All practice questions

Question 1401Question →

If the expression $\frac{10x^2 + x - 3}{2x - 1}$ is equivalent to $ax + b$ for all $x \neq 0.5$ , where $a$ and $b$ are constants, what is the value of $a + b$ ?

Show answer & explanation

Answer: 8

Answer

The value of

a + b

is 8.

To find the equivalent expression, factor the numerator:

10x^2 + x - 3 = (2x - 1)(5x + 3)

. For all

x \neq 0.5

, the denominator

2x - 1

is non-zero, so we can divide out the common factor

(2x - 1)

to get

5x + 3

. Matching

5x + 3

with

ax + b

gives

a = 5

and

b = 3

. The sum of these constants is

5 + 3 = 8

.

Step-by-Step Solution

1

Factor the numerator of the expression

10x^2 + x - 3 = (2x - 1)(5x + 3)

Factoring the quadratic trinomial allows us to identify common factors that can be simplified.

2

Simplify the rational expression

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

(for

x \neq 0.5

)

Since

x \neq 0.5

, the term

2x - 1

is non-zero and can be canceled from both the numerator and the denominator.

3

Identify the values of

a

and

b

a = 5

and

b = 3

By comparing the simplified expression

5x + 3

to the form

ax + b

, the coefficients of corresponding terms must be equal.

4

Calculate the sum of

a

and

b

5 + 3 = 8

The question asks for the sum of the constants

a

and

b

.

Key Concept

Simplifying rational expressions by factoring and coefficient matching

Question 1402Question →

For $x > 1$ , which of the following expressions is equivalent to $\frac{x - x^{-1}}{x^{1/2} - x^{-1/2}}$ ?

x^{1/2} - x^{-1/2}

x + 1

x^{1/2} + x^{-1/2}

x^{3/2} + x^{1/2}

Show answer & explanation

Answer: $x^{1/2} + x^{-1/2}$

Answer

x^{1/2} + x^{-1/2}

The expression

x^{1/2} + x^{-1/2}

is correct because the numerator

x - x^{-1}

can be factored using the difference of squares identity,

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

, where

a = x^{1/2}

and

b = x^{-1/2}

. This yields

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

. Substituting this back into the original fraction and canceling the common factor of

x^{1/2} - x^{-1/2}

in both the numerator and denominator simplifies the expression directly to

x^{1/2} + x^{-1/2}

.

Step-by-Step Solution

1

Recognize the numerator

x - x^{-1}

as a difference of squares in terms of the base variables

x^{1/2}

and

x^{-1/2}

.

Write

x - x^{-1}

as

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

.

Since the denominator is

x^{1/2} - x^{-1/2}

, factoring the numerator as a difference of squares allows us to identify a common factor that can be canceled.

2

Substitute the factored numerator back into the original expression and cancel the common factor of

x^{1/2} - x^{-1/2}

from the numerator and the denominator.

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

For all

x > 1

, the term

x^{1/2} - x^{-1/2}

is non-zero, so we can divide both the numerator and the denominator by this common term to simplify the expression.

Key Concept

Factoring algebraic expressions using the difference of squares identity with fractional exponents.

Alternative Method

Convert the fractional and negative exponents into standard algebraic fractions:

\frac{x - \frac{1}{x}}{\sqrt{x} - \frac{1}{\sqrt{x}}}

. Find common denominators for both the numerator and denominator to write the expression as

\frac{\frac{x^2 - 1}{x}}{\frac{x - 1}{\sqrt{x}}}

. Next, multiply the numerator by the reciprocal of the denominator:

\frac{(x - 1)(x + 1)}{x} \cdot \frac{\sqrt{x}}{x - 1}

. Cancel the common factor of

x - 1

to get

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

. Distributing

\sqrt{x}

and dividing each term by

x

gives

\frac{x\sqrt{x}}{x} + \frac{\sqrt{x}}{x} = \sqrt{x} + \frac{1}{\sqrt{x}}

, which is equivalent to

x^{1/2} + x^{-1/2}

.

Estimated Time:1m 30s

Question 1403Question →

If $\frac{3}{5}(5x - 10) - \frac{1}{2}(4x - 6) = 12$ , what is the value of $2x - 5$ ?

13

15

25

37

Show answer & explanation

Answer: 25

Answer

The correct answer is 25.

The correct answer is 25. Distributing the fractions yields

3x - 6 - 2x + 3 = 12

, which simplifies to

x - 3 = 12

, so

x = 15

. Substituting this into

2x - 5

gives

2(15) - 5 = 25

.

Step-by-Step Solution

1

Distribute the coefficients to the terms inside the parentheses.

(3x - 6) - (2x - 3) = 12

To simplify the equation by removing the parentheses.

2

Combine like terms on the left side of the equation.

x - 3 = 12

To group the variable terms and constant terms together.

3

Isolate the variable

x

by adding 3 to both sides of the equation.

x = 15

To find the value of

x

.

4

Substitute

x = 15

into the expression

2x - 5

.

2(15) - 5 = 25

To find the final value requested by the question.

Key Concept

Solving linear equations in one variable with grouping symbols and fractions, and evaluating an algebraic expression.

Estimated Time:1m 30s

Question 1404Question →

Consider the system of equations below.

\begin{aligned} 2x - y &= 4 \\ x + 2y &= 7 \end{aligned}

If

(x, y)

is the solution to the system of equations, what is the value of

x + y

?

1

5

8

11

Show answer & explanation

Answer: 5

Answer

The value of

x + y

is 5.

To solve the system, we can express

y

in terms of

x

from the first equation:

y = 2x - 4

. Substituting this into the second equation gives

x + 2(2x - 4) = 7

. Distributing the 2 gives

x + 4x - 8 = 7

, which simplifies to

5x - 8 = 7

. Adding 8 to both sides yields

5x = 15

, which gives

x = 3

. Substituting

x = 3

back into the first equation yields

2(3) - y = 4 \implies 6 - y = 4 \implies y = 2

. Thus, the value of

x + y

is

3 + 2 = 5

.

Step-by-Step Solution

1

Express

y

in terms of

x

using the first equation.

y = 2x - 4

This sets up the system for solving by substitution.

2

Substitute the expression for

y

into the second equation and solve for

x

.

x + 2(2x - 4) = 7 \implies 5x - 8 = 7 \implies 5x = 15 \implies x = 3

This isolates and solves for the variable

x

.

3

Substitute the value of

x

back into the expression for

y

, then compute

x + y

.

y = 2(3) - 4 = 2

, and

x + y = 3 + 2 = 5

This finds the value of

y

and computes the final required sum.

Key Concept

Solving a system of linear equations in two variables to find the value of a linear combination.

Question 1405Question →

In the $xy$ -plane, the line $y = mx$ , where $m$ is a positive constant, is tangent to the parabola $y = x^2 + 9$ . What is the value of $m$ ?

3

6

12

36

Show answer & explanation

Answer: 6

Answer

The value of

m

is 6.

Equating the equations of the line and the parabola yields the quadratic equation

x^2 - mx + 9 = 0

. For the line to be tangent to the parabola, this system must have exactly one real solution, meaning the discriminant

b^2 - 4ac

must equal

0

. Substituting

a = 1

,

b = -m

, and

c = 9

into the discriminant formula gives

(-m)^2 - 4(1)(9) = 0

, which simplifies to

m^2 - 36 = 0

. Solving for the positive constant

m

gives

6

.

Step-by-Step Solution

1

Set the equation of the line equal to the equation of the parabola to find their intersection points.

mx = x^2 + 9

The intersection points of the system correspond to the values of

x

where both equations have the same

y

-value.

2

Rewrite the equation in standard quadratic form

ax^2 + bx + c = 0

by subtracting

mx

from both sides.

x^2 - mx + 9 = 0

This allows the identification of the coefficients

a

,

b

, and

c

to compute the discriminant.

3

Identify the coefficients and set the discriminant

b^2 - 4ac

equal to 0.

(-m)^2 - 4(1)(9) = 0

A line is tangent to a parabola if and only if the system has exactly one real solution, which corresponds to a discriminant of zero.

4

Solve the equation for the positive constant

m

.

m^2 - 36 = 0 \implies m^2 = 36 \implies m = 6

Taking the square root of both sides gives

m = \pm 6

. Since

m

is specified as a positive constant, we choose

m = 6

.

Key Concept

Solving nonlinear systems of equations where a line is tangent to a parabola by setting the discriminant of the resulting quadratic equation to zero.

Question 1406Question →

A circle and a line are graphed in the $xy$ -plane. The circle is defined by the equation $x^2 + y^2 = 25$ , and the line is defined by the equation $y = 3$ . If the line intersects the circle at the point $(x, 3)$ , where $x > 0$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is 4.

Substituting

y = 3

into the circle equation

x^2 + y^2 = 25

yields

x^2 + 3^2 = 25

. Simplifying this gives

x^2 + 9 = 25

, which simplifies to

x^2 = 16

. Taking the square root of both sides gives

x = 4

or

x = -4

. Since it is given that

x > 0

, the value of

x

must be

4

.

Step-by-Step Solution

1

Substitute the value of

y = 3

into the circle's equation.

x^2 + 3^2 = 25

Since the line is

y = 3

, any point of intersection must satisfy this y-coordinate. Substituting it into the circle's equation allows us to solve for the x-coordinate.

2

Simplify the equation and isolate

x^2

.

x^2 = 16

Squaring

3

gives

9

, and subtracting

9

from both sides of the equation

x^2 + 9 = 25

isolates

x^2

.

3

Solve for

x

and apply the constraint

x > 0

.

x = 4

Taking the square root of both sides of

x^2 = 16

gives

x = 4

or

x = -4

. Since the problem states

x > 0

, the only valid solution is

4

.

Key Concept

Solving systems of nonlinear equations using substitution

Question 1407Question →

An artist creates custom oil paintings and charcoal sketches.

* Each oil painting, $x$ , requires $4$ hours of setup time and $2$ hours of finishing time.
* Each charcoal sketch, $y$ , requires $1$ hour of setup time and $3$ hours of finishing time.

Each week, the artist can spend at most $20$ hours on setup time and at most $30$ hours on finishing time. Which of the following systems of inequalities represents this situation?

\begin{aligned} 4x + y &\ge 20 \\ 2x + 3y &\ge 30 \end{aligned}

\begin{aligned} 4x + 2y &\le 20 \\ x + 3y &\le 30 \end{aligned}

\begin{aligned} 4x + y &\le 20 \\ 2x + 3y &\le 30 \end{aligned}

\begin{aligned} 4x + y &\le 30 \\ 2x + 3y &\le 20 \end{aligned}

Show answer & explanation

Answer: $\begin{aligned} 4x + y &\le 20 \\ 2x + 3y &\le 30 \end{aligned}$

Answer

The system of inequalities consisting of the inequalities

4x + y \le 20

and

2x + 3y \le 30

The variable

x

represents the number of oil paintings and

y

represents the number of charcoal sketches. Setup time constraint: each oil painting requires

4

hours and each charcoal sketch requires

1

hour. Thus, the total setup time is

4x + y

hours. Since the artist can spend at most

20

hours on setup time, this is represented by the inequality

4x + y \le 20

. Finishing time constraint: each oil painting requires

2

hours and each charcoal sketch requires

3

hours. Thus, the total finishing time is

2x + 3y

hours. Since the artist can spend at most

30

hours on finishing time, this is represented by the inequality

2x + 3y \le 30

. Combining these gives the correct system.

Step-by-Step Solution

1

Identify the variables and setup times for each type of artwork to write the setup time inequality.

The total setup time is

4x + y

hours. Since the limit is at most

20

hours, the inequality is

4x + y \le 20

.

To establish the mathematical representation of the setup time constraint.

2

Identify the variables and finishing times for each type of artwork to write the finishing time inequality.

The total finishing time is

2x + 3y

hours. Since the limit is at most

30

hours, the inequality is

2x + 3y \le 30

.

To establish the mathematical representation of the finishing time constraint.

3

Combine the two inequalities into a system.

The system of inequalities is

4x + y \le 20

and

2x + 3y \le 30

.

To find the system that represents both constraints simultaneously.

Key Concept

Translating real-world constraints into a system of linear inequalities in two variables.

Question 1408Question →

An architect is tracking the height of a new skyscraper under construction. The total height $H$ , in feet, of the building $w$ weeks after construction of the main frames began is modeled by the equation $H = 120 + 15w$ . According to the model, by how many feet does the height of the building increase each week?

Show answer & explanation

Answer: 15

Answer

The height of the building increases by 15 feet each week.

In the linear model

H = 120 + 15w

, the variable

H

represents the total height of the building in feet and the variable

w

represents the number of weeks since construction began. The coefficient of the variable

w

, which is

15

, is the slope of the equation and represents the constant rate of change of the height per week. Therefore, the height of the building increases by

15

feet each week.

Step-by-Step Solution

1

Analyze the linear model equation.

The equation

H = 120 + 15w

represents the total height

H

as a function of the number of weeks

w

, where

15

is the coefficient of the variable

w

.

To determine what each part of the linear equation represents in the given context.

2

Interpret the coefficient of the independent variable

w

.

The coefficient of

w

is the slope of the linear equation, which is

15

.

The slope represents the constant rate of change, which is the weekly increase in height.

Key Concept

Interpreting the slope of a linear equation in context

Question 1409Question →

In the system of equations below, $k$ is a constant.

\begin{aligned} kx - 3y &= 12 \\ (k-5)x - 2y &= 10 \end{aligned}

If the system has no solution, what is the value of $k$ ?

-15

-10

5

15

Show answer & explanation

Answer: 15

Answer

15

The correct answer is

15

. For a system of two linear equations to have no solution, the equations must represent parallel lines. This occurs when the coefficients of the variables are proportional but the constant terms are not. In this system, the ratio of the

x

-coefficients,

\frac{k}{k-5}

, must equal the ratio of the

y

-coefficients,

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

. Solving the equation

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

by cross-multiplying gives

2k = 3(k-5)

, which simplifies to

2k = 3k - 15

. Subtracting

3k

from both sides gives

-k = -15

, so

k = 15

. Since the ratio of the constants is

\frac{12}{10} = 1.2

, which is not equal to

1.5

, this value of

k

ensures the lines are parallel and distinct.

Step-by-Step Solution

1

Set up the condition for a system of linear equations to have no solution.

The coefficients of

x

and

y

must be proportional, but not equal to the ratio of the constant terms:

\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}

.

If the coefficients are proportional, the lines have the same slope. If the constant terms are not in the same proportion, the lines have different intercepts, meaning they are parallel and distinct, and thus never intersect.

2

Substitute the coefficients from the given equations into the proportion.

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

and verify that

\frac{-3}{-2} = 1.5 \neq \frac{12}{10} = 1.2

.

This establishes the relationship between the coefficients of

x

and

y

required to make the lines parallel, while ensuring they are not identical lines.

3

Solve the proportion for

k

.

\frac{k}{k-5} = \frac{3}{2} \Rightarrow 2k = 3(k-5) \Rightarrow 2k = 3k - 15 \Rightarrow k = 15

.

Cross-multiplying and isolating the variable

k

yields the value of the constant.

Key Concept

For a system of linear equations to have no solution, the lines represented by the equations must be parallel and distinct, meaning their slopes are equal but their y-intercepts are different.

Estimated Time:1m 30s

Question 1410Question →

If the expression $\frac{4x^2 + kx - 7}{2x - 1}$ is equivalent to $2x + 5 - \frac{2}{2x - 1}$ for all $x \neq \frac{1}{2}$ , where $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

8

The correct answer is 8. Combining the expression

2x + 5 - \frac{2}{2x - 1}

into a single fraction requires finding a common denominator of

2x - 1

. Multiplying the linear term

2x + 5

by

\frac{2x - 1}{2x - 1}

and subtracting 2 yields

\frac{(2x + 5)(2x - 1) - 2}{2x - 1}

. Expanding and simplifying the numerator gives

4x^2 + 8x - 7

. Equating this to the numerator of the original expression,

4x^2 + kx - 7

, shows that the coefficient of the linear term,

k

, must be equal to 8.

Step-by-Step Solution

1

Multiply the linear expression by the denominator to prepare for combining the terms.

(2x + 5)(2x - 1) = 4x^2 + 8x - 5

To combine all terms under a single common denominator of

2x - 1

, the non-fractional terms must be multiplied by the denominator.

2

Subtract the numerator of the fractional term from the expanded product.

(4x^2 + 8x - 5) - 2 = 4x^2 + 8x - 7

This completes the subtraction of the fraction over the common denominator, resulting in a single rational expression.

3

Compare the resulting numerator to the numerator of the original expression to find the value of the constant.

k = 8

For the two rational expressions to be equivalent, their numerators must be equal for all values of

x

. Thus, the coefficient of

x

in both expressions must match.

Key Concept

Equivalence of rational expressions through finding a common denominator

Question 1411Question →

A subscription-based meal kit service charges a monthly membership fee of $\$35$ plus $\$8.50$ per meal. A non-member can purchase the same meals for $\$12.00$ each, but must pay a flat monthly delivery fee of $\$7$ . How many meals must be purchased in a month for the total monthly cost for a member to be equal to the total monthly cost for a non-member?

Show answer & explanation

Answer: 8

Answer

The total monthly cost is equal for a member and a non-member when 8 meals are purchased in a month.

The correct answer of 8 is found by setting the member cost expression,

35 + 8.5m

, equal to the non-member cost expression,

12m + 7

, and solving for the number of meals,

m

.

Step-by-Step Solution

1

Represent the total monthly cost for a member and a non-member using equations where

m

is the number of meals purchased.

Member cost is represented by

35 + 8.50m

, and non-member cost is represented by

12.00m + 7

.

To set up an algebraic representation of the cost structures.

2

Equate the two cost expressions to find the number of meals where the costs are equal.

35 + 8.50m = 12.00m + 7

To find the value of

m

that makes both costs equivalent.

3

Isolate the variable

m

on one side of the equation.

3.50m = 28

Subtracting

8.50m

and

7

from both sides simplifies the equation by keeping variable terms on one side and constants on the other.

4

Divide both sides of the equation by

3.50

to solve for

m

.

m = 8

Dividing

28

by

3.50

yields the final number of meals.

Key Concept

Solving linear equations in one variable with variable terms on both sides of the equation

Estimated Time:1m 30s

Question 1412Question →

Before the completion of the transcontinental telegraph line in 1861, sending a message across North America took weeks by mail carrier. To build this new network, labor crews faced not only the daunting task of transporting thousands of heavy wooden poles across rugged mountain ranges but also __________.

Which choice completes the text so that it conforms to the conventions of Standard English?

to string copper wire over vast, arid plains.

stringing copper wire over vast, arid plains.

the constant difficulty of stringing copper wire over vast, arid plains.

they had to string copper wire over vast, arid plains.

Show answer & explanation

Answer: the constant difficulty of stringing copper wire over vast, arid plains.

Answer

the constant difficulty of stringing copper wire over vast, arid plains.

The sentence uses the correlative conjunction pair 'not only... but also...' to link two challenges faced by the labor crews. The first challenge is phrased as a noun phrase: 'the daunting task of transporting thousands of heavy wooden poles across rugged mountain ranges.' To maintain parallel structure, the second challenge must also be structured as a noun phrase. The option stating 'the constant difficulty of stringing copper wire over vast, arid plains' matches this structure perfectly by using a noun modified by an adjective and followed by a prepositional phrase starting with 'of'.

Step-by-Step Solution

1

Identify the grammatical structure of the sentence.

The sentence contains the correlative conjunctions 'not only... but also...', which require parallel grammatical structures on either side.

Correlative conjunctions must link grammatically identical elements to maintain parallel structure.

2

Analyze the grammatical structure of the first element linked by 'not only'.

The phrase 'the daunting task of transporting...' is a noun phrase led by the noun 'task'.

Determining the grammatical category of the first element dictates the category needed for the second element.

3

Evaluate the choices to find the one that matches this noun phrase structure.

The phrase 'the constant difficulty of stringing...' is a noun phrase led by the noun 'difficulty', matching the pattern of the first element.

Selecting the matching noun phrase structure completes the parallel comparison in accordance with standard English conventions.

Key Concept

Parallel Structure

Question 1413Question →

Although the panel of independent judges, which was appointed to evaluate the submissions for the annual architectural competition, initially requested more time, _______ completed the deliberations ahead of schedule.

Which choice completes the text so that it conforms to the conventions of Standard English?

it

they

them

its

Show answer & explanation

Answer: it

Answer

it

The correct option is 'it'. The pronoun refers back to 'the panel of independent judges.' In Standard English, 'panel' is a singular collective noun. An intervening relative clause ('which was appointed...') does not change the singular nature of the antecedent. Thus, the singular subject pronoun 'it' is correct to act as the subject of the verb 'completed.'

Step-by-Step Solution

1

Identify the antecedent of the pronoun to be placed in the blank.

The antecedent is the noun phrase 'the panel of independent judges.'

The pronoun must agree in number and case with its antecedent.

2

Determine the grammatical number of the antecedent.

The noun 'panel' is a singular collective noun, even though it contains the prepositional phrase 'of independent judges.'

Prepositional phrases intervening between the subject/antecedent and the pronoun do not change the number of the noun.

3

Determine the grammatical case required by the sentence structure.

The blank serves as the subject of the verb phrase 'completed the deliberations,' so a subject case pronoun is required.

The pronoun is performing the action in the dependent clause.

4

Select the option that is singular and in the subject case.

The pronoun 'it' is singular and in the subject case, matching the singular antecedent 'panel.'

This maintains pronoun-antecedent agreement in number and case.

Key Concept

Pronoun-Antecedent Agreement with Collective Nouns

Question 1414Question →

A bounded region in the $xy$ -plane is defined by the following system of inequalities:

\begin{aligned} y &\ge \frac{1}{2}x + 2 \\ y &\le 2x - 1 \\ y &\le -x + 11 \end{aligned}

What is the maximum $y$ -coordinate of any point within this region?

Show answer & explanation

Answer: 7

Answer

The maximum y-coordinate of any point in the bounded region is 7.

The solution region is a triangle with vertices at

(2, 3)

,

(4, 7)

, and

(6, 5)

. Since all inequalities are less-than-or-equal-to or greater-than-or-equal-to, the boundary points are included in the solution set. The maximum y-value occurs at the vertex

(4, 7)

, which gives a maximum y-coordinate of 7.

Step-by-Step Solution

1

Determine the equations of the boundary lines.

The boundary equations are

y = \frac{1}{2}x + 2

,

y = 2x - 1

, and

y = -x + 11

.

These equations represent the boundaries of the system of inequalities.

2

Find the intersection point of

y = 2x - 1

and

y = \frac{1}{2}x + 2

.

Solving

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

yields

x = 2

, which gives

y = 3

. The intersection is

(2, 3)

.

This is one vertex of the bounded region.

3

Find the intersection point of

y = 2x - 1

and

y = -x + 11

.

Solving

2x - 1 = -x + 11

yields

x = 4

, which gives

y = 7

. The intersection is

(4, 7)

.

This is the second vertex of the bounded region.

4

Find the intersection point of

y = -x + 11

and

y = \frac{1}{2}x + 2

.

Solving

-x + 11 = \frac{1}{2}x + 2

yields

x = 6

, which gives

y = 5

. The intersection is

(6, 5)

.

This is the third vertex of the bounded region.

5

Compare the y-coordinates of the three vertices of the bounded region.

The y-coordinates of the vertices

(2, 3)

,

(4, 7)

, and

(6, 5)

are 3, 7, and 5, respectively. The maximum value is 7.

The maximum value of a linear coordinate over a bounded convex region occurs at one of its vertices.

Key Concept

Finding the vertices of a bounded region defined by a system of linear inequalities and optimizing a coordinate value.

Question 1415Question →

A system of equations consists of the equations $y = -x^2 + 2x + 7$ and $y = 6x + k$ , where $k$ is a constant. If the system has two distinct real solutions, what is the greatest integer value of $k$ ?

Show answer & explanation

Answer: 10

Answer

The correct answer is 10. The greatest integer value of the constant that allows the system to have two distinct real solutions is 10.

To find the number of solutions to the system, equate the two equations:

-x^2 + 2x + 7 = 6x + k

. Rearranging this equation into standard quadratic form gives

x^2 + 4x + (k - 7) = 0

. For the system to have two distinct real solutions, the discriminant of this quadratic equation must be strictly greater than zero. The discriminant is calculated as

b^2 - 4ac = 4^2 - 4(1)(k - 7) = 16 - 4k + 28 = 44 - 4k

. Setting this greater than zero yields

44 - 4k > 0

, which simplifies to

k < 11

. The greatest integer value of

k

that is strictly less than 11 is 10.

Step-by-Step Solution

1

Equate the expressions for

y

from both equations.

-x^2 + 2x + 7 = 6x + k

At the points of intersection, the

y

-values of both equations must be equal.

2

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

x^2 + 4x + (k - 7) = 0

Standard form is required to calculate the discriminant of the quadratic equation.

3

Write the expression for the discriminant

\Delta = b^2 - 4ac

using the coefficients from the quadratic equation.

\Delta = 4^2 - 4(1)(k - 7) = 44 - 4k

The discriminant determines the number of real solutions to the quadratic equation.

4

Set the discriminant to be strictly greater than 0 and solve the inequality for

k

.

44 - 4k > 0 \implies k < 11

For the system to have two distinct real solutions, the discriminant must be positive.

5

Determine the greatest integer value of

k

that satisfies the inequality

k < 11

.

10

The largest integer strictly less than 11 is 10.

Key Concept

Using the discriminant of a quadratic equation derived from a nonlinear system to determine the number of real solutions.

Question 1416Question →

\begin{aligned} y &= x^2 - 7 \\ y &= 2x + 1 \end{aligned}

If $(x, y)$ is a solution to the system of equations above and $y > 0$ , what is the value of $y$ ?

4

5

-3

9

Show answer & explanation

Answer: 9

Answer

9

Substituting the expression for

y

from the second equation into the first equation yields

2x + 1 = x^2 - 7

. Subtracting

2x

and

1

from both sides results in the standard form quadratic equation

x^2 - 2x - 8 = 0

. Factoring this equation gives

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

, which yields the solutions

x = 4

and

x = -2

. Substituting these

x

-values back into the linear equation

y = 2x + 1

gives the corresponding

y

-values:

y = 2(4) + 1 = 9

and

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

. Since the system specifies the constraint

y > 0

, the correct value of

y

must be

9

.

Step-by-Step Solution

1

Substitute the expression for

y

from the linear equation into the quadratic equation.

2x + 1 = x^2 - 7

This eliminates the variable

y

and leaves a single quadratic equation in terms of

x

.

2

Rearrange the quadratic equation into standard form

ax^2 + bx + c = 0

.

x^2 - 2x - 8 = 0

Moving all terms to one side allows us to solve the quadratic equation by factoring.

3

Factor the quadratic equation.

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

, which gives

x = 4

or

x = -2

.

Finding the roots of the quadratic equation provides the possible

x

-coordinates of the solution points.

4

Substitute the

x

-values back into the linear equation

y = 2x + 1

to find the corresponding

y

-values.

For

x = 4

,

y = 2(4) + 1 = 9

. For

x = -2

,

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

.

This gives the complete coordinate pairs

(4, 9)

and

(-2, -3)

for the system's solutions.

5

Apply the constraint

y > 0

to identify the correct solution.

Since

9 > 0

and

-3 < 0

, the correct value of

y

is

9

.

Only the solution

(4, 9)

satisfies the condition that

y

must be positive.

Key Concept

Solving linear-quadratic systems of equations by substitution and evaluating solutions under given constraints.

Question 1417Question →

A botanist tracks the growth of a bamboo plant. The height $h$ , in inches, of the plant $d$ days after the tracking began can be modeled by the equation $h = 4.5d + 12$ . Which of the following is the best interpretation of $12$ in this context?

The height of the bamboo plant, in inches, 1 day after the tracking began.

The initial height of the bamboo plant, in inches, when the tracking began.

The average growth of the bamboo plant, in inches per day.

The number of days the tracking lasted.

Show answer & explanation

Answer: The initial height of the bamboo plant, in inches, when the tracking began.

Answer

The initial height of the bamboo plant, in inches, when the tracking began.

In the linear equation

h = 4.5d + 12

, the constant term

12

represents the y-intercept. This corresponds to the value of the function

h

when the independent variable

d

is equal to

0

. Since

d

is the number of days since tracking began,

d = 0

is the start of tracking, meaning

12

represents the initial height of the bamboo plant, in inches.

Step-by-Step Solution

1

Identify the component of the linear equation

h = 4.5d + 12

to be interpreted.

The number to interpret is the constant term

12

.

We must determine the contextual meaning of the constant value in the linear model.

2

Find the mathematical meaning of the constant in the slope-intercept form

y = mx + b

.

The constant term

12

represents the y-intercept of the linear equation, which occurs when the independent variable

d = 0

.

In a linear function, the constant represents the value of the dependent variable when the independent variable is zero.

3

Translate the mathematical definition to the real-world context of the problem.

Since

d

represents the number of days after tracking began,

d = 0

represents the start of tracking. At

d = 0

,

h = 12

, which is the initial height of the bamboo plant in inches.

Relating the y-intercept to the context yields the initial state of the measured quantity.

Key Concept

Interpreting the y-intercept of a linear model in context

Estimated Time:45s

Question 1418Question →

Which punctuation mark should be written in the blank to grammatically connect the two independent clauses in the passage?

Fill in the blanks below

For years, nineteenth-century colonial historians argued that the massive, complex stone walls of Great Zimbabwe must have been built by ancient foreign civilizations local Shona oral traditions, however, had long maintained that the structures were built by their own ancestors during the Kingdom of Zimbabwe's golden age.

Show answer & explanation

Answer

A semicolon should be placed in the blank because it correctly links two independent clauses without a coordinating conjunction.

A semicolon is the correct punctuation mark because it links two closely related independent clauses. The second clause presents a contrasting view with the transition 'however', making the semicolon the ideal choice to show the close relationship between the thoughts.

Step-by-Step Solution

1

Analyze the grammatical structure of the text surrounding the blank.

The text before the blank is an independent clause, and the text after the blank is also an independent clause containing the transition word 'however'.

Identifying clause types is necessary to determine the correct punctuation conventions to apply.

2

Determine the punctuation mark needed to connect the two independent clauses without creating a run-on sentence or a comma splice.

A semicolon is the correct punctuation mark to join these two independent clauses.

Under Standard English conventions, a semicolon is used to link two closely related independent clauses when no coordinating conjunction is present.

Key Concept

Using semicolons to connect independent clauses

Estimated Time:1m 0s

Question 1419Question →

A local coffee shop tracks its remaining syrup inventory at the end of each day. On day 3, the shop has 20 liters of syrup remaining. On day 7, the shop has 12 liters of syrup remaining. If the amount of syrup decreases at a constant daily rate, on which day will the shop have exactly 6 liters of syrup remaining?

4

7

10

31

Show answer & explanation

Answer: 10

Answer

10

To find the day when

6

liters of syrup remain, we first find the constant rate of decrease (slope) of the syrup. The slope is calculated as the change in syrup volume divided by the change in days:

m = \frac{12 - 20}{7 - 3} = -2

liters per day. Using the point-slope formula with the point

(3, 20)

, the relationship is

y - 20 = -2(x - 3)

, which simplifies to

y = -2x + 26

. Setting the remaining syrup

y

to

6

gives

6 = -2x + 26

. Solving for

x

yields

2x = 20

, so

x = 10

. Thus, the shop will have exactly

6

liters of syrup remaining on day

10

.

Step-by-Step Solution

1

Calculate the constant daily rate of decrease (slope) of the syrup inventory.

The rate of decrease is

-2

liters per day.

The slope

m

of a linear relationship between day

x

and remaining syrup

y

is given by

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 20}{7 - 3} = \frac{-8}{4} = -2

.

2

Determine the linear equation modeling the remaining syrup as a function of the day.

The linear equation is

y = -2x + 26

.

Using the point-slope form with the point

(3, 20)

and slope

m = -2

, we have

y - 20 = -2(x - 3)

. Distributing and simplifying gives

y - 20 = -2x + 6

, which simplifies to

y = -2x + 26

.

3

Find the day when the remaining syrup is exactly

6

liters.

The shop will have exactly

6

liters of syrup remaining on day

10

.

Substitute

y = 6

into the equation and solve for

x

:

6 = -2x + 26 \Rightarrow -20 = -2x \Rightarrow x = 10

.

Key Concept

Linear Functions and Graphs

Alternative Method

Instead of finding the full linear equation, we can use the rate of change directly. The syrup decreases by

2

liters per day. From day 7, where there are

12

liters remaining, we need to reach

6

liters remaining. This is a decrease of

12 - 6 = 6

liters. Since the rate of decrease is

2

liters per day, it will take

\frac{6}{2} = 3

additional days. Adding

3

days to day 7 gives

7 + 3 = 10

.

Estimated Time:1m 30s

Question 1420Question →

To survive nocturnal hunting in shallow waters, the Hawaiian bobtail squid (*Euprymna scolopes*) hosts bioluminescent bacteria inside a specialized light organ. The squid provides these microbes with nutrients and a secure habitat ________ the bacteria emit light that mimics moonlight, effectively camouflaging the squid from predators below.

Which choice completes the text so that it conforms to the conventions of Standard English?

, in turn,

in turn

; in turn,

although, in turn,

Show answer & explanation

Answer: ; in turn,

Answer

The correct choice is the option using a semicolon followed by a transition phrase ('; in turn,') to separate the two independent clauses.

The sentence consists of two independent clauses: 'The squid provides these microbes with nutrients and a secure habitat' and 'the bacteria emit light that mimics moonlight...' To link these two independent clauses grammatically, a semicolon is required when no coordinating conjunction is used. The transitional phrase 'in turn' is correctly followed by a comma to establish the relationship between the two clauses.

Step-by-Step Solution

1

Analyze the structure of the clauses before and after the blank.

The first clause ('The squid provides these microbes with nutrients and a secure habitat') is independent. The second clause ('the bacteria emit light that mimics moonlight...') is also independent.

Identifying clause types determines what punctuation is grammatically required to link them.

2

Evaluate the grammatical rules for linking two independent clauses.

Two independent clauses must be joined by a period, a semicolon, a colon, or a comma paired with a coordinating conjunction (FANBOYS).

Using a comma alone or no punctuation at all results in grammatical errors such as comma splices or run-ons.

3

Test the available options against these rules.

The option with the semicolon ('; in turn,') correctly separates the independent clauses, whereas the comma option creates a comma splice, the option with no punctuation creates a run-on, and the subordinating option creates a sentence fragment.

Selecting the option that correctly adheres to Standard English conventions for clause boundaries.

Key Concept

Linking independent clauses

Estimated Time:1m 15s

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