All practice questions

Question 1821Question →

Which of the following is equivalent to the expression $2a(3a - b)^2 - 3b(2a^2 - ab) - a^2(18a - 15b)$ for all real values of $a$ and $b$ ?

9a^2b + 5ab^2

-33a^2b + 5ab^2

-3a^2b - ab^2

-3a^2b + 5ab^2

-3a^2b + 5ab

Show answer & explanation

Answer: $-3a^2b + 5ab^2$

Answer

The simplified expression is

-3a^2b + 5ab^2

.

The correct expression is

-3a^2b + 5ab^2

. Expanding the three parts yields

18a^3 - 12a^2b + 2ab^2

,

-6a^2b + 3ab^2

, and

-18a^3 + 15a^2b

. Summing these parts together cancels out the cubic

a^3

terms (

18a^3 - 18a^3 = 0

), combines the

a^2b

terms (

-12a^2b - 6a^2b + 15a^2b = -3a^2b

), and combines the

ab^2

terms (

2ab^2 + 3ab^2 = 5ab^2

).

Step-by-Step Solution

1

Expand the first term

2a(3a - b)^2

18a^3 - 12a^2b + 2ab^2

First expand the squared binomial

(3a - b)^2 = 9a^2 - 6ab + b^2

, then distribute the

2a

to each term.

2

Expand the second term

-3b(2a^2 - ab)

-6a^2b + 3ab^2

Distribute

-3b

to both terms inside the parentheses, ensuring that the negative sign is applied to both terms.

3

Expand the third term

-a^2(18a - 15b)

-18a^3 + 15a^2b

Distribute

-a^2

to both terms inside the parentheses, changing the sign of both terms.

4

Combine the expanded expressions and group like terms

(18a^3 - 18a^3) + (-12a^2b - 6a^2b + 15a^2b) + (2ab^2 + 3ab^2)

Grouping terms with identical variable parts allows them to be added or subtracted.

5

Perform the final simplification

-3a^2b + 5ab^2

The

a^3

terms cancel out, the

a^2b

terms combine to

-3a^2b

, and the

ab^2

terms combine to

5ab^2

.

Key Concept

Simplifying algebraic expressions by expanding parentheses, applying exponent rules, and combining like terms.

Question 1822Question →

For all real values of $x$ where the expression is defined, which of the following inequality expressions represents the complete set of values that satisfy the inequality below?

\frac{|x - 2| - 3}{|x + 1| - 2} \leq 0

-3 \leq x \leq -1

or

1 \leq x \leq 5

-1 \leq x \leq 5

-3 < x \leq -1

or

1 < x \leq 5

1 < x \leq 5

x < -3

or

-1 \leq x < 1

or

x \geq 5

Show answer & explanation

Answer: $-3 < x \leq -1$ or $1 < x \leq 5$

Answer

-3 < x \leq -1

or

1 < x \leq 5

The correct answer represents the union of the two valid cases where the numerator and denominator have opposite signs. Specifically, the numerator

|x - 2| - 3

is non-positive when

-1 \leq x \leq 5

, and the denominator

|x + 1| - 2

is positive when

x < -3

or

x > 1

. Their intersection is the interval

1 < x \leq 5

. Alternatively, the numerator is non-negative when

x \leq -1

or

x \geq 5

, and the denominator is negative when

-3 < x < 1

. Their intersection is the interval

-3 < x \leq -1

. Combining these yields the solution set

-3 < x \leq -1

or

1 < x \leq 5

.

Step-by-Step Solution

1

Analyze the conditions required for a rational expression to be non-positive.

The fraction

\frac{N}{D} \leq 0

is satisfied when the numerator

N

and denominator

D

have opposite signs, while ensuring the denominator is not equal to zero (

D \neq 0

). Thus, we must evaluate two cases: Case 1 where

N \leq 0

and

D > 0

, and Case 2 where

N \geq 0

and

D < 0

.

This establishes the logical framework for solving the rational inequality.

2

Determine the critical intervals for the numerator,

N = |x - 2| - 3

.

Solve

|x - 2| - 3 \leq 0 \implies |x - 2| \leq 3 \implies -3 \leq x - 2 \leq 3 \implies -1 \leq x \leq 5

. Thus, the numerator is non-positive on

[-1, 5]

and non-negative on

(-\infty, -1] \cup [5, \infty)

.

This identifies the values of

x

where the numerator changes sign.

3

Determine the critical intervals for the denominator,

D = |x + 1| - 2

, and identify restrictions.

Solve

|x + 1| - 2 < 0 \implies |x + 1| < 2 \implies -2 < x + 1 < 2 \implies -3 < x < 1

. Thus, the denominator is negative on

(-3, 1)

and positive on

(-\infty, -3) \cup (1, \infty)

. The boundary points

x = -3

and

x = 1

make the denominator zero and must be excluded from the domain.

This identifies the values of

x

where the denominator changes sign or makes the expression undefined.

4

Evaluate Case 1 and Case 2, then combine the solutions.

For Case 1 (

N \leq 0

and

D > 0

), find the intersection of

[-1, 5]

and

(-\infty, -3) \cup (1, \infty)

, which yields

1 < x \leq 5

. For Case 2 (

N \geq 0

and

D < 0

), find the intersection of

(-\infty, -1] \cup [5, \infty)

and

(-3, 1)

, which yields

-3 < x \leq -1

. Combining these intervals gives the final solution set:

-3 < x \leq -1

or

1 < x \leq 5

.

This determines the overall solution set that satisfies the starting inequality.

Key Concept

Solving rational inequalities containing absolute value expressions by finding critical intervals.

Estimated Time:3m 0s

Question 1823Question →

A charity concert sells floor tickets and balcony tickets. The price of a floor ticket is $5$ dollars more than twice the price of a balcony ticket. If the concert organizers sell $80$ balcony tickets and $50$ floor tickets, their total revenue is $R$ dollars. Under a new promotional structure, the price of a balcony ticket is discounted by $25\%$ , the price of a floor ticket is increased by $10\%$ , and the organizers sell $120$ balcony tickets. In terms of the original price of a balcony ticket, $b$ , which of the following expressions represents the number of floor tickets they must sell under the new promotional structure to achieve the same total revenue, $R$ ?

\frac{1500b + 2500}{22b + 55}

\frac{900b + 2500}{22b - 55}

\frac{900b + 2500}{22b + 55}

\frac{2500 - 1100b}{55 - 22b}

\frac{900b + 5000}{22b + 110}

Show answer & explanation

Answer: $\frac{900b + 2500}{22b + 55}$

Answer

\frac{900b + 2500}{22b + 55}

The correct expression is derived by first writing the original revenue in terms of the balcony ticket price

b

, which yields

R = 180b + 250

. Under the new pricing, the balcony ticket costs

0.75b

and the floor ticket costs

1.1(2b + 5) = 2.2b + 5.5

. Selling

120

balcony tickets generates

90b

in revenue. Setting the new total revenue equal to

R

gives

90b + F(2.2b + 5.5) = 180b + 250

. Solving for the number of floor tickets,

F

, results in

F = \frac{90b + 250}{2.2b + 5.5}

. Multiplying the numerator and denominator by

10

to clear the decimals yields the correct expression.

Step-by-Step Solution

1

Express the original ticket prices and revenue in terms of

b

.

Balcony ticket price =

b

. Floor ticket price =

2b + 5

. Original revenue

R = 80b + 50(2b + 5) = 180b + 250

.

To set up the baseline total revenue equation using the algebraic descriptions.

2

Determine the new pricing for both balcony and floor tickets.

New balcony ticket price =

0.75b

. New floor ticket price =

1.1(2b + 5) = 2.2b + 5.5

.

To apply the

25\%

discount and

10\%

increase to the original ticket prices.

3

Set up the equation equating the new promotional revenue to the original revenue

R

.

New Revenue =

120(0.75b) + F(2.2b + 5.5) = 180b + 250

, where

F

is the number of floor tickets.

To represent the condition that the total revenue remains the same under the new structure.

4

Simplify the equation and isolate the variable

F

.

90b + F(2.2b + 5.5) = 180b + 250 \implies F(2.2b + 5.5) = 90b + 250 \implies F = \frac{90b + 250}{2.2b + 5.5}

.

To solve for the number of floor tickets algebraically.

5

Clear decimals from the rational expression.

F = \frac{10(90b + 250)}{10(2.2b + 5.5)} = \frac{900b + 2500}{22b + 55}

.

To match the standard fraction format of the options by multiplying the numerator and denominator by

10

.

Key Concept

Translating verbal descriptions into multi-step algebraic systems and isolating a target variable from rational equations.

Estimated Time:3m 0s

Question 1824Question →

For the complex number $z = \frac{4 + 2i}{3 - i}$ , where $i$ is the imaginary unit such that $i^2 = -1$ , what is the imaginary part of $z$ ?

\frac{5}{4}

\frac{6}{5}

\frac{1}{5}

1

-2

Show answer & explanation

Answer: $1$

Answer

The imaginary part of

z

is

1

.

To find the imaginary part of

z

, multiply the numerator and denominator of

\frac{4 + 2i}{3 - i}

by the complex conjugate of the denominator, which is

3 + i

. This simplifies the denominator to

9 - i^2 = 10

and the numerator to

12 + 10i + 2i^2 = 10 + 10i

. Dividing each term by

10

yields

1 + i

, which has an imaginary part of

1

.

Step-by-Step Solution

1

Identify the complex conjugate of the denominator.

The complex conjugate of the denominator,

3 - i

, is

3 + i

.

To divide complex numbers, we multiply both the numerator and denominator by the conjugate of the denominator to make the denominator a real number.

2

Multiply the numerator and denominator of the fraction by the complex conjugate.

z = \frac{(4 + 2i)(3 + i)}{(3 - i)(3 + i)}

Multiplying by

\frac{3 + i}{3 + i}

is equivalent to multiplying by

1

, which preserves the value of the complex number.

3

Expand and simplify the numerator and denominator.

Numerator:

(4 + 2i)(3 + i) = 12 + 4i + 6i + 2i^2 = 12 + 10i + 2(-1) = 10 + 10i

. Denominator:

(3 - i)(3 + i) = 9 - i^2 = 9 - (-1) = 10

.

Apply the distributive property (FOIL method) and substitute

i^2 = -1

to simplify the expression.

4

Divide each term in the numerator by the denominator to find the imaginary part.

z = \frac{10 + 10i}{10} = 1 + i

, so the imaginary part is

1

.

Rewrite the fraction in standard form

a + bi

, where the imaginary part is

b

(the coefficient of

i

).

Key Concept

Simplifying a quotient of complex numbers by multiplying both the numerator and the denominator by the complex conjugate of the denominator.

Question 1825Question →

The cubic polynomial $6x^3 - 19x^2 + 11x + 6$ can be factored completely over the integers in the form $(x - a)(bx - c)(dx + e)$ , where $a$ , $b$ , $c$ , $d$ , and $e$ are positive integers. What is the value of $a + b + c + d + e$ ?

Show answer & explanation

Answer: 11

Answer

The sum of the coefficients and constants from the factored form is 11.

By applying the Rational Root Theorem, we find the root

x = 2

, which gives the factor

(x - 2)

. Dividing the original cubic expression by

(x - 2)

yields

6x^2 - 7x - 3

. Factoring this quadratic expression by grouping yields

(2x - 3)(3x + 1)

. Writing the completely factored form as

(x - 2)(2x - 3)(3x + 1)

and comparing it to

(x - a)(bx - c)(dx + e)

where

a, b, c, d, e

are positive integers results in

a = 2

,

b = 2

,

c = 3

,

d = 3

, and

e = 1

. The sum

a + b + c + d + e

is equal to 11.

Step-by-Step Solution

1

Identify a rational root of

6x^3 - 19x^2 + 11x + 6

x = 2

is a root, meaning

(x - 2)

is a factor.

Applying the Rational Root Theorem and testing potential integer root values.

2

Divide

6x^3 - 19x^2 + 11x + 6

by

(x - 2)

The quotient is the quadratic polynomial

6x^2 - 7x - 3

.

To reduce the cubic polynomial to a quadratic expression that can be factored using standard trinomial methods.

3

Factor the quadratic trinomial

6x^2 - 7x - 3

(2x - 3)(3x + 1)

Finding two numbers that multiply to

-18

and add to

-7

(which are

-9

and

2

) and factoring by grouping.

4

Match the factored form

(x - 2)(2x - 3)(3x + 1)

to

(x - a)(bx - c)(dx + e)

a = 2, b = 2, c = 3, d = 3, e = 1

Since

a, b, c, d,

and

e

must be positive integers, the constant terms and signs uniquely determine the mapping of each factor.

5

Calculate the sum of

a, b, c, d,

and

e

2 + 2 + 3 + 3 + 1 = 11

To find the final numeric value requested by the question.

Key Concept

Factoring cubic polynomials by finding rational roots and factoring quadratic trinomials by grouping.

Estimated Time:2m 30s

Question 1826Question →

When solving the radical equation $\sqrt{3x+10} + \sqrt{x+2} = 2$ for all real values of $x$ , squaring both sides yields two potential solutions. What is the numerical value of the extraneous solution?

Fill in the blanks below

The numerical value of the extraneous solution is .

Show answer & explanation

Answer

The extraneous solution is

x = 2

.

The value

x = 2

is the extraneous solution because substituting it into the original equation yields

\sqrt{16} + \sqrt{4} = 6 \neq 2

, showing it does not satisfy the original equation despite being generated by the algebraic squaring process.

Step-by-Step Solution

1

Isolate one of the radical terms in the equation.

\sqrt{3x+10} = 2 - \sqrt{x+2}

Isolating one radical simplifies the squaring process.

2

Square both sides of the equation.

3x+10 = 4 - 4\sqrt{x+2} + (x+2)

, which simplifies to

2x+4 = -4\sqrt{x+2}

.

Squaring eliminates the isolated radical.

3

Divide both sides by 2 and square both sides again.

x+2 = -2\sqrt{x+2} \Rightarrow (x+2)^2 = 4(x+2)

, which simplifies to

x^2 - 4 = 0

.

This eliminates the remaining radical term.

4

Solve the quadratic equation for

x

.

x = 2

or

x = -2

Factoring

(x-2)(x+2) = 0

gives the potential solutions.

5

Check both potential solutions in the original equation.

For

x = -2

:

\sqrt{4} + \sqrt{0} = 2

, which is true. For

x = 2

:

\sqrt{16} + \sqrt{4} = 6 \neq 2

, which is false.

This identifies which solution is extraneous.

Key Concept

Solving radical equations and verifying solutions to identify extraneous roots

Estimated Time:2m 0s

Question 1827Question →

For all real numbers $x$ such that $x \neq 0$ and $x \neq -1$ , which of the following is equivalent to the expression $\frac{1}{x} + \frac{1}{x+1}$ ?

\frac{2}{2x+1}

\frac{2}{x^2+x}

\frac{2x+1}{x^2+1}

\frac{2x+1}{x^2+x}

\frac{3}{x+1}

Show answer & explanation

Answer: $\frac{2x+1}{x^2+x}$

Answer

The correct expression is

\frac{2x+1}{x^2+x}

.

The correct expression is obtained by finding a common denominator of

x(x+1)

, rewriting the numerators as

x+1

and

x

, and then adding them to get

2x+1

over the expanded denominator

x^2+x

.

Step-by-Step Solution

1

Find a common denominator for the two rational terms.

The common denominator for

\frac{1}{x}

and

\frac{1}{x+1}

is

x(x+1)

, which expands to

x^2+x

.

To add fractions with different denominators, they must share a common denominator.

2

Rewrite each fraction with the common denominator by multiplying their numerators and denominators by the appropriate factor.

\frac{1}{x} = \frac{x+1}{x(x+1)}

and

\frac{1}{x+1} = \frac{x}{x(x+1)}

Multiplying the numerator and denominator of each fraction by the missing factor of the common denominator preserves the value of the fraction.

3

Add the numerators together over the common denominator and simplify.

\frac{(x+1) + x}{x(x+1)} = \frac{2x+1}{x^2+x}

Combining the numerators and expanding the denominator gives the simplified equivalent expression.

Key Concept

Adding rational expressions by finding a common denominator

Question 1828Question →

If $f(x) = 3x - 2$ and $g(x) = x^2 + 1$ , what is the value of $g(f(2))$ ?

Show answer & explanation

Answer: 17

Answer

The value of

g(f(2))

is 17.

To evaluate the composite function

g(f(2))

, we first find the value of the inner function

f(2) = 3(2) - 2 = 4

. Then, we substitute this output as the input for the outer function to get

g(4) = 4^2 + 1 = 17

.

Step-by-Step Solution

1

Substitute

x = 2

into the expression for

f(x)

to evaluate the inner function.

f(2) = 4

In the composition

g(f(2))

, the inner function

f(x)

must be evaluated first at the input value of 2.

2

Substitute the result from Step 1, which is 4, into the expression for

g(x)

.

g(4) = 17

The output of the inner function becomes the input for the outer function, so

g(f(2)) = g(4) = 4^2 + 1 = 17

.

Key Concept

Function Composition

Estimated Time:45s

Question 1829Question →

For what value of $a$ does the equation $\frac{3}{4}(x - 2) - \frac{1}{3}(2x + a) = \frac{1}{12}x - 5$ have infinitely many solutions for $x$ ?

Show answer & explanation

Answer: 10.5

Answer

10.5

Expanding the left side of the equation yields

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

. Combining the coefficients of

x

gives

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

. The equation becomes

\frac{1}{12}x - (\frac{3}{2} + \frac{a}{3}) = \frac{1}{12}x - 5

. For a linear equation to have infinitely many solutions, the variable coefficients must be equal and the constant terms must also be equal. Therefore, we equate the constants:

-\frac{3}{2} - \frac{a}{3} = -5

. Multiplying all terms by

-6

to clear the denominators yields

9 + 2a = 30

, which simplifies to

2a = 21

, or

a = 10.5

.

Step-by-Step Solution

1

Expand and simplify the left side of the equation by distributing the fraction coefficients.

\frac{1}{12}x - \frac{3}{2} - \frac{a}{3}

Distributing

\frac{3}{4}

and

-\frac{1}{3}

across their respective parentheses and combining the

x

terms allows us to compare the coefficients on both sides.

2

Equate the constant terms from both sides of the equation.

-\frac{3}{2} - \frac{a}{3} = -5

A linear equation of the form

Ax + B = Cx + D

has infinitely many solutions if and only if

A = C

and

B = D

. Since both

A

and

C

are

\frac{1}{12}

, we set the constant terms equal.

3

Isolate the variable

a

and solve.

a = 10.5

Adding

\frac{3}{2}

to both sides gives

-\frac{a}{3} = -3.5

. Multiplying both sides by

-3

yields the final value.

Key Concept

Solving linear equations with infinitely many solutions by equating coefficients and constant terms.

Estimated Time:2m 0s

Question 1830Question →

A chemist is preparing $120\text{ milliliters}$ of a chemical mixture with an overall acid concentration of $27.5\%$ by volume. To do this, she mixes Solution $X$ ( $10\%$ acid by volume), Solution $Y$ ( $25\%$ acid by volume), and Solution $Z$ ( $40\%$ acid by volume). The chemist decides that the volume of Solution $Y$ must be exactly $20\text{ milliliters}$ less than twice the volume of Solution $X$ used in the mixture. What is the positive difference, in milliliters, between the volume of Solution $Z$ and the volume of Solution $X$ in the final mixture?

Show answer & explanation

Answer: 20

Answer

The positive difference between the volume of Solution Z and Solution X is 20 milliliters.

The correct answer is 20, because solving the system of equations yields a volume of 30 milliliters for Solution X and 50 milliliters for Solution Z. The positive difference between these two volumes is 20 milliliters.

Step-by-Step Solution

1

Define variables for the volume of each solution used in the mixture.

Let

x

be the volume of Solution

X

,

y

be the volume of Solution

Y

, and

z

be the volume of Solution

Z

(all in milliliters).

Defining variables allows for translating the word problem's conditions into algebraic equations.

2

Translate the given information into a system of three linear equations.

Equation 1 (Total Volume):

x + y + z = 120

. Equation 2 (Total Acid):

0.10x + 0.25y + 0.40z = 33

(since

27.5\%

of

120\text{ ml}

is

33\text{ ml}

). Equation 3 (Volume Relation):

y = 2x - 20

.

These equations model the constraints and quantities described in the problem.

3

Express

z

in terms of

x

by substituting the expression for

y

into the total volume equation.

x + (2x - 20) + z = 120 \implies 3x - 20 + z = 120 \implies z = 140 - 3x

.

This substitution reduces the system to two variables (

x

and

z

), making it easier to solve.

4

Substitute the expressions for

y

and

z

in terms of

x

into the acid equation and solve for

x

.

0.10x + 0.25(2x - 20) + 0.40(140 - 3x) = 33 \implies 0.10x + 0.50x - 5 + 56 - 1.20x = 33 \implies -0.60x + 51 = 33 \implies -0.60x = -18 \implies x = 30

.

This isolates the single variable

x

so that its value can be calculated.

5

Determine the volume of Solution

Z

and calculate the final positive difference.

z = 140 - 3(30) = 50

. The positive difference between the volume of Solution

Z

and Solution

X

is

|50 - 30| = 20

.

This directly answers the question's requirement for the difference between the two solution volumes.

Key Concept

Translating word problems into a system of three linear equations and solving them using substitution.

Question 1831Question →

An algebra student is asked to simplify a polynomial expression by factoring it completely. The student is given the expression $2x^6 - 32x^2$ . Which of the following is the completely factored form of the expression?

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

2x^2(x^3 - 16)

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

32x^2(x - 2)(x + 2)(x^2 + 4)

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

Show answer & explanation

Answer: $2x^2(x - 2)(x + 2)(x^2 + 4)$

Answer

The completely factored form of the expression is

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

.

The correct answer is

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

. First, find the greatest common factor (GCF) of

2x^6

and

32x^2

, which is

2x^2

. Factoring this out gives

2x^2(x^4 - 16)

. Next, recognize that

x^4 - 16

is a difference of squares since

x^4 = (x^2)^2

and

16 = 4^2

. This factors into

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

. Finally, the term

x^2 - 4

is also a difference of squares and factors into

(x - 2)(x + 2)

, while the sum of squares

x^2 + 4

cannot be factored further over the real numbers. Thus, the completely factored form is

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

.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) of the two terms in

2x^6 - 32x^2

.

The GCF is

2x^2

, so the expression becomes

2x^2(x^4 - 16)

.

Factoring out the GCF simplifies the polynomial and reveals a difference of squares.

2

Recognize that

x^4 - 16

is a difference of squares and factor it.

Since

x^4 = (x^2)^2

and

16 = 4^2

, the expression factors as

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

.

The difference of squares formula

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

applies directly to

x^4 - 16

.

3

Check if any remaining factors can be factored further, and factor them.

The term

x^2 - 4

is another difference of squares (

x^2 - 2^2

), which factors into

(x - 2)(x + 2)

. The term

x^2 + 4

is a sum of squares and cannot be factored over the real numbers. The completely factored expression is

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

.

To factor completely, all factorable terms must be broken down into their prime polynomial factors.

Key Concept

Factoring polynomials completely using the greatest common factor and the difference of squares.

Question 1832Question →

An art gallery owner wants to display 5 paintings in a row on a wall. The owner selects these 5 paintings from a collection of 4 different landscape paintings and 4 different portrait paintings. The display must meet the following guidelines:

1. No two landscape paintings can be placed next to each other.
2. At least 2 landscape paintings must be displayed.

How many different arrangements of 5 paintings are possible?

Show answer & explanation

Answer: 2016

Answer

2016

The correct answer is 2016. By breaking down the problem into two mutually exclusive cases based on the number of landscape paintings (either 3 landscapes and 2 portraits, or 2 landscapes and 3 portraits), we can find the valid arrangements for each. For 3 landscapes, the only valid layout is

L-P-L-P-L

, yielding

P(4,3) \times P(4,2) = 288

arrangements. For 2 landscapes, there are

\binom{4}{2} = 6

layout configurations, each yielding

P(4,2) \times P(4,3) = 288

arrangements, for a total of 1728. Adding these two cases gives

288 + 1728 = 2016

.

Step-by-Step Solution

1

Determine the possible number of landscape paintings (

k

) that can be displayed.

k = 2

or

k = 3

Since at least 2 landscapes must be displayed,

k \ge 2

. Since no two landscapes can be adjacent in a 5-painting row, we cannot have 4 landscapes (as that would require at least 3 portraits to separate them, making the total count at least 7 paintings). Thus,

k

can only be 2 or 3.

2

Calculate the arrangements for the case with 3 landscape paintings and 2 portrait paintings.

288 arrangements

For 3 landscapes (

L

) and 2 portraits (

P

) to have no adjacent landscapes, the only possible pattern of positions is

L-P-L-P-L

. The number of ways to choose and arrange 3 landscapes from 4 is

P(4, 3) = 4 \times 3 \times 2 = 24

. The number of ways to choose and arrange 2 portraits from 4 is

P(4, 2) = 4 \times 3 = 12

. The total arrangements for this case is

24 \times 12 = 288

.

3

Calculate the arrangements for the case with 2 landscape paintings and 3 portrait paintings.

1728 arrangements

For 2 landscapes (

L

) and 3 portraits (

P

) to have no adjacent landscapes, we place the 3 portraits first:

\_ P \_ P \_ P \_

. We choose 2 of the 4 available spaces for the landscapes in

\binom{4}{2} = 6

ways. For each pattern, the number of ways to choose and arrange 2 landscapes from 4 is

P(4, 2) = 12

, and the number of ways to choose and arrange 3 portraits from 4 is

P(4, 3) = 24

. The total arrangements for this case is

6 \times 12 \times 24 = 1728

.

4

Sum the arrangements from both cases.

2016 arrangements

Since the two cases are mutually exclusive, we add their individual counts:

288 + 1728 = 2016

.

Key Concept

Permutations and Combinations with Constraints

Question 1833Question →

What is the value of $x$ that satisfies the equation $\frac{2}{3}x - \frac{1}{4} = \frac{5}{12}$ ?

\frac{1}{4}

\frac{9}{16}

\frac{4}{9}

1

12

Show answer & explanation

Answer: $1$

Answer

1

The correct answer is 1. Adding

\frac{1}{4}

to both sides of the equation

\frac{2}{3}x - \frac{1}{4} = \frac{5}{12}

gives

\frac{2}{3}x = \frac{5}{12} + \frac{3}{12} = \frac{8}{12}

. Simplifying

\frac{8}{12}

yields

\frac{2}{3}

, so

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

. Multiplying both sides by

\frac{3}{2}

isolates

x

, giving

x = 1

.

Step-by-Step Solution

1

Add

\frac{1}{4}

to both sides of the equation to isolate the variable term.

\frac{2}{3}x = \frac{5}{12} + \frac{1}{4}

To solve for

x

, terms containing

x

must be isolated on one side of the equation.

2

Find a common denominator to add the fractions on the right side.

\frac{2}{3}x = \frac{5}{12} + \frac{3}{12} = \frac{8}{12} = \frac{2}{3}

Fractions must have the same denominator to be added.

3

Multiply both sides of the equation by the reciprocal of the coefficient of

x

, which is

\frac{3}{2}

.

x = \frac{2}{3} \cdot \frac{3}{2} = 1

Multiplying a coefficient by its reciprocal yields

1

, isolating the variable

x

.

Key Concept

Solving single-variable linear equations involving fractions by isolating the variable using inverse operations.

Question 1834Question →

A manufacturer of custom planners determines that the setup cost for a production run is $100$ dollars, and each planner costs $6$ dollars to produce. The planners sell for $10$ dollars each, except for the first $10$ planners sold, which are discounted by $2$ dollars each. If the manufacturer wants to achieve a net profit of exactly $300$ dollars for a single production run, how many planners must they produce and sell?

Show answer & explanation

Answer: 105

Answer

The manufacturer must produce and sell

105

planners to achieve a net profit of

300

dollars.

The correct answer is found by setting up the profit equation:

\text{Profit} = \text{Total Revenue} - \text{Total Cost}

. The cost is

100 + 6x

and the revenue is

10(8) + 10(x-10) = 10x - 20

. Equating their difference to

300

yields

(10x - 20) - (100 + 6x) = 300

, which simplifies to

4x - 120 = 300

. Solving for

x

results in

105

.

Step-by-Step Solution

1

Define the variable for the number of planners.

Let

x

be the number of planners produced and sold, where

x \geq 10

.

Establishing the variable is necessary to set up algebraic expressions for cost and revenue.

2

Write the total cost expression.

\text{Total Cost} = 100 + 6x

The cost combines the fixed setup fee of

100

dollars and the variable cost of

6

dollars per planner.

3

Write the total revenue expression.

\text{Total Revenue} = 10(8) + 10(x - 10) = 10x - 20

The first

10

planners sell for

8

dollars each, and the remaining

x - 10

planners sell for the regular price of

10

dollars each.

4

Set up the profit equation and solve for

x

.

(10x - 20) - (100 + 6x) = 300 \implies 4x - 120 = 300 \implies 4x = 420 \implies x = 105

Profit is the difference between total revenue and total cost, which must equal the target profit of

300

dollars.

Key Concept

Translating real-world pricing and cost constraints into a single-variable linear equation.

Question 1835Question →

For $x \neq 0$ , the expression $\frac{(x^2)^5}{x^k}$ simplifies to $x^6$ . What is the value of the exponent $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of the exponent

k

is 4.

Applying the power of a power rule to the numerator yields

(x^2)^5 = x^{10}

. Next, applying the quotient rule to divide

x^{10}

by

x^k

yields

x^{10-k}

. Setting this equal to the simplified term

x^6

leads to the exponent equation

10 - k = 6

. Solving this equation gives the final result

k = 4

.

Step-by-Step Solution

1

Simplify the numerator expression

(x^2)^5

x^{10}

Multiply the exponents when raising a power to another power:

(x^a)^b = x^{ab}

.

2

Simplify the division of the two exponential expressions

\frac{x^{10}}{x^k}

x^{10-k}

Subtract the exponent of the denominator from the exponent of the numerator:

\frac{x^a}{x^b} = x^{a-b}

.

3

Solve the linear equation for

k

using the target exponent 6

k = 4

Equating the exponent

10-k

to

6

gives

10-k=6

. Subtracting 10 from both sides gives

-k = -4

, so

k = 4

.

Key Concept

Applying properties of exponents in algebraic expressions, specifically the power of a power rule and the quotient rule.

Question 1836Question →

For what value of the constant $a$ does the linear equation $\frac{a(x - 2)}{3} - \frac{3x - 1}{2} = -\frac{5x + 7}{6}$ have no real solution for $x$ ?

Show answer & explanation

Answer: 2

Answer

The constant

a

must be equal to

2

for the equation to have no solution.

A linear equation in the form

Ax + B = Cx + D

has no solution if the coefficients of the variable are equal (

A = C

) but the constant terms are different (

B \neq D

). Multiplying the given equation by the least common denominator, 6, clears the fractions and yields

2a(x - 2) - 3(3x - 1) = -(5x + 7)

. Expanding both sides results in

2ax - 4a - 9x + 3 = -5x - 7

, which simplifies to

(2a - 9)x + (3 - 4a) = -5x - 7

. Equating the coefficients of

x

gives

2a - 9 = -5

, which solves to

a = 2

. Substituting

a = 2

back into the constants yields a left-side constant of

-5

and a right-side constant of

-7

. Since

-5 \neq -7

, the variable terms cancel out while leaving an inequality, meaning the equation has no solution when

a = 2

.

Step-by-Step Solution

1

Clear the denominators by multiplying the entire equation by the least common denominator, which is 6.

2a(x - 2) - 3(3x - 1) = -(5x + 7)

Multiplying by the LCD simplifies the rational expressions into polynomial terms.

2

Distribute and expand the terms on both sides of the equation.

2ax - 4a - 9x + 3 = -5x - 7

Expanding the terms allows us to group variable terms and constant terms together.

3

Group the

x

terms and constant terms on the left side.

(2a - 9)x + (3 - 4a) = -5x - 7

Structuring the equation in the standard form

Ax + B = Cx + D

makes it easier to compare coefficients.

4

Set the coefficients of

x

on both sides equal to each other.

2a - 9 = -5

For a linear equation to have no solution, the variable terms must cancel out, meaning their coefficients must be identical.

5

Solve for the parameter

a

and verify the constant terms are unequal.

2a = 4 \implies a = 2

. Constant check:

3 - 4(2) = -5

, and

-5 \neq -7

.

If the constant terms were equal, the equation would have infinitely many solutions instead of no solution.

Key Concept

Identifying the parameter value that results in a linear equation having no solution by equating variable coefficients and ensuring constant terms are unequal.

Question 1837Question →

For each of the given quadratic equations, solve for $x$ by factoring. Match each quadratic equation on the left to its correct solution set on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The equation

2x^2 + 5x = 3

matches the solution set

\left\{-3, \frac{1}{2}\right\}

; the equation

3x^2 - 10x + 8 = 0

matches the solution set

\left\{\frac{4}{3}, 2\right\}

; and the equation

x(x + 2) = 15

matches the solution set

\left\{-5, 3\right\}

.

Each equation matches its corresponding solution set through distributing terms if necessary, rewriting the equation in standard quadratic form

ax^2 + bx + c = 0

, factoring the quadratic trinomial over the integers, and then using the zero product property to solve for

x

.

Step-by-Step Solution

1

For the equation

2x^2 + 5x = 3

, rewrite in standard form

ax^2 + bx + c = 0

by subtracting 3 from both sides to get

2x^2 + 5x - 3 = 0

.

The equation is rewritten as

2x^2 + 5x - 3 = 0

.

Before factoring a quadratic equation, all terms must be moved to one side so the other side is equal to zero.

2

Factor the trinomial

2x^2 + 5x - 3

by grouping. Find two integers that multiply to

2 \times (-3) = -6

and add to

5

. These integers are

6

and

-1

. Rewrite the middle term and factor by grouping:

2x^2 + 6x - x - 3 = 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3) = 0

.

The equation becomes

(2x - 1)(x + 3) = 0

.

Factoring allows us to apply the zero product property to find the solutions.

3

Set each factor of

(2x - 1)(x + 3) = 0

to zero and solve for

x

:

2x - 1 = 0 \Rightarrow x = \frac{1}{2}

and

x + 3 = 0 \Rightarrow x = -3

.

The solutions are

x = \frac{1}{2}

and

x = -3

, forming the solution set

\left\{-3, \frac{1}{2}\right\}

.

By the zero product property, if the product of two factors is zero, at least one of the factors must be zero.

4

For the equation

3x^2 - 10x + 8 = 0

, factor the trinomial by finding two integers that multiply to

3 \times 8 = 24

and add to

-10

. These integers are

-6

and

-4

. Rewrite the middle term and factor by grouping:

3x^2 - 6x - 4x + 8 = 3x(x - 2) - 4(x - 2) = (3x - 4)(x - 2) = 0

.

The equation becomes

(3x - 4)(x - 2) = 0

.

The equation is already in standard form, so we can directly proceed with factoring.

5

Set each factor of

(3x - 4)(x - 2) = 0

to zero and solve for

x

:

3x - 4 = 0 \Rightarrow x = \frac{4}{3}

and

x - 2 = 0 \Rightarrow x = 2

.

The solutions are

x = \frac{4}{3}

and

x = 2

, forming the solution set

\left\{\frac{4}{3}, 2\right\}

.

Solving each linear factor yields the roots of the quadratic equation.

6

For the equation

x(x + 2) = 15

, first distribute

x

to get

x^2 + 2x = 15

, then subtract 15 from both sides to write in standard form:

x^2 + 2x - 15 = 0

.

The equation is rewritten as

x^2 + 2x - 15 = 0

.

Distributing and moving terms sets the quadratic to zero, which is necessary for factoring.

7

Factor the quadratic

x^2 + 2x - 15 = 0

by finding two integers that multiply to

-15

and add to

2

. These integers are

5

and

-3

, yielding

(x + 5)(x - 3) = 0

.

The equation becomes

(x + 5)(x - 3) = 0

.

Factoring a quadratic trinomial with a leading coefficient of 1 involves finding numbers that sum to the linear coefficient and multiply to the constant term.

8

Set each factor of

(x + 5)(x - 3) = 0

to zero and solve for

x

:

x + 5 = 0 \Rightarrow x = -5

and

x - 3 = 0 \Rightarrow x = 3

.

The solutions are

x = -5

and

x = 3

, forming the solution set

\left\{-5, 3\right\}

.

Solving the resulting linear equations gives the roots of the original quadratic equation.

Key Concept

Solving Quadratic Equations by Factoring

Alternative Method

Instead of factoring, the solutions to these quadratic equations can be verified by substituting the values in the solution sets back into the original equations, or by applying the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

after rewriting them in standard form.

Estimated Time:2m 0s

Question 1838Question →

Which of the following is the completely factored form of the expression $9x^2 - 36$ ?

(3x - 6)^2

(3x - 6)(3x + 6)

9(x - 2)(x + 2)

9(x^2 - 4)

9(x - 2)^2

Show answer & explanation

Answer: $9(x - 2)(x + 2)$

Answer

The correct factored form is

9(x - 2)(x + 2)

The expression

9x^2 - 36

has a greatest common factor of

9

. Factoring out

9

yields

9(x^2 - 4)

. The term inside the parentheses,

x^2 - 4

, is a difference of squares that can be factored as

(x - 2)(x + 2)

. Combining these gives the completely factored form

9(x - 2)(x + 2)

.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) of the terms in the expression.

The GCF of

9x^2

and

36

is

9

. Factoring it out gives

9(x^2 - 4)

.

Factoring out the GCF simplifies the remaining polynomial and is the first step in completely factoring an expression.

2

Recognize and factor the quadratic expression inside the parentheses.

The term

x^2 - 4

is a difference of squares of the form

a^2 - b^2

, where

a = x

and

b = 2

. Factoring it yields

(x - 2)(x + 2)

.

A difference of squares

a^2 - b^2

always factors into

(a - b)(a + b)

.

3

Combine the factored terms to write the completely factored expression.

Combining the GCF and the factored binomials gives

9(x - 2)(x + 2)

.

This represents the expression written as a product of its prime polynomial factors.

Key Concept

Factoring polynomials by first extracting the greatest common factor (GCF) and then applying the difference of squares identity.

Alternative Method

Instead of factoring out the greatest common factor first, the expression can be factored as a difference of squares directly:

9x^2 - 36 = (3x)^2 - 6^2 = (3x - 6)(3x + 6)

. Then, a common factor of

3

can be factored out from each binomial:

3(x - 2) \cdot 3(x + 2) = 9(x - 2)(x + 2)

.

Estimated Time:45s

Question 1839Question →

If the algebraic expression $\frac{(x^a y^2)^{-3} (x^2 y^b)^2}{(x^{-3} y^{-1})^{-2}}$ simplifies to $x^4 y^4$ for all non-zero real numbers $x$ and $y$ , where $a$ and $b$ are integers, what is the value of $a + b$ ?

Show answer & explanation

Answer: 4

Answer

The value of

a + b

is

4

.

Applying exponent rules to the expression

\frac{(x^a y^2)^{-3} (x^2 y^b)^2}{(x^{-3} y^{-1})^{-2}}

yields

\frac{x^{-3a}y^{-6} \cdot x^4 y^{2b}}{x^6 y^2} = \frac{x^{-3a+4} y^{2b-6}}{x^6 y^2} = x^{-3a-2} y^{2b-8}

. Equating these exponents to the target expression

x^4 y^4

gives the system

-3a - 2 = 4

and

2b - 8 = 4

. Solving these equations gives

a = -2

and

b = 6

, resulting in a sum of

a + b = 4

.

Step-by-Step Solution

1

Simplify the terms in the numerator.

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

and

(x^2 y^b)^2 = x^4 y^{2b}

Apply the power of a product rule:

(u^m v^n)^p = u^{mp} v^{np}

.

2

Multiply the simplified terms in the numerator.

x^{-3a+4} y^{2b-6}

Apply the product rule of exponents by adding exponents of like bases:

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

.

3

Simplify the denominator.

(x^{-3} y^{-1})^{-2} = x^6 y^2

Apply the power of a product rule.

4

Divide the numerator by the denominator.

x^{-3a-2} y^{2b-8}

Apply the quotient rule of exponents by subtracting denominator exponents from numerator exponents:

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

.

5

Set up equations by equating the simplified exponents to the exponents in the target expression

x^4 y^4

.

-3a - 2 = 4

and

2b - 8 = 4

For the expressions to be equivalent for all non-zero real numbers, the corresponding exponents of

x

and

y

must be equal.

6

Solve the linear equations for the integer constants

a

and

b

.

a = -2

and

b = 6

Isolate the variables:

-3a = 6 \implies a = -2

, and

2b = 12 \implies b = 6

.

7

Find the sum of

a

and

b

.

4

Add the values of the constants:

-2 + 6 = 4

.

Key Concept

Properties of exponents (product, quotient, and power rules) in multi-step algebraic simplification

Question 1840Question →

For all real numbers $m$ and $n$ , the expression $4m(2m - 3n)^2 - n(5m^2 - 36mn)$ can be written in the form $am^3 + bm^2n + cmn^2$ , where $a$ , $b$ , and $c$ are real constants. What is the value of the coefficient $b$ ?

Show answer & explanation

Answer: -53

Answer

The value of the coefficient

b

is

-53

.

Expanding the entire expression yields

16m^3 - 53m^2n + 72mn^2

. Comparing this to the template form

am^3 + bm^2n + cmn^2

shows that

b = -53

.

Step-by-Step Solution

1

Expand the squared binomial

(2m - 3n)^2

4m^2 - 12mn + 9n^2

Apply the identity

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

to expand the expression inside the parentheses.

2

Distribute

4m

through the trinomial

16m^3 - 48m^2n + 36mn^2

Multiply each term of

4m^2 - 12mn + 9n^2

by

4m

using properties of exponents.

3

Distribute

-n

across

(5m^2 - 36mn)

-5m^2n + 36mn^2

Multiply

-n

by both terms inside the parentheses, paying attention to sign rules.

4

Group and combine the like terms

16m^3 + (-48m^2n - 5m^2n) + (36mn^2 + 36mn^2)

Identify terms with the same variables and exponents to combine them.

5

Combine the coefficients of the like terms

16m^3 - 53m^2n + 72mn^2

Perform arithmetic on the coefficients:

-48 - 5 = -53

for the

m^2n

terms and

36 + 36 = 72

for the

mn^2

terms.

6

Identify the coefficient

b

-53

Compare the simplified polynomial to the form

am^3 + bm^2n + cmn^2

to find the value of

b

.

Key Concept

Simplifying algebraic expressions by expanding binomials, distributing variables and signs, and combining like terms.

Alternative Method

Instead of expanding the entire expression, focus only on terms that produce

m^2n

: from the first part,

4m \times (-12mn) = -48m^2n

, and from the second part,

-n \times 5m^2 = -5m^2n

. Adding these gives

-53m^2n

.

Estimated Time:1m 30s

Answer

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