All practice questions

Question 1921Question →

For a certain value of $x$ , the expression $\frac{2}{5}x + 3$ is equal to $11$ . What is the value of $2x + 5$ ?

11.4

21

27

45

85

Show answer & explanation

Answer: 45

Answer

45

To find the value of

2x + 5

, we first isolate the variable

x

in the equation

\frac{2}{5}x + 3 = 11

. Subtracting

3

from both sides gives

\frac{2}{5}x = 8

. Multiplying both sides by the reciprocal of the coefficient,

\frac{5}{2}

, yields

x = 8 \times \frac{5}{2} = 20

. Finally, substituting

x = 20

into the expression

2x + 5

gives

2(20) + 5 = 40 + 5 = 45

.

Step-by-Step Solution

1

Set up the equation based on the text and subtract 3 from both sides of the equation to isolate the variable term.

\frac{2}{5}x = 8

To solve for

x

, we first need to isolate the term containing

x

.

2

Multiply both sides of the equation by

\frac{5}{2}

to solve for

x

.

x = 20

Multiplying by the reciprocal of the coefficient cancels the fraction, leaving

x

isolated.

3

Substitute

x = 20

into the expression

2x + 5

.

2(20) + 5 = 45

The question asks for the value of

2x + 5

, so we evaluate it using the solved value of

x

.

Key Concept

Solving Linear Equations

Alternative Method

We can also multiply the entire equation by

5

first to eliminate the fraction:

2x + 15 = 55

. Subtracting

15

from both sides gives

2x = 40

. Since the target expression is

2x + 5

, we can simply add

5

to both sides of

2x = 40

to get

2x + 5 = 45

, avoiding the need to solve for

x

directly.

Estimated Time:45s

Question 1922Question →

A line with a positive slope passes through the point $(0, -4)$ and is tangent to the circle $x^2 + y^2 = 4$ . If this same line is also tangent to the parabola $y = x^2 + k$ , what is the value of the constant $k$ ?

Show answer & explanation

Answer: -3.25

Answer

The constant

k

must be

-3.25

.

By writing the equation of the line passing through

(0, -4)

as

y = mx - 4

and applying the condition that it is tangent to the circle

x^2 + y^2 = 4

, we find the positive slope is

m = \sqrt{3}

. Substituting this tangent line

y = \sqrt{3}x - 4

into the parabola equation

y = x^2 + k

yields the quadratic equation

x^2 - \sqrt{3}x + (k + 4) = 0

. For the line to be tangent to the parabola, the discriminant of this equation must be zero, which gives

3 - 4(k + 4) = 0

, leading to the final value

k = -3.25

.

Step-by-Step Solution

1

Set up the equation of the line passing through

(0, -4)

and substitute it into the equation of the circle.

x^2 + (mx - 4)^2 = 4 \implies (1 + m^2)x^2 - 8mx + 12 = 0

This establishes the quadratic equation representing the intersection points of the line and the circle.

2

Solve for the slope

m

by setting the discriminant of the intersection equation to zero.

(-8m)^2 - 4(1 + m^2)(12) = 0 \implies 64m^2 - 48 - 48m^2 = 0 \implies 16m^2 = 48 \implies m = \sqrt{3}

Since the line is tangent to the circle, the intersection equation must have exactly one real solution, meaning its discriminant is zero. The positive root is selected because the slope is specified as positive.

3

Equate the line

y = \sqrt{3}x - 4

and the parabola

y = x^2 + k

to analyze their intersection.

x^2 - \sqrt{3}x + (k + 4) = 0

This sets up the quadratic equation for the intersection points between the tangent line and the parabola.

4

Solve for

k

by setting the discriminant of this second quadratic equation to zero.

(-\sqrt{3})^2 - 4(1)(k + 4) = 0 \implies 3 - 4k - 16 = 0 \implies -13 - 4k = 0 \implies k = -3.25

For the line to be tangent to the parabola, the system must have exactly one solution, meaning the discriminant of their intersection equation must be zero.

Key Concept

Systems of Linear and Non-Linear Equations

Question 1923Question →

Let the functions $f$ and $g$ be defined by $f(x) = \sqrt{3x + 4}$ and $g(x) = x^2 - 2$ . What is the positive value of $x$ for which the composite function value $f(g(x)) = 5$ ?

Show answer & explanation

Answer: 3

Answer

3

The correct answer is 3. We set

f(g(x)) = 5

, which translates to

\sqrt{3g(x) + 4} = 5

. Squaring both sides yields

3g(x) + 4 = 25

. Solving for

g(x)

gives

3g(x) = 21

, which simplifies to

g(x) = 7

. Substituting

g(x) = x^2 - 2

into this equation gives

x^2 - 2 = 7

, which simplifies to

x^2 = 9

. Taking the square root of both sides gives

x = 3

or

x = -3

. Since the question specifies the positive value of

x

, the correct answer is 3.

Step-by-Step Solution

1

Substitute the inner function

g(x)

into the outer function

f(x)

to express

f(g(x)) = 5

\sqrt{3g(x) + 4} = 5

Setting up the initial composite equation to isolate the inner function

g(x)

2

Square both sides of the equation and isolate

g(x)

3g(x) + 4 = 25 \implies 3g(x) = 21 \implies g(x) = 7

Eliminating the radical allows us to determine the required output value of

g(x)

3

Substitute the formula for

g(x)

into the equation and solve for

x

x^2 - 2 = 7 \implies x^2 = 9 \implies x = 3

or

x = -3

Finding the input values of

x

that satisfy the condition for

g(x)

4

Select the positive value from the solution set

x = 3

The question specifically requests the positive value of

x

Key Concept

Evaluating and solving composite function equations by working from the outside function inward

Alternative Method

Instead of solving from the outside in, we can first construct the symbolic expression for

f(g(x))

by substituting

g(x)

directly:

f(g(x)) = \sqrt{3(x^2 - 2) + 4} = \sqrt{3x^2 - 6 + 4} = \sqrt{3x^2 - 2}

. Setting this equal to 5 yields

\sqrt{3x^2 - 2} = 5

. Squaring both sides gives

3x^2 - 2 = 25 \implies 3x^2 = 27 \implies x^2 = 9 \implies x = \pm 3

. The positive solution is 3.

Estimated Time:1m 30s

Question 1924Question →

An online store sells digital songs for $\$1.20$ each. A customer uses a discount code to get $\$3.00$ off the total purchase. If the customer's total cost after using the discount code is $\$15.00$ , how many digital songs did the customer purchase?

Show answer & explanation

Answer: 15

Answer

The customer purchased 15 songs.

The word problem translates directly to the linear equation

1.20s - 3.00 = 15.00

, where

s

is the number of songs. Adding

3.00

to both sides of the equation yields

1.20s = 18.00

. Dividing both sides by

1.20

yields

s = 15

.

Step-by-Step Solution

1

Define the variable and set up the equation.

Let

s

be the number of digital songs purchased. The cost of

s

songs is

1.20s

. Subtracting the discount of

\$3.00

gives the equation:

1.20s - 3.00 = 15.00

.

We must represent the cost of the songs and the discount algebraically to equal the final payment.

2

Isolate the variable term.

1.20s = 18.00

Adding

3.00

to both sides of the equation simplifies the equation by canceling the subtraction of

3.00

.

3

Solve for the variable.

s = 15

Dividing both sides of the equation by

1.20

isolates

s

to find the total number of songs.

Key Concept

Translating real-world scenarios with linear relationships into algebraic equations and solving them.

Question 1925Question →

For all positive real numbers

x

and

y

, the expression

\frac{(3x^2 y^{-3})^3 \cdot (2x^{-1} y^2)^2}{(6x^3 y^{-2})^2}

can be simplified to the form

A x^a y^b

, where

A

,

a

, and

b

are integers. What is the value of the sum

A + a + b

?

Show answer & explanation

Answer: 0

Answer

The value of the sum

A + a + b

is 0.

By applying the rules of exponents systematically, the expression simplifies to

3 x^{-2} y^{-1}

. Comparing this to

A x^a y^b

yields

A = 3

,

a = -2

, and

b = -1

. The sum is

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

.

Step-by-Step Solution

1

Simplify the first term in the numerator

27x^6y^{-9}

Apply the power of a product rule and power of a power rule to

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

.

2

Simplify the second term in the numerator

4x^{-2}y^4

Apply the power of a product rule and power of a power rule to

(2x^{-1} y^2)^2

.

3

Multiply the simplified terms in the numerator together

108x^4y^{-5}

Multiply coefficients and add the exponents of like bases.

4

Simplify the denominator

36x^6y^{-4}

Apply the power of a product rule and power of a power rule to

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

.

5

Divide the numerator by the denominator

3x^{-2}y^{-1}

Divide the coefficients and subtract the denominator exponents from the numerator exponents for like bases.

6

Sum the constants

A

,

a

, and

b

0

Identify

A = 3

,

a = -2

,

b = -1

from the expression

3x^{-2}y^{-1}

, and calculate

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

.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 1926Question →

If $2.5(x - 4) = 15$ , what is the value of $x$ ?

Show answer & explanation

Answer: 10

Answer

The value of

x

is

10

.

To solve the equation

2.5(x - 4) = 15

, divide both sides by

2.5

to obtain

x - 4 = 6

. Then, add

4

to both sides to get the final solution of

x = 10

. Alternatively, distribute

2.5

to get

2.5x - 10 = 15

, add

10

to both sides to get

2.5x = 25

, and divide by

2.5

to get

x = 10

.

Step-by-Step Solution

1

Divide both sides of the equation by

2.5

.

x - 4 = 6

To isolate the parenthetical term on the left side of the equation.

2

Add

4

to both sides of the equation.

x = 10

To isolate the variable

x

.

Key Concept

Solving multi-step linear equations using inverse operations.

Alternative Method

Distribute

2.5

to get

2.5x - 10 = 15

. Add

10

to both sides of the equation to get

2.5x = 25

. Divide both sides by

2.5

to find

x = 10

.

Estimated Time:45s

Question 1927Question →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $x^2 + y^2 = 10$ . The line $y = 3x$ intersects the circle at a point $(x, y)$ in the first quadrant. What is the value of $x + y$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is 4.

Substituting

y = 3x

into the equation

x^2 + y^2 = 10

yields

x^2 + (3x)^2 = 10

. Simplifying this expression gives

x^2 + 9x^2 = 10

, which combines to

10x^2 = 10

. Dividing both sides by 10 results in

x^2 = 1

. Since the point of intersection lies in the first quadrant, we take the positive square root, so

x = 1

. Substituting

x = 1

back into

y = 3x

gives

y = 3(1) = 3

. The sum of the coordinates is

x + y = 1 + 3 = 4

.

Step-by-Step Solution

1

Substitute the linear expression for

y

into the equation of the circle.

x^2 + (3x)^2 = 10

Substituting one equation into another allows us to solve for a single variable.

2

Simplify the equation and solve for

x

.

10x^2 = 10 \Rightarrow x^2 = 1 \Rightarrow x = 1

Combining like terms gives

10x^2 = 10

, which simplifies to

x^2 = 1

. Since the intersection point is in the first quadrant,

x

must be positive.

3

Substitute the value of

x

back into the linear equation to find

y

.

y = 3(1) = 3

Using the value of

x

in the linear equation determines the corresponding

y

-coordinate.

4

Calculate the sum of the coordinates

x + y

.

1 + 3 = 4

The question asks for the value of the sum of the coordinates of the intersection point.

Key Concept

Solving a system of linear and non-linear (circular) equations using substitution.

Question 1928Question →

A local bakery sells chocolate chip cookies for $1.50 each and oatmeal raisin cookies for$ 1.25 each. The table below shows the number of cookies sold on Monday and Tuesday:

Day	Chocolate Chip	Oatmeal Raisin
Monday	$80$	$60$
Tuesday	$95$	$70$

If the sales data is represented by the matrix $S = \begin{bmatrix} 80 & 60 \\ 95 & 70 \end{bmatrix}$ and the prices are represented by the matrix $P = \begin{bmatrix} 1.50 \\ 1.25 \end{bmatrix}$ , which of the following matrices represents the total revenue, in dollars, generated on Monday and Tuesday, respectively?

\begin{bmatrix} 190.00 \\ 223.75 \end{bmatrix}

\begin{bmatrix} 120.00 & 75.00 \\ 142.50 & 87.50 \end{bmatrix}

\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}

\begin{bmatrix} 238.75 \\ 177.50 \end{bmatrix}

\begin{bmatrix} 185.00 \\ 220.00 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}$

Answer

The matrix representing the revenue is

\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}

The correct matrix is obtained by performing standard matrix multiplication of

S

(dimensions

2 \times 2

) and

P

(dimensions

2 \times 1

). The resulting matrix has dimensions

2 \times 1

. The first element is the sum of the products of the first row of

S

and the column of

P

:

80 \times 1.50 + 60 \times 1.25 = 120 + 75 = 195

. The second element is the sum of the products of the second row of

S

and the column of

P

:

95 \times 1.50 + 70 \times 1.25 = 142.50 + 87.50 = 230

. This gives the matrix containing the values 195.00 and 230.00.

Step-by-Step Solution

1

Set up the matrix multiplication of sales and prices

Multiply the

2 \times 2

sales matrix

S

by the

2 \times 1

price matrix

P

:

S \times P = \begin{bmatrix} 80 & 60 \\ 95 & 70 \end{bmatrix} \begin{bmatrix} 1.50 \\ 1.25 \end{bmatrix}

Matrix multiplication aligns the quantities sold of each item with their respective unit prices to calculate the total revenue per day.

2

Perform the matrix multiplication for each row

Monday's revenue is

80 \times 1.50 + 60 \times 1.25 = 120 + 75 = 195

. Tuesday's revenue is

95 \times 1.50 + 70 \times 1.25 = 142.50 + 87.50 = 230

.

Multiplying the elements of each row in the first matrix by the corresponding elements in the column of the second matrix, and then summing them, calculates the dot product for each day.

3

Write the results in a

2 \times 1

matrix

The final product matrix is

\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}

.

The product of a

2 \times 2

matrix and a

2 \times 1

matrix is a

2 \times 1

matrix containing the total values for each row.

Key Concept

Matrix Multiplication in Word Problems

Question 1929Question →

For the functions $f(x) = |x - 2| - 3$ and $g(x) = x^2 - 2x$ , what is the sum of all real values of $x$ for which $g(f(x)) = 0$ ?

2

4

8

12

-8

Show answer & explanation

Answer: 8

Answer

The sum of all real values of

x

is 8.

To solve

g(f(x)) = 0

, we substitute

f(x)

as the variable in

g(x)

, obtaining

(f(x))^2 - 2f(x) = 0

. Factoring this equation yields

f(x)(f(x) - 2) = 0

, giving two cases:

f(x) = 0

or

f(x) = 2

. For the first case,

|x - 2| - 3 = 0

simplifies to

|x - 2| = 3

, which has solutions

x = 5

and

x = -1

. For the second case,

|x - 2| - 3 = 2

simplifies to

|x - 2| = 5

, which has solutions

x = 7

and

x = -3

. Summing these four solutions gives

5 + (-1) + 7 + (-3) = 8

.

Step-by-Step Solution

1

Substitute

f(x)

into

g(x)

to set up the composite equation

g(f(x)) = 0

.

g(f(x)) = (f(x))^2 - 2f(x) = 0

To evaluate a function composition, the output of the inner function

f(x)

becomes the input of the outer function

g(x)

.

2

Factor the quadratic equation in terms of

f(x)

.

f(x)(f(x) - 2) = 0

, which means

f(x) = 0

or

f(x) = 2

.

Factoring allows us to split the quadratic equation into two separate linear cases.

3

Substitute

f(x) = |x - 2| - 3

into the first case,

f(x) = 0

, and solve for

x

.

|x - 2| - 3 = 0 \Rightarrow |x - 2| = 3 \Rightarrow x - 2 = 3

or

x - 2 = -3

, yielding

x = 5

and

x = -1

.

An absolute value equation

|u| = c

has two cases:

u = c

and

u = -c

when

c \geq 0

.

4

Substitute

f(x) = |x - 2| - 3

into the second case,

f(x) = 2

, and solve for

x

.

|x - 2| - 3 = 2 \Rightarrow |x - 2| = 5 \Rightarrow x - 2 = 5

or

x - 2 = -5

, yielding

x = 7

and

x = -3

.

We must solve the second case using both positive and negative scenarios for the absolute value.

5

Calculate the sum of all four real solutions.

5 + (-1) + 7 + (-3) = 8

The question asks for the sum of all real values of

x

that satisfy the composite equation.

Key Concept

Function Composition and Absolute Value Equations

Alternative Method

Instead of expanding the algebraic expressions, substitute a temporary variable

u = f(x)

into

g(u) = 0

to get

u^2 - 2u = 0

, which easily factors to

u = 0

or

u = 2

. Then, substitute

f(x)

back to solve

|x - 2| - 3 = 0

and

|x - 2| - 3 = 2

.

Estimated Time:1m 30s

Question 1930Question →

In the quadratic equation $a x^2 + b x + c = 0$ , the coefficients $a$ , $b$ , and $c$ are real numbers, and $a = \frac{2}{3}$ . If one of the roots of this equation is $x = \frac{1}{2} - i\sqrt{2}$ , where $i = \sqrt{-1}$ , what is the value of the product $bc$ ?

-1

-\frac{7}{9}

-\frac{1}{2}

0

\frac{7}{9}

Show answer & explanation

Answer: $-1$

Answer

-1

For any quadratic equation with real coefficients, if a complex number is a root, its conjugate must also be a root. Thus, the two roots are

x_1 = \frac{1}{2} - i\sqrt{2}

and

x_2 = \frac{1}{2} + i\sqrt{2}

. Using Vieta's formulas, the sum of the roots is

x_1 + x_2 = 1 = -\frac{b}{a}

, which gives

b = -\frac{2}{3}

. The product of the roots is

x_1 x_2 = \frac{1}{4} - 2i^2 = \frac{9}{4} = \frac{c}{a}

, which gives

c = \frac{3}{2}

. The product

bc

is therefore

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

.

Step-by-Step Solution

1

Find the second root of the quadratic equation.

The second root is

x_2 = \frac{1}{2} + i\sqrt{2}

.

Since the quadratic equation has real coefficients, complex roots must occur in conjugate pairs.

2

Calculate the sum of the roots to find the coefficient

b

.

b = -\frac{2}{3}

The sum of the roots is

x_1 + x_2 = (\frac{1}{2} - i\sqrt{2}) + (\frac{1}{2} + i\sqrt{2}) = 1

. By Vieta's formulas,

x_1 + x_2 = -\frac{b}{a}

. Substituting

a = \frac{2}{3}

gives

1 = -\frac{b}{2/3} \implies b = -\frac{2}{3}

.

3

Calculate the product of the roots to find the coefficient

c

.

c = \frac{3}{2}

The product of the roots is

x_1 x_2 = (\frac{1}{2} - i\sqrt{2})(\frac{1}{2} + i\sqrt{2}) = (\frac{1}{2})^2 - (i\sqrt{2})^2 = \frac{1}{4} - 2i^2

. Since

i^2 = -1

, the product is \frac{1}{4} - 2(-1) = \frac{9}{4}

. By Vieta's formulas,

x_1 x_2 = \frac{c}{a}

. Substituting

a = \frac{2}{3}

gives \frac{9}{4} = \frac{c}{2/3} \implies c = \frac{3}{2}

.

4

Calculate the product

bc

.

bc = -1

Multiplying the values of

b

and

c

gives

bc = (-\frac{2}{3})(\frac{3}{2}) = -1

.

Key Concept

Relating the roots of a quadratic equation with real coefficients to its coefficients via Vieta's formulas and the Complex Conjugate Root Theorem.

Question 1931Question →

A line and a parabola intersect at a point $(x, y)$ in the first quadrant of the standard $(x, y)$ coordinate plane. If the equation of the line is $y = x + 1$ and the equation of the parabola is $y = (x - 1)^2$ , what is the value of $x + y$ ?

3

5

1

7

9

Show answer & explanation

Answer: 7

Answer

7

To find the intersection point, we set the two equations equal to each other:

(x - 1)^2 = x + 1

. Expanding the left side gives

x^2 - 2x + 1 = x + 1

. Subtracting

x + 1

from both sides yields

x^2 - 3x = 0

. Factoring gives

x(x - 3) = 0

, so

x = 0

or

x = 3

. The first quadrant requires positive coordinates, so we choose

x = 3

. Substituting this back into either equation gives

y = 3 + 1 = 4

. Thus, the intersection point is

(3, 4)

, and the sum of the coordinates is

3 + 4 = 7

.

Step-by-Step Solution

1

Set the two equations equal to find the x-coordinates of the intersection points.

(x - 1)^2 = x + 1

Since both equations are equal to

y

, their right-hand sides must be equal at the points of intersection.

2

Expand the quadratic expression and simplify the equation.

x^2 - 2x + 1 = x + 1 \implies x^2 - 3x = 0

Expanding the binomial

(x-1)^2

yields

x^2 - 2x + 1

. Subtracting

x

and

1

from both sides simplifies the equation to standard quadratic form.

3

Factor the quadratic equation to solve for

x

.

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

or

x = 3

Factoring out the greatest common factor

x

allows us to find the roots of the equation.

4

Determine which solution lies in the first quadrant and calculate the corresponding y-coordinate.

x = 3 \implies y = 3 + 1 = 4

The first quadrant requires both coordinates to be strictly positive. The solution

x = 0

gives

(0, 1)

, which is on the y-axis. Thus, we must use

x = 3

, which gives

(3, 4)

.

5

Calculate the sum

x + y

.

3 + 4 = 7

The question asks for the value of

x + y

for the first-quadrant intersection point.

Key Concept

Solving systems of linear and non-linear equations by substitution, factoring quadratic equations, and applying coordinate plane quadrant constraints.

Estimated Time:1m 0s

Question 1932Question →

A logistics company determines that its daily operating cost, $C$ (in dollars), for a delivery truck satisfies the inequality $\frac{a(2C - 3)}{3} + \frac{5}{4} \leq \frac{C - a}{2} - \frac{7}{8}$ , where $a$ is a constant regional fuel efficiency parameter such that $a < -2$ . Which of the following represents the range of possible operating costs $C$ ?

C \geq \frac{51 - 12a}{12 - 16a}

C \leq \frac{51 - 12a}{12 - 16a}

C \geq \frac{4a + 89}{28 - 16a}

C \geq \frac{228a + 21}{12 - 16a}

C \leq \frac{36a + 51}{16a + 12}

Show answer & explanation

Answer: $C \geq \frac{51 - 12a}{12 - 16a}$

Answer

The range of possible operating costs is

C \geq \frac{51 - 12a}{12 - 16a}

.

To solve the inequality, we first eliminate the denominators by multiplying the entire inequality by 24, resulting in

8a(2C - 3) + 30 \leq 12(C - a) - 21

. Expanding both sides and gathering all terms with

C

on the left gives

(16a - 12)C \leq 12a - 51

. Because

a < -2

, the coefficient

16a - 12

is negative. Dividing by a negative number reverses the inequality direction, giving

C \geq \frac{12a - 51}{16a - 12}

. Multiplying the numerator and denominator by

-1

yields the correct solution.

Step-by-Step Solution

1

Clear the denominators by multiplying all terms by the least common multiple of 3, 4, 2, and 8, which is 24.

8a(2C - 3) + 30 \leq 12(C - a) - 21

Eliminating fractions simplifies the algebraic manipulation of the linear inequality.

2

Expand both sides of the inequality.

16aC - 24a + 30 \leq 12C - 12a - 21

Distributing terms allows grouping the variable

C

and the constants.

3

Isolate the terms containing

C

on the left side and all other terms on the right side.

16aC - 12C \leq 12a - 51

Grouping like terms is necessary to solve for

C

.

4

Factor out

C

on the left side.

(16a - 12)C \leq 12a - 51

This isolates the variable

C

with a single coefficient.

5

Determine the sign of the coefficient

(16a - 12)

based on the condition

a < -2

.

Since

a < -2

, we have

16a < -32

, which implies

16a - 12 < -44

. Thus, the coefficient is negative.

Knowing whether the coefficient is positive or negative determines whether the inequality sign must flip upon division.

6

Divide both sides by

(16a - 12)

and reverse the inequality sign.

C \geq \frac{12a - 51}{16a - 12}

Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

7

Simplify the resulting fraction by multiplying the numerator and denominator by

-1

.

C \geq \frac{51 - 12a}{12 - 16a}

This yields the simplified final expression matching the target choice.

Key Concept

Solving linear inequalities involving fractions and variable parameters, with strict application of the inequality sign-flip rule when dividing by a negative algebraic term.

Estimated Time:3m 0s

Question 1933Question →

A chemist is preparing a mixture. The volume of acid, $V$ in liters, required for a specific reaction satisfies the equation:

\frac{3}{5}(2V - 7) - \frac{1}{2}(V + 4) = \frac{1}{10}(3V - 2)

What is the value of the expression $4V + 3$ ?

15

23

27

47

63

Show answer & explanation

Answer: 63

Answer

The value of the expression is 63.

The correct answer is 63. Multiplying both sides of the equation by the least common denominator of 10 eliminates the fractions and yields

6(2V - 7) - 5(V + 4) = 3V - 2

. Distributing the factors results in

12V - 42 - 5V - 20 = 3V - 2

, which simplifies to

7V - 62 = 3V - 2

. Subtracting

3V

and adding 62 to both sides produces

4V = 60

, which gives

V = 15

. Substituting 15 into the expression

4V + 3

results in

4(15) + 3 = 63

.

Step-by-Step Solution

1

Multiply the entire equation by the least common denominator (LCD) to eliminate the fractions.

6(2V - 7) - 5(V + 4) = 1(3V - 2)

The denominators are 5, 2, and 10, so the LCD is 10.

2

Distribute the constants on the left side of the equation.

12V - 42 - 5V - 20 = 3V - 2

Distributing 6 to

(2V - 7)

yields

12V - 42

, and distributing

-5

to

(V + 4)

yields

-5V - 20

.

3

Combine like terms on the left side of the equation.

7V - 62 = 3V - 2

Combining the variable terms gives

12V - 5V = 7V

, and combining the constant terms gives

-42 - 20 = -62

.

4

Isolate the variable term by performing inverse operations.

4V = 60

Subtract

3V

from both sides to get

4V

, and add 62 to both sides to get 60.

5

Solve for

V

by dividing both sides of the equation.

V = 15

Dividing 60 by 4 yields 15.

6

Substitute the value of

V

into the requested expression.

4(15) + 3 = 63

The question asks for the value of the expression

4V + 3

, not the value of

V

.

Key Concept

Solving multi-step linear equations containing fractions by finding a common denominator, distributing terms correctly, and evaluating algebraic expressions.

Estimated Time:2m 0s

Question 1934Question →

For all real numbers $p$ and $q$ , which of the following is equivalent to the expression $2p^2(p - 3q) - 3q(p^2 - q^2) - (2p^3 - 5p^2q)$ ?

-14p^2q + 3q^3

-4p^2q - 3q^3

-4p^2q + 3q^3

-4p^2q + 3q^2

-14p^2q - 3q^3

Show answer & explanation

Answer: $-4p^2q + 3q^3$

Answer

-4p^2q + 3q^3

The correct answer is obtained by distributing all factors and negative signs, then combining the coefficients of the like terms. This yields

-4p^2q + 3q^3

.

Step-by-Step Solution

1

Distribute the term

2p^2

into the first parenthesis,

-3q

into the second parenthesis, and the negative sign into the third parenthesis.

2p^2(p - 3q) = 2p^3 - 6p^2q

-3q(p^2 - q^2) = -3p^2q + 3q^3

-(2p^3 - 5p^2q) = -2p^3 + 5p^2q

Expanding the terms removes the parentheses and prepares the expression for combining like terms.

2

Combine the expanded parts into a single expression and group the like terms together.

(2p^3 - 2p^3) + (-6p^2q - 3p^2q + 5p^2q) + 3q^3

Grouping terms with identical variable parts makes combining coefficients straightforward.

3

Simplify the coefficients for each group of like terms.

0p^3 + (-6 - 3 + 5)p^2q + 3q^3 = -4p^2q + 3q^3

Simplifying the combined coefficients yields the final, simplified expression.

Key Concept

Simplifying Expressions and Combining Like Terms

Question 1935Question →

A geometric sequence has a first term of $\frac{1}{4}$ and a common ratio of $\frac{1}{2}$ . What is the 4th term of this sequence?

\frac{1}{32}

\frac{1}{64}

\frac{1}{512}

64

\frac{2}{3}

Show answer & explanation

Answer: $\frac{1}{32}$

Answer

\frac{1}{32}

The correct answer is found by substituting

a_1 = \frac{1}{4}

,

r = \frac{1}{2}

, and

n = 4

into the geometric sequence formula

a_n = a_1 \cdot r^{n-1}

. This gives

a_4 = \frac{1}{4} \cdot (\frac{1}{2})^3 = \frac{1}{4} \cdot \frac{1}{8} = \frac{1}{32}

.

Step-by-Step Solution

1

Identify the given components of the geometric sequence.

The first term is

a_1 = \frac{1}{4}

and the common ratio is

r = \frac{1}{2}

. We want to find the 4th term, so

n = 4

.

Before performing any calculations, we must identify the values of the variables needed for the geometric sequence formula.

2

Substitute the values into the general formula for the

n

-th term of a geometric sequence,

a_n = a_1 \cdot r^{n-1}

.

a_4 = \frac{1}{4} \cdot \left(\frac{1}{2}\right)^{4-1} = \frac{1}{4} \cdot \left(\frac{1}{2}\right)^3

This formula relates the

n

-th term to the first term and the common ratio.

3

Evaluate the exponent first, and then multiply the fractions.

a_4 = \frac{1}{4} \cdot \frac{1}{8} = \frac{1}{32}

Following the order of operations, we raise the common ratio to the 3rd power before multiplying by the first term.

Key Concept

Finding the

n

-th term of a geometric sequence using the general formula

a_n = a_1 \cdot r^{n-1}

.

Question 1936Question →

If the polynomial $2x^2 + kx - 12$ can be factored as the product of two binomials with integer coefficients, what is the greatest possible integer value of $k$ ?

Show answer & explanation

Answer: 23

Answer

The greatest possible integer value of

k

is 23.

The greatest possible value of

k

is 23. This is found by setting the factored form of the expression as

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

. We find that

ab = -12

and

k = 2b + a

. The possible integer pairs for

(a, b)

that multiply to

-12

are tested to find the maximum sum of

2b + a

. The maximum is achieved when

b = 12

and

a = -1

, yielding

2(12) - 1 = 23

.

Step-by-Step Solution

1

Set up the algebraic representation for the binomial factors with integer coefficients.

The polynomial must factor into the form

(2x + a)(x + b)

where

a

and

b

are integers.

Since the leading coefficient is 2 (a prime number), the coefficients of the linear terms in the binomial factors must be 2 and 1 to obtain a product of

2x^2

.

2

Expand the product of the binomials to relate the parameters

a

and

b

to the original expression.

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

, which implies

ab = -12

and

k = 2b + a

.

By equating the coefficients of corresponding terms in the expanded expression and the original polynomial, we establish relationships for the constant term and the linear coefficient.

3

Analyze the factors of

-12

to find the integer values of

a

and

b

that maximize the linear term coefficient.

Choosing

b = 12

and

a = -1

gives

ab = -12

and results in

k = 2(12) + (-1) = 23

.

To maximize

2b + a

, we choose the largest possible positive factor of

-12

for the term multiplied by 2, which is 12, paired with the corresponding negative factor

-1

for

a

.

Key Concept

Factoring quadratic trinomials of the form

ax^2 + bx + c

with

a > 1

Question 1937Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined as $z = \frac{(1 + 2i)^3}{2 - i}$ . What is the absolute value of $z$ ?

Show answer & explanation

Answer: 5

Answer

5

The complex number

z

simplifies to

-4 - 3i

. The absolute value of a complex number

a + bi

is defined as

\sqrt{a^2 + b^2}

, which gives

\sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = 5

. Alternatively, using properties of the modulus,

|z| = \frac{|1 + 2i|^3}{|2 - i|} = \frac{(\sqrt{1^2 + 2^2})^3}{\sqrt{2^2 + (-1)^2}} = \frac{(\sqrt{5})^3}{\sqrt{5}} = 5

.

Step-by-Step Solution

1

Expand the numerator

(1 + 2i)^3

-11 - 2i

To begin simplifying the complex fraction by expanding the cubed binomial in the numerator.

2

Divide the result by the denominator

2 - i

by multiplying both numerator and denominator by the complex conjugate

2 + i

-4 - 3i

To express the quotient in standard complex form

a + bi

.

3

Find the absolute value of the simplified complex number

z = -4 - 3i

5

To calculate the final magnitude of the complex number using

|a + bi| = \sqrt{a^2 + b^2}

.

Key Concept

Complex number operations, complex division, and absolute value (modulus) of complex numbers

Question 1938Question →

A projectile is launched from ground level with an initial upward velocity of $v$ meters per second. The height of the projectile in meters after $t$ seconds is modeled by the equation $h(t) = -\frac{9}{2}t^2 + vt$ . A sensor is placed at a height of $32$ meters. What is the minimum integer value of $v$ for which the projectile is at or above the height of the sensor for a duration of at least $4$ seconds?

Show answer & explanation

Answer: 30

Answer

The minimum integer value of the initial upward velocity

v

is 30.

The height condition

h(t) \ge 32

leads to the quadratic inequality

9t^2 - 2vt + 64 \le 0

. The roots

t_1

and

t_2

of

9t^2 - 2vt + 64 = 0

represent the boundary times. The duration above

32

meters is the difference of the roots:

t_2 - t_1 = \frac{\sqrt{b^2 - 4ac}}{|a|} = \frac{\sqrt{4v^2 - 2304}}{9}

. Setting this to be at least

4

seconds gives

\frac{\sqrt{4v^2 - 2304}}{9} \ge 4

. Squaring both sides after multiplying by

9

yields

4v^2 - 2304 \ge 1296

, which simplifies to

4v^2 \ge 3600

, or

v^2 \ge 900

. Taking the positive square root gives

v \ge 30

, meaning the minimum integer value is

30

.

Step-by-Step Solution

1

Write the inequality representing the condition that the projectile is at or above the sensor height.

9t^2 - 2vt + 64 \le 0

The height function

h(t) = -\frac{9}{2}t^2 + vt

must be greater than or equal to

32

meters.

2

Use the formula for the difference of roots to represent the time duration.

t_2 - t_1 = \frac{\sqrt{4v^2 - 2304}}{9}

The duration of time the projectile is above the sensor is the difference between the two times at which the projectile crosses the sensor height.

3

Set the duration to be at least

4

seconds and solve for

v

.

v \ge 30

Solving the inequality

\frac{\sqrt{4v^2 - 2304}}{9} \ge 4

yields the range of initial velocities.

4

Find the smallest integer in the solution set.

30

Since the velocity must be an integer and at least

30

, the minimum possible integer value is

30

.

Key Concept

Solving quadratic inequalities and finding the difference of roots using the discriminant.

Question 1939Question →

A company offers two monthly data storage plans. Plan A costs $32.00 plus$ 0.08 per gigabyte of data stored. Plan B costs $15.00 plus$ 0.14 per gigabyte for the first 150 gigabytes of data stored, and $0.12 per gigabyte for all data stored beyond the first 150 gigabytes. For how many gigabytes of data stored in a single month would the monthly cost under both plans be exactly the same?

Show answer & explanation

Answer: 350

Answer

The monthly costs under both plans are exactly the same when the data stored is 350 gigabytes.

Equating the two cost functions yields

32 + 0.08g = 15 + 0.14(150) + 0.12(g - 150)

, which simplifies to

32 + 0.08g = 18 + 0.12g

. Solving for

g

gives

0.04g = 14

, which results in

g = 350

.

Step-by-Step Solution

1

Determine the cost function for Plan A.

C_A(g) = 32 + 0.08g

Plan A charges a flat fee of

32.00 and

0.08 per gigabyte.

2

Determine the cost function for Plan B when the usage exceeds 150 gigabytes.

C_B(g) = 15 + 0.14(150) + 0.12(g - 150) = 18 + 0.12g

Plan B charges

15.00 flat,

0.14 per gigabyte for the first 150 gigabytes, and $0.12 per gigabyte for any additional usage.

3

Equate the two cost functions and solve for

g

.

32 + 0.08g = 18 + 0.12g \implies 14 = 0.04g \implies g = 350

To find the usage where both plans cost the same, set their cost functions equal and isolate the variable

g

.

Key Concept

Solving linear equations in real-life contexts involving piecewise rates

Question 1940Question →

For all real numbers $x$ such that $x \neq 0$ and $x \neq 3$ , what is the real solution to the equation $\frac{x}{x-3} + \frac{2}{x} = \frac{3}{x-3}$ ?

Show answer & explanation

Answer: -2

Answer

The correct answer is -2.

Subtracting

\frac{x}{x-3}

from both sides yields

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

. Since

3-x = -(x-3)

, the right side simplifies to

-1

for all

x \neq 3

. The equation becomes

\frac{2}{x} = -1

, which gives

x = -2

. Since

-2

does not violate the domain constraints, it is the correct solution.

Step-by-Step Solution

1

Subtract

\frac{x}{x-3}

from both sides of the equation.

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

To group terms with common denominators on the same side.

2

Simplify the fraction on the right side of the equation.

\frac{2}{x} = -1

The numerator

3-x

is the negative of the denominator

x-3

, so their quotient is

-1

for all

x \neq 3

.

3

Solve the simplified equation

\frac{2}{x} = -1

for

x

.

x = -2

Multiplying both sides by

x

yields

2 = -x

, and dividing by

-1

gives

x = -2

.

4

Check the solution against the domain restrictions

x \neq 0

and

x \neq 3

.

The solution

x = -2

is valid.

Since

-2

is neither

0

nor

3

, it does not cause any denominator in the original equation to equal zero.

Key Concept

Solving rational equations by isolating terms with common denominators and checking for extraneous solutions.

Answer

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