machine X's work is

5 R_X = 5(\frac{y}{2} - 28) = 2.5y - 140

. Since the total work is

336

, we have

(2.5y - 140) + y = 336 \implies 3.5y = 476 \implies y = 136

.

Estimated Time:3m 0s

Question 1875Question →

Let the functions $f$ and $g$ be defined by $f(x) = \sqrt{25 - x^2}$ and $g(x) = \frac{1}{\sqrt{x^2 - 9}}$ for all real numbers $x$ where the expressions are defined. What is the number of integers in the domain of the composite function $h(x) = g(f(x))$ ?

Show answer & explanation

Answer: 7

Answer

The number of integers in the domain of the composite function

h(x) = g(f(x))

is 7.

The composite function

h(x) = g(f(x))

is defined when

x

is in the domain of

f

and

f(x)

is in the domain of

g

. The domain of

f(x) = \sqrt{25 - x^2}

is

[-5, 5]

. The domain of

g(u) = \frac{1}{\sqrt{u^2 - 9}}

is

u > 3

or

u < -3

. Substituting

f(x)

for

u

gives

\sqrt{25 - x^2} > 3

, which simplifies to

x^2 < 16

or

-4 < x < 4

. The intersection of

[-5, 5]

and

(-4, 4)

is

(-4, 4)

. The integers in this interval are

-3, -2, -1, 0, 1, 2, 3

, which total 7 integers.

Step-by-Step Solution

1

Determine the domain of the inner function

f(x) = \sqrt{25 - x^2}

.

The domain is

[-5, 5]

.

The term inside the square root must be greater than or equal to zero for the function to yield real values:

25 - x^2 \ge 0

.

2

Determine the domain of the outer function

g(u) = \frac{1}{\sqrt{u^2 - 9}}

.

The domain is

(-\infty, -3) \cup (3, \infty)

.

The expression inside the square root in the denominator must be strictly positive:

u^2 - 9 > 0

.

3

Apply the domain constraint of the outer function to the outputs of the inner function.

-4 < x < 4

.

We require

f(x) > 3

or

f(x) < -3

. Since the range of

f(x)

is non-negative,

f(x) < -3

has no solutions. Thus, we solve

\sqrt{25 - x^2} > 3

, which squares to

25 - x^2 > 9

, or

x^2 < 16

.

4

Find the intersection of the inner function's domain and the composite constraint.

The composite domain is

(-4, 4)

.

The input

x

must satisfy both the domain of

f

(

-5 \le x \le 5

) and the composition constraint (

-4 < x < 4

).

5

List and count the integers within the composite domain

(-4, 4)

.

There are 7 integers.

The integers strictly between

-4

and

4

are

-3, -2, -1, 0, 1, 2, 3

.

Key Concept

Domain of a composite function

Estimated Time:2m 0s

Question 1876Question →

A box contains only red, blue, and green marbles. The ratio of red marbles to blue marbles is $2:3$ , and the ratio of blue marbles to green marbles is $3:5$ . If one marble is drawn at random from the box, the probability of drawing a red marble is $P$ . If $6$ green marbles are added to the box and no other marbles are removed, the probability of drawing a red marble becomes $Q$ . Given that $P - Q = \frac{3}{115}$ , how many blue marbles were originally in the box?

3

8

12

15

20

Show answer & explanation

Answer: 12

Answer

12

The correct answer is 12. By expressing the original counts of red, blue, and green marbles as

2k

,

3k

, and

5k

, the total number of marbles is

10k

. The initial probability of drawing a red marble is

P = \frac{2k}{10k} = \frac{1}{5}

. When 6 green marbles are added, the new total becomes

10k + 6

, yielding a new probability

Q = \frac{2k}{10k + 6} = \frac{k}{5k + 3}

. Setting their difference to

\frac{3}{115}

results in

\frac{1}{5} - \frac{k}{5k + 3} = \frac{3}{115}

. Simplifying this rational equation gives

25k + 15 = 115

, which solves to

k = 4

. Therefore, the original number of blue marbles is

3k = 3(4) = 12

.

Step-by-Step Solution

1

Define the number of red, blue, and green marbles in terms of a variable

k

.

Let red marbles =

2k

, blue marbles =

3k

, and green marbles =

5k

. The initial total number of marbles is

2k + 3k + 5k = 10k

.

The given ratios are red:blue = 2:3 and blue:green = 3:5, which combine to a continuous ratio of red:blue:green = 2:3:5.

2

Express the initial probability

P

of drawing a red marble.

P = \frac{2k}{10k} = \frac{1}{5}

.

Probability is the number of favorable outcomes (red marbles) divided by the total number of outcomes.

3

Express the new probability

Q

of drawing a red marble after adding 6 green marbles.

The number of green marbles becomes

5k + 6

, so the new total is

10k + 6

. Thus,

Q = \frac{2k}{10k + 6}

.

Adding 6 green marbles increases the total marble count by 6 while the number of red marbles remains

2k

.

4

Set up the equation using the given difference

P - Q = \frac{3}{115}

and solve for

k

.

\frac{1}{5} - \frac{2k}{10k + 6} = \frac{3}{115}

\frac{1}{5} - \frac{k}{5k + 3} = \frac{3}{115}

\frac{(5k + 3) - 5k}{5(5k + 3)} = \frac{3}{115}

\frac{3}{25k + 15} = \frac{3}{115}

25k + 15 = 115 \implies 25k = 100 \implies k = 4

.

Solving the rational equation gives the multiplier

k

that determines the actual number of marbles.

5

Calculate the original number of blue marbles.

Blue marbles =

3k = 3(4) = 12

.

The question asks for the original number of blue marbles, which is represented by

3k

.

Key Concept

Basic Probability and Counting Methods

Estimated Time:3m 0s

Question 1877Question →

What is the complete set of real numbers $x$ that satisfy the inequality $\frac{2 - x}{3} - \frac{3x - 1}{4} \geq 2$ ?

x \geq -1

x \leq -\frac{19}{13}

x \geq \frac{5}{4}

x \leq -1

x \leq \frac{9}{13}

Show answer & explanation

Answer: $x \leq -1$

Answer

The complete set of real numbers satisfying the inequality is

x \leq -1

.

To solve the inequality

\frac{2 - x}{3} - \frac{3x - 1}{4} \geq 2

, we first eliminate the denominators by multiplying both sides by their least common multiple, which is

12

. This yields

4(2 - x) - 3(3x - 1) \geq 24

. Distributing the terms gives

8 - 4x - 9x + 3 \geq 24

. Combining like terms results in

11 - 13x \geq 24

. Subtracting

11

from both sides yields

-13x \geq 13

. Finally, dividing by

-13

requires reversing the inequality sign, leading to the solution

x \leq -1

.

Step-by-Step Solution

1

Multiply all terms on both sides of the inequality by

12

, which is the least common multiple of the denominators

3

and

4

.

4(2 - x) - 3(3x - 1) \geq 24

Multiplying by a positive number preserves the inequality direction while eliminating the fractions to simplify the equation.

2

Distribute the coefficients

4

and

-3

to the terms inside the parentheses.

8 - 4x - 9x + 3 \geq 24

Applying the distributive property expands the expression. Note that multiplying

-3

by

-1

results in

+3

.

3

Combine like terms on the left side of the inequality.

11 - 13x \geq 24

Grouping the constants (

8 + 3 = 11

) and the variable terms (

-4x - 9x = -13x

) simplifies the inequality.

4

Subtract

11

from both sides of the inequality.

-13x \geq 13

Isolating the variable term on the left side by moving the constant term to the right side.

5

Divide both sides of the inequality by

-13

and reverse the direction of the inequality sign.

x \leq -1

Dividing by a negative number requires reversing the inequality sign from

\geq

to

\leq

.

Key Concept

Solving linear inequalities by clearing fractions, distributing terms correctly, and reversing the inequality sign when multiplying or dividing by a negative number.

Estimated Time:2m 0s

Question 1878Question →

If $x$ and $y$ are positive real numbers, the expression

\frac{(x^{1/2} + y^{1/2})^2 - (x^{1/2} - y^{1/2})^2}{(2x^{-1/4} y^{3/4})^{-2}}

can be simplified to the form $k y^p$ . What is the value of the sum $k + p$ ?

1

10

15

17

18

Show answer & explanation

Answer: 18

Answer

18

Expanding the numerator yields

4x^{1/2}y^{1/2}

. Simplifying the denominator using exponent rules yields

\frac{1}{4}x^{1/2}y^{-3/2}

. Dividing the numerator by the denominator gives

\frac{4}{1/4} \cdot x^{1/2 - 1/2} \cdot y^{1/2 - (-3/2)} = 16y^2

, which corresponds to

k = 16

and

p = 2

. The sum is

16 + 2 = 18

.

Step-by-Step Solution

1

Expand and simplify the numerator.

4x^{1/2}y^{1/2}

Expand both squared binomials:

(x^{1/2} + y^{1/2})^2 = x + 2x^{1/2}y^{1/2} + y

and

(x^{1/2} - y^{1/2})^2 = x - 2x^{1/2}y^{1/2} + y

. Subtracting the second expression from the first yields

(x + 2x^{1/2}y^{1/2} + y) - (x - 2x^{1/2}y^{1/2} + y) = 4x^{1/2}y^{1/2}

.

2

Simplify the denominator using exponent rules.

\frac{1}{4}x^{1/2}y^{-3/2}

Apply the power of a product rule

(ab)^n = a^n b^n

to distribute the exponent of

-2

:

(2x^{-1/4}y^{3/4})^{-2} = 2^{-2} \cdot (x^{-1/4})^{-2} \cdot (y^{3/4})^{-2}

. This simplifies to

\frac{1}{4} \cdot x^{(-1/4)(-2)} \cdot y^{(3/4)(-2)} = \frac{1}{4}x^{1/2}y^{-3/2}

.

3

Divide the simplified numerator by the simplified denominator.

16y^2

Divide the coefficients and subtract the exponents of like bases:

\frac{4x^{1/2}y^{1/2}}{\frac{1}{4}x^{1/2}y^{-3/2}} = \left(\frac{4}{1/4}\right) x^{1/2 - 1/2} y^{1/2 - (-3/2)} = 16 x^0 y^{1/2 + 3/2} = 16y^2

.

4

Identify the values of

k

and

p

and calculate their sum.

18

Comparing

16y^2

to the form

k y^p

gives

k = 16

and

p = 2

. Therefore, the sum is

k + p = 16 + 2 = 18

.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 1879Question →

If $x$ is a real number, what is the sum of all distinct real solutions to the rational equation $\frac{3}{(x-2)^2} - \frac{2}{x-2} = 1$ ?

3

-3

-2

2

4

Show answer & explanation

Answer: 2

Answer

The sum of all distinct real solutions is 2.

The correct answer is obtained by multiplying the equation by the least common denominator,

(x-2)^2

, which yields the quadratic equation

x^2 - 2x - 3 = 0

. Factoring this equation gives two solutions:

3

and

-1

. Since neither value makes the original denominator equal to zero, both are valid solutions. Summing these values gives a result of

2

.

Step-by-Step Solution

1

Identify the domain constraints of the rational equation.

The denominator

x-2

cannot be zero, so the domain constraint is

x \neq 2

.

Division by zero is undefined.

2

Eliminate the denominators by multiplying the entire equation by the least common denominator, which is

(x-2)^2

.

3 - 2(x-2) = (x-2)^2

This transforms the rational equation into a polynomial equation.

3

Expand both sides of the equation and combine like terms.

3 - 2x + 4 = x^2 - 4x + 4 \implies 7 - 2x = x^2 - 4x + 4

Standard algebraic expansion simplifies the equation into a quadratic form.

4

Rearrange the quadratic equation into standard form

ax^2 + bx + c = 0

.

x^2 - 2x - 3 = 0

Setting the equation to zero allows us to solve for

x

by factoring.

5

Factor the quadratic equation.

(x-3)(x+1) = 0 \implies x = 3

or

x = -1

Factoring finds the values of

x

that satisfy the quadratic equation.

6

Verify if the potential solutions satisfy the domain constraint and calculate their sum.

Both

3

and

-1

are within the domain (

x \neq 2

). The sum is

3 + (-1) = 2

.

Extraneous solutions must be discarded before calculating the final sum.

Key Concept

Solving rational equations by clearing denominators and verifying domain constraints

Question 1880Question →

Two courier drones, Drone A and Drone B, fly in opposite directions along a straight path from the same distribution center. Drone A departs at 10:00 AM and flies at a constant speed of $40\text{ miles per hour}$ . Drone B departs from the same location at 10:30 AM and flies in the opposite direction at a constant speed of $60\text{ miles per hour}$ . At what time will the two drones be exactly $220\text{ miles}$ apart?

12:00 PM

12:12 PM

12:30 PM

12:42 PM

12:54 PM

Show answer & explanation

Answer: 12:30 PM

Answer

12:30 PM

The correct answer is 12:30 PM. To find this, define

t

as the travel time of Drone B in hours. Since Drone A departs 30 minutes (

0.5

hours) earlier, its travel time is

t + 0.5

hours. Using the relationship that the sum of the distances traveled in opposite directions equals the total distance, we set up the equation

40(t + 0.5) + 60t = 220

. Solving this yields

100t + 20 = 220

, which simplifies to

100t = 200

and

t = 2\text{ hours}

. Adding 2 hours to Drone B's departure time of 10:30 AM results in 12:30 PM.

Step-by-Step Solution

1

Define variables for the travel times of both drones relative to their departure times.

Let

t

be the number of hours Drone B travels after departing at 10:30 AM. Since Drone A departs 30 minutes (

0.5

hours) earlier at 10:00 AM, Drone A's travel time is

t + 0.5

hours.

Establishing a single variable for time allows us to write a single-variable linear equation for the total distance covered.

2

Express the distance traveled by each drone using the formula

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

and sum them to equal the total separation distance.

The distance traveled by Drone A is

40(t + 0.5)

miles, and the distance traveled by Drone B is

60t

miles. The equation is:

40(t + 0.5) + 60t = 220

.

Since the drones fly in opposite directions from the same starting point, the total distance between them is the sum of their individual distances.

3

Solve the equation for

t

.

40t + 20 + 60t = 220 \implies 100t + 20 = 220 \implies 100t = 200 \implies t = 2\text{ hours}

.

Solving for

t

gives the number of hours Drone B travels before they are

220\text{ miles}

apart.

4

Determine the final time by adding the calculated travel time to the corresponding departure time.

Drone B departs at 10:30 AM and travels for

2\text{ hours}

.

10:30\text{ AM} + 2\text{ hours} = 12:30\text{ PM}

. (Alternatively, Drone A departs at 10:00 AM and travels for

2 + 0.5 = 2.5\text{ hours}

, which also yields 12:30 PM).

Adding the elapsed time to the initial departure time gives the clock time when the condition is met.

Key Concept

Translating relative motion and time-offset word problems into linear equations

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept