All practice questions

Question 101Question →

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

Show answer & explanation

Answer: 10.6

Answer

The weight of the other package is 10.6 pounds.

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

C(w) = 2.5w + 4

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

16.50

dollars, subtracting this from the total cost of

47.00

dollars to get

30.50

dollars, and solving

2.5w + 4 = 30.50

for the weight.

Step-by-Step Solution

1

Set up the linear system from the given costs

3k + b = 11.50

and

8k + b = 24.00

To determine the relationship between weight and shipping cost.

2

Solve for the slope

k

k = 2.50

Subtracting the first equation from the second eliminates

b

.

3

Solve for the intercept

b

b = 4.00

Substitute

k = 2.50

back into the first equation.

4

Determine the cost of the 5-pound package

C(5) = 16.50

dollars

Evaluate the linear expression

2.50(5) + 4.00

.

5

Determine the remaining cost for the second package

C(w_2) = 30.50

dollars

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

47.00 - 16.50

).

6

Solve the linear equation for the second package's weight

w_2 = 10.6

Solve

2.50w_2 + 4.00 = 30.50

for

w_2

.

Key Concept

Solving Linear Equations

Question 102Question →

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

Show answer & explanation

Answer: 45

Answer

Jordan walked for

45

minutes.

The correct answer is

45

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

120

steps per minute and a starting baseline of

2{}500

steps. Let

m

represent the number of minutes Jordan walked. The equation is

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

. Subtracting

2{}500

from both sides gives

120m = 5{}400

, and dividing by

120

yields

m = 45

.

Step-by-Step Solution

1

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

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

The total steps are the sum of the starting steps (

2{}500

) and the product of the rate (

120

steps/min) and time (

m

minutes).

2

Isolate the variable term by subtracting

2{}500

from both sides of the equation.

120m = 5{}400

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

3

Solve for

m

by dividing both sides of the equation by

120

.

m = 45

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

Key Concept

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

Alternative Method

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

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

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

5{}400 \div 120 = 45

minutes).

Estimated Time:45s

Question 103Question →

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

Show answer & explanation

Answer: -21

Answer

The final value of the expression is

-21

.

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

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

. Distributing and combining terms yields

-x - 13 = 1

, which gives

x = -14

. Substituting this value into the expression

2x + 7

yields

-21

.

Step-by-Step Solution

1

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

15

) to eliminate all fractions.

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

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

2

Distribute the coefficients (

9

and

-5

) to their respective terms inside the parentheses.

9x - 18 - 10x + 5 = 1

Distributive property allows us to remove parentheses. Note that distributing

-5

to

-1

results in

+5

.

3

Combine the variable terms (

9x

and

-10x

) and the constant terms (

-18

and

5

) on the left side of the equation.

-x - 13 = 1

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

4

Isolate the variable term

-x

by adding

13

to both sides, then solve for

x

by multiplying by

-1

.

x = -14

Adding

13

yields

-x = 14

, and multiplying by

-1

isolates

x

to find its value.

5

Substitute

x = -14

into the given expression

2x + 7

.

2(-14) + 7 = -21

The question asks for the value of the expression

2x + 7

, not just

x

itself.

Key Concept

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

Alternative Method

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

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

. Combining the

x

terms gives

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

. Combining the constant terms gives

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

. The equation becomes

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

. Multiplying the entire equation by

15

yields

-x - 13 = 1

, which gives

x = -14

, and substituting into

2x + 7

yields

-21

.

Estimated Time:2m 0s

Question 104Question →

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

Show answer & explanation

Answer: 40

Answer

The firm ran advertisements on SocialMedia for 40 days.

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

S + E = 60

and

15S = 0.5(25E) + 350

. Substituting the first equation into the second yields

15S = 12.5(60 - S) + 350

. Solving this linear equation results in

S = 40

days.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

S

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

E

be the number of days they ran advertisements on SearchEngine.

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

2

Express the relationship for the total number of days.

S + E = 60

, which simplifies to

E = 60 - S

.

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

3

Translate the cost relationship statement into an algebraic equation.

15S = 12.5E + 350

The cost of running advertisements on SocialMedia is

15S

. The cost on SearchEngine is

25E

. Half of the SearchEngine cost is

12.5E

. Adding

350

to half of the SearchEngine cost gives the SocialMedia cost.

4

Substitute the expression for

E

into the cost equation and solve for

S

.

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

.

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

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:2m 0s

Question 105Question →

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

Show answer & explanation

Answer: -3

Answer

The coefficient of the linear term,

B

, is

-3

.

Subtracting the area of the square,

x^2 - 4x + 4

, from the area of the rectangle,

6x^2 - 7x - 3

, yields

5x^2 - 3x - 7

. Thus, the coefficient

B

of the

x

term is

-3

.

Step-by-Step Solution

1

Calculate the area of the rectangle.

Area =

6x^2 - 7x - 3

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

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

.

2

Calculate the area of the square.

Area =

x^2 - 4x + 4

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

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

.

3

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

Remaining Area =

5x^2 - 3x - 7

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

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

.

4

Identify the coefficient

B

.

B = -3

In the standard quadratic form

Ax^2 + Bx + C

, the coefficient of the linear term

x

is

B

, which corresponds to

-3

in the polynomial

5x^2 - 3x - 7

.

Key Concept

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

Question 106Question →

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

Show answer & explanation

Answer: -7

Answer

The value of

y

that satisfies the equation is

-7

.

Distributing the 3 yields

3y - 12 = 5y + 2

. Subtracting

3y

from both sides gives

-12 = 2y + 2

. Subtracting 2 from both sides gives

-14 = 2y

. Dividing by 2 results in

y = -7

.

Step-by-Step Solution

1

Distribute the 3 on the left side of the equation.

3y - 12 = 5y + 2

To simplify the expression by expanding the parentheses.

2

Subtract

3y

from both sides of the equation.

-12 = 2y + 2

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

3

Subtract 2 from both sides of the equation.

-14 = 2y

To isolate the variable term.

4

Divide both sides by 2.

y = -7

To solve for

y

.

Key Concept

Solving linear equations by distributing and isolating the variable.

Estimated Time:45s

Question 107Question →

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

Show answer & explanation

Answer: 7

Answer

The number of integers in the domain of the composite function

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

is 7.

The composite function

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

is defined when

x

is in the domain of

f

and

f(x)

is in the domain of

g

. The domain of

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

is

[-5, 5]

. The domain of

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

is

u > 3

or

u < -3

. Substituting

f(x)

for

u

gives

\sqrt{25 - x^2} > 3

, which simplifies to

x^2 < 16

or

-4 < x < 4

. The intersection of

[-5, 5]

and

(-4, 4)

is

(-4, 4)

. The integers in this interval are

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

, which total 7 integers.

Step-by-Step Solution

1

Determine the domain of the inner function

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

.

The domain is

[-5, 5]

.

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

25 - x^2 \ge 0

.

2

Determine the domain of the outer function

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

.

The domain is

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

.

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

u^2 - 9 > 0

.

3

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

-4 < x < 4

.

We require

f(x) > 3

or

f(x) < -3

. Since the range of

f(x)

is non-negative,

f(x) < -3

has no solutions. Thus, we solve

\sqrt{25 - x^2} > 3

, which squares to

25 - x^2 > 9

, or

x^2 < 16

.

4

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

The composite domain is

(-4, 4)

.

The input

x

must satisfy both the domain of

f

(

-5 \le x \le 5

) and the composition constraint (

-4 < x < 4

).

5

List and count the integers within the composite domain

(-4, 4)

.

There are 7 integers.

The integers strictly between

-4

and

4

are

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

.

Key Concept

Domain of a composite function

Estimated Time:2m 0s

Question 108Question →

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

Show answer & explanation

Answer: 8

Answer

8

To find

h(h(2))

, we first calculate the value of the inner function,

h(2) = 5 - 3(2) = 5 - 6 = -1

. We then substitute this result back into the function to evaluate the outer function:

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

. Therefore, the correct value is 8.

Step-by-Step Solution

1

Evaluate the inner function

h(2)

h(2) = -1

To evaluate a nested function composition of the form

h(h(x))

, first calculate the value of the inner function at the given input.

2

Evaluate the outer function

h(-1)

using the result from Step 1

h(h(2)) = h(-1) = 8

Substitute the output of the inner function,

-1

, as the new input for the outer function

h(x)

.

Key Concept

Evaluating the composition of a function with itself

Question 109Question →

A security passcode consists of a sequence of 4 digits chosen from the digits 0 through 9. To be valid, a passcode must contain at least one repeated digit, but no digit can appear more than twice. Additionally, the passcode cannot end with an odd digit. How many different valid security passcodes can be created?

Show answer & explanation

Answer: 2295

Answer

There are 2,295 different valid security passcodes.

The correct answer of 2,295 is found by subtracting all invalid passcodes from the total possible passcodes. The total number of 4-digit passcodes ending in an even digit is 5,000. The invalid passcodes are those with all distinct digits (2,520), those with a digit repeated three times (180), and those with all four digits identical (5). Subtracting these gives 5,000 - 2,520 - 180 - 5 = 2,295.

Step-by-Step Solution

1

Calculate the total number of 4-digit passcodes ending in an even digit.

5,000

The last digit must be even (0, 2, 4, 6, or 8) to not be odd, giving 5 choices. The first three digits can be any of the 10 digits from 0 through 9. By the Fundamental Counting Principle, the total number of passcodes is 10 * 10 * 10 * 5 = 5,000.

2

Calculate the number of invalid passcodes where all 4 digits are distinct.

2,520

For all 4 digits to be distinct, the last digit must be chosen from the 5 even digits. The remaining 3 positions must be filled with 3 distinct digits chosen from the remaining 9 digits. There are 9 * 8 * 7 = 504 ways to choose these. This gives 504 * 5 = 2,520 passcodes.

3

Calculate the number of invalid passcodes where a single digit is repeated three times.

180

If the repeated digit is the last digit, there are 5 choices for that digit, and the single distinct digit (9 choices) can be placed in any of the 3 remaining positions, giving 5 * 9 * 3 = 135 passcodes. If the repeated digit is not the last digit, the three identical digits occupy the first three positions, and the last digit (5 choices) is distinct from them (9 choices), giving 5 * 9 = 45 passcodes. In total, 135 + 45 = 180 passcodes.

4

Calculate the number of invalid passcodes where all four digits are identical.

5

Since the last digit must be even, all four digits must be the same even digit (0000, 2222, 4444, 6666, or 8888), which gives 5 passcodes.

5

Subtract the invalid passcodes from the total number of passcodes.

2,295

Subtracting the passcodes with all distinct digits (2,520) and those with a digit repeated three or four times (180 + 5 = 185) from the total of 5,000 gives 5,000 - 2,520 - 185 = 2,295.

Key Concept

Complementary counting using permutations and partition analysis

Alternative Method

Instead of complementary counting, count the valid cases directly: Case A where the digit frequencies are [2, 1, 1] (which yields 2,160 codes) and Case B where the digit frequencies are [2, 2] (which yields 135 codes). Adding these yields 2,160 + 135 = 2,295 codes.

Estimated Time:3m 0s

Question 110Question →

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 - 12y + 27 = 0$ . A parabola that opens downward has its vertex at $(0, k)$ and is defined by the equation $y = -x^2 + k$ . If the system of equations consisting of this circle and parabola has exactly three distinct real solution points, what is the value of $k$ ?

Show answer & explanation

Answer: 9

Answer

The value of

k

is 9.

The correct value of

k

is 9 because when

k=9

, the system of equations reduces to a quadratic in

y

with roots

y=9

and

y=4

. Both roots satisfy the real-number constraint

y \leq 9

for the parabola

x^2 = 9-y

, producing three distinct real solutions:

(0, 9)

,

(\sqrt{5}, 4)

, and

(-\sqrt{5}, 4)

.

Step-by-Step Solution

1

Complete the square for the circle's equation.

x^2 + (y-6)^2 = 9

To identify the circle's center at

(0, 6)

and radius

R=3

for geometric interpretation.

2

Express

x^2

in terms of

y

using the parabola's equation.

x^2 = k - y

To substitute into the circle's equation and eliminate the

x

variable.

3

Substitute

x^2

into the circle's equation and simplify.

y^2 - 13y + (k+27) = 0

To create a quadratic equation in

y

representing the y-coordinates of the intersection points.

4

Set

y = k

in the quadratic equation.

k^2 - 12k + 27 = 0

, which factors as

(k-3)(k-9) = 0

An intersection must lie on the y-axis (

x=0

, which means

y=k

) to yield an odd number of intersection points.

5

Verify which candidate value of

k

yields exactly three real solutions.

For

k=3

, the solutions are restricted because

y=10

gives no real

x

value, resulting in only 1 solution. For

k=9

, the roots

y=9

and

y=4

both yield real

x

values, resulting in exactly 3 solutions:

(0, 9)

,

(\sqrt{5}, 4)

, and

(-\sqrt{5}, 4)

.

The algebraic condition for real

x

coordinates is

x^2 = k - y \geq 0

, so we must verify that the roots

y

satisfy

y \leq k

.

Key Concept

Solving systems of non-linear equations algebraically and analyzing the number of real intersection points under coordinate constraints.

Alternative Method

Geometrically, a parabola opening downward with its vertex on the y-axis will intersect a circle centered on the y-axis in exactly three points if and only if its vertex is at the top of the circle and its curvature is less than that of the circle at that point. Completing the square for the circle

x^2 + y^2 - 12y + 27 = 0

gives

x^2 + (y-6)^2 = 9

, which shows the top point of the circle is

(0, 9)

. Thus, the vertex of the downward-opening parabola must be at

(0, 9)

, meaning

k = 9

. We then algebraically verify that this curvature indeed allows two other real intersections.

Estimated Time:3m 0s

Question 111Question →

Let the complex number $z$ be defined as $z = (4 - 3i)(1 + 2i) + 5i^{14}$ , where $i = \sqrt{-1}$ . What is the real part of $z$ ?

Show answer & explanation

Answer: 5

Answer

The real part of the complex number

z

is

5

.

First, expand the product

(4 - 3i)(1 + 2i)

to get

4 + 8i - 3i - 6i^2

. Replacing

i^2

with

-1

gives

10 + 5i

. Next, simplify

5i^{14}

. Since

i^{14} = (i^4)^3 \cdot i^2 = 1^3 \cdot (-1) = -1

, the term becomes

-5

. Adding the components together gives

z = (10 + 5i) - 5 = 5 + 5i

. The real part of this complex number is

5

.

Step-by-Step Solution

1

Expand the product of the complex binomials

(4 - 3i)(1 + 2i)

10 + 5i

Applying the distributive property gives

4 + 8i - 3i - 6i^2

. Substituting

i^2 = -1

simplifies the expression to

4 + 5i + 6 = 10 + 5i

.

2

Simplify the power of the imaginary unit in

5i^{14}

-5

Since the powers of

i

cycle every 4 terms,

i^{14} = i^{12} \cdot i^2 = 1 \cdot (-1) = -1

. Therefore,

5i^{14} = 5(-1) = -5

.

3

Add the simplified terms together to find

z

5 + 5i

Adding the real and imaginary parts of the terms yields

z = (10 + 5i) + (-5) = 5 + 5i

.

4

Determine the real part of

z

5

A complex number is written in the form

a + bi

, where

a

represents the real part. For

5 + 5i

, the real part is

5

.

Key Concept

Complex multiplication and simplification of powers of the imaginary unit

Question 112Question →

When the polynomial $12x^2 + 11x - 15$ is factored completely into the form $(ax + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are integers such that $a > c > 0$ , what is the value of the constant term $d$ ?

Show answer & explanation

Answer: 5

Answer

The value of the constant term

d

is

5

.

Factoring the trinomial

12x^2 + 11x - 15

completely gives

(4x - 3)(3x + 5)

. Applying the constraint

a > c > 0

means the factor with the larger

x

-coefficient must be written first in the template

(ax + b)(cx + d)

. This yields

a = 4

,

b = -3

,

c = 3

, and

d = 5

. Thus, the constant term

d

is

5

.

Step-by-Step Solution

1

Find the factor pair for the AC method.

We need two numbers that multiply to

12 \times (-15) = -180

and add up to

11

. The numbers are

20

and

-9

.

This allows us to split the linear middle term to factor by grouping.

2

Rewrite the polynomial and factor by grouping.

12x^2 + 20x - 9x - 15 = 4x(3x + 5) - 3(3x + 5) = (4x - 3)(3x + 5)

.

Grouping the first two terms and the last two terms reveals a common binomial factor of

(3x + 5)

.

3

Apply the given inequality constraints to match the template.

Comparing

(4x - 3)(3x + 5)

to

(ax + b)(cx + d)

with

a > c > 0

yields

a = 4

,

b = -3

,

c = 3

, and

d = 5

.

Since the lead coefficient

4

is greater than

3

, the factor

(4x - 3)

must correspond to

(ax + b)

.

Key Concept

Factoring quadratic trinomials of the form

Ax^2 + Bx + C

using the grouping (AC) method.

Question 113Question →

A chemist wants to create $100$ milliliters of a $42.5\%$ acid solution by mixing three different acid solutions: a $10\%$ acid solution, a $20\%$ acid solution, and an $80\%$ acid solution. She decides that the volume of the $20\%$ acid solution used must be exactly $3$ times the volume of the $10\%$ acid solution used. What is the difference, in milliliters, between the volume of the $80\%$ acid solution and the volume of the $10\%$ acid solution used in the final mixture?

Show answer & explanation

Answer: 25

Answer

The difference between the volume of the

80\%

acid solution and the volume of the

10\%

acid solution is

25

milliliters.

Solving the system of equations yields that

15

milliliters of the

10\%

solution,

45

milliliters of the

20\%

solution, and

40

milliliters of the

80\%

solution are needed. The difference between the volume of the

80\%

solution and the

10\%

solution is

40 - 15 = 25

milliliters.

Step-by-Step Solution

1

Define variables for the volume of each acid solution.

Let

x

be the volume of the

10\%

solution,

y

be the volume of the

20\%

solution, and

z

be the volume of the

80\%

solution.

This establishes algebraic representations for the unknowns.

2

Set up a system of linear equations based on the relationships given in the problem statement.

x + y + z = 100

(total volume)

y = 3x

(relationship between the

20\%

and

10\%

solutions)

0.10x + 0.20y + 0.80z = 42.5

(total acid content)

Translating word problems to mathematical equations allows us to solve for the variables systematically.

3

Reduce the system to a single equation in terms of

x

by substituting

y = 3x

and expressing

z

in terms of

x

.

4x + z = 100 \implies z = 100 - 4x

Substitute both into the acid equation:

0.10x + 0.20(3x) + 0.80(100 - 4x) = 42.5

Substitution simplifies the system of equations to a single linear equation with one variable.

4

Solve the simplified linear equation for

x

.

0.70x + 80 - 3.20x = 42.5 \implies -2.50x = -37.5 \implies x = 15

This determines the volume of the

10\%

acid solution.

5

Calculate the volume of the

80\%

solution,

z

.

z = 100 - 4(15) = 40

This determines the volume of the

80\%

acid solution.

6

Find the difference between

z

and

x

.

z - x = 40 - 15 = 25

The question asks for the difference between these two volumes.

Key Concept

Translating and solving systems of linear equations from verbal descriptions (mixture problems).

Estimated Time:2m 30s

Question 114Question →

A craft shop sells handmade candles. The price of a large candle is $3$ dollars more than twice the price of a small candle. If a large candle costs $15$ dollars, what is the price, in dollars, of a small candle?

Show answer & explanation

Answer: 6

Answer

The price of a small candle is

6

dollars.

By letting

s

represent the price of a small candle, the price of a large candle is

2s + 3

. Since the large candle costs

15

dollars, we write the equation

2s + 3 = 15

. Subtracting

3

from both sides gives

2s = 12

, and dividing by

2

yields

s = 6

. Therefore, the price of a small candle is

6

dollars.

Step-by-Step Solution

1

Define the variable and translate the verbal description into an algebraic expression.

Let

s

be the price of a small candle. The price of a large candle is expressed as

2s + 3

.

Translating 'twice the price of a small candle' to

2s

and '3 more than' to

+ 3

allows us to represent the large candle's cost algebraically.

2

Formulate an equation by setting the expression equal to the known cost of the large candle.

2s + 3 = 15

The problem states that the large candle costs

15

dollars.

3

Solve the equation for the variable

s

.

s = 6

Subtracting

3

from both sides gives

2s = 12

. Dividing both sides by

2

isolates

s

, resulting in

6

.

Key Concept

Translating verbal statements into linear equations and solving for a single variable.

Alternative Method

We can solve the problem by working backward from the price of the large candle. Since the large candle (

15

dollars) is

3

dollars more than twice the small candle's price, we subtract

3

dollars to find twice the price of the small candle:

15 - 3 = 12

dollars. Then, since

12

dollars is twice the price of the small candle, we divide by

2

to find the price of a single small candle:

12 \div 2 = 6

dollars.

Estimated Time:45s

Question 115Question →

A quadratic equation of the form $a x^2 + b x + c = 0$ , where $a$ , $b$ , and $c$ are real constants and $a > 0$ , has a discriminant of $37$ . If the sum of the roots of this equation is $5.5$ and the product of the roots is $5.25$ , what is the value of the coefficient $a$ ?

Show answer & explanation

Answer: 2

Answer

The value of the coefficient

a

is

2

.

By applying Vieta's formulas, we can write

b = -5.5a

and

c = 5.25a

. Plugging these into the discriminant formula gives

D = (-5.5a)^2 - 4a(5.25a) = 30.25a^2 - 21a^2 = 9.25a^2

. Setting the discriminant to

37

results in

9.25a^2 = 37 \implies a^2 = 4

. Since the problem specifies

a > 0

, taking the positive square root gives

a = 2

.

Step-by-Step Solution

1

Express the coefficients

b

and

c

in terms of

a

using Vieta's formulas.

b = -5.5a

and

c = 5.25a

The sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

.

2

Substitute the expressions for

b

and

c

into the discriminant formula

D = b^2 - 4ac

.

D = 9.25a^2

Substituting the terms yields

D = (-5.5a)^2 - 4a(5.25a) = 30.25a^2 - 21a^2 = 9.25a^2

.

3

Equate the discriminant expression to

37

and solve for

a

.

a = 2

Since

D = 37

, we write

9.25a^2 = 37 \implies a^2 = 4

. Because

a

must be positive, we find

a = 2

.

Key Concept

Using the properties of quadratic roots (Vieta's formulas) and the definition of the discriminant to solve for coefficients.

Question 116Question →

What is the maximum integer value of $x$ that satisfies the inequality $\frac{3(2 - 3x)}{4} - \frac{2(2x + 5)}{3} \geq \frac{1 - x}{2} + \frac{7}{6}$ ?

Show answer & explanation

Answer: -2

Answer

The maximum integer value of

x

that satisfies the inequality is

-2

.

Multiplying the inequality by the common denominator 12 and simplifying yields the inequality

-37x \ge 42

. Dividing by

-37

requires reversing the inequality sign, which gives

x \le -\frac{42}{37}

. The value of

-\frac{42}{37}

is approximately

-1.135

. The largest integer less than or equal to

-1.135

is

-2

.

Step-by-Step Solution

1

Multiply both sides of the inequality by the least common multiple of the denominators (12).

9(2 - 3x) - 8(2x + 5) \geq 6(1 - x) + 14

This eliminates the fractions and simplifies the algebraic manipulation.

2

Expand the terms on both sides of the inequality.

18 - 27x - 16x - 40 \geq 6 - 6x + 14

Expanding the terms allows us to combine like terms.

3

Combine the constant and variable terms on each side.

-43x - 22 \geq 20 - 6x

This simplifies the inequality to a standard linear form.

4

Add

6x

and

22

to both sides to isolate the variable term on the left.

-37x \geq 42

Grouping variable terms on one side and constant terms on the other prepares for the final division.

5

Divide both sides by

-37

and reverse the direction of the inequality sign.

x \leq -\frac{42}{37}

Dividing an inequality by a negative number requires flipping the inequality sign.

6

Find the largest integer that is less than or equal to

-\frac{42}{37}

.

-2

Since

-\frac{42}{37} \approx -1.135

, the integers less than or equal to this value are

-2, -3, -4, \dots

, of which

-2

is the greatest.

Key Concept

Solving multi-step linear inequalities with rational coefficients, applying the inequality sign-flip rule, and finding boundary integer conditions.

Alternative Method

Instead of clearing the fractions first, you can group all terms containing

x

on one side and the constant terms on the other side by finding a common denominator for only the variables and only the constants. However, clearing the fractions first is generally less prone to errors.

Estimated Time:2m 0s

Question 117Question →

For all real numbers $x \geq -5$ , the functions $f$ and $g$ are defined by $f(x) = x^2 - 4x$ and $g(x) = \sqrt{x + 5}$ . If $f(g(k)) = 12$ , what is the real value of $k$ ?

Show answer & explanation

Answer: 31

Answer

The correct answer is 31.

Substituting

g(k) = \sqrt{k+5}

into

f(x) = x^2 - 4x

gives the equation

(\sqrt{k+5})^2 - 4\sqrt{k+5} = 12

. Simplifying and isolating the radical yields

k - 7 = 4\sqrt{k+5}

. Squaring both sides results in

(k-7)^2 = 16(k+5)

, which simplifies to

k^2 - 30k - 31 = 0

. Factoring this equation gives

(k-31)(k+1) = 0

, yielding potential solutions of

k = 31

and

k = -1

. Checking these solutions reveals that

k = -1

is extraneous because

f(g(-1)) = -4 \neq 12

. Therefore, the only valid real solution is

k = 31

.

Step-by-Step Solution

1

Express the composition

f(g(k))

using the given functions.

f(g(k)) = (g(k))^2 - 4(g(k))

To evaluate a composite function, substitute the inner function

g(k)

as the input into the outer function

f

.

2

Substitute

g(k) = \sqrt{k+5}

and set the composite expression equal to 12.

(\sqrt{k+5})^2 - 4\sqrt{k+5} = 12

This sets up the equation to solve for the unknown variable

k

.

3

Isolate the radical term on one side of the equation.

k - 7 = 4\sqrt{k+5}

Simplifying

(\sqrt{k+5})^2

to

k+5

and moving terms helps isolate the radical before squaring.

4

Square both sides of the equation to eliminate the square root.

(k-7)^2 = 16(k+5)

Squaring is the inverse operation of the square root, which removes the radical.

5

Expand both sides and rewrite the equation in standard quadratic form.

k^2 - 30k - 31 = 0

Expanding

(k-7)^2

to

k^2 - 14k + 49

and

16(k+5)

to

16k + 80

, then moving all terms to one side, allows us to solve the resulting quadratic equation.

6

Factor the quadratic equation.

(k-31)(k+1) = 0

Factoring allows us to find the potential roots easily.

7

Solve for the potential values of

k

.

k = 31

or

k = -1

Setting each factor to zero gives the candidate solutions.

8

Verify both potential values in the original equation to check for extraneous solutions.

k = 31

is valid;

k = -1

is extraneous.

Squaring both sides can introduce extraneous solutions. Evaluating

f(g(-1))

yields

-4

, not

12

, while evaluating

f(g(31))

yields

12

.

Key Concept

Function composition involves substituting one function into another, and solving equations with radicals requires checking for extraneous solutions.

Question 118Question →

When the product of the polynomials $(x^3 - 3x^2 + 2x - 4)$ and $(ax^2 + bx + c)$ is subtracted from $2x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16$ , the resulting polynomial is equal to $0$ for all real values of $x$ . What is the value of $4a + 2b - c$ ?

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. By equating the corresponding coefficients of the product

(x^3 - 3x^2 + 2x - 4)(ax^2 + bx + c)

to the polynomial

2x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16

, we find

a = 2

from the

x^5

terms,

c = 4

from the constant terms, and

b = 1

from the

x^4

terms. Substituting these values into

4a + 2b - c

yields

4(2) + 2(1) - 4 = 6

.

Step-by-Step Solution

1

Relate the product of the polynomials to the given polynomial expression.

(x^3 - 3x^2 + 2x - 4)(ax^2 + bx + c) = 2x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16

Since subtracting the product from the given polynomial results in a polynomial that is always

0

, the product must be identically equal to that polynomial.

2

Equate the leading coefficients to find the value of

a

.

a = 2

The highest-degree term of the product is

x^3 \cdot ax^2 = ax^5

, which must equal the highest-degree term on the right side,

2x^5

.

3

Equate the constant terms to find the value of

c

.

-4c = -16 \implies c = 4

The constant term of the product is

-4 \cdot c = -4c

, which must equal the constant term on the right side,

-16

.

4

Equate the coefficients of the

x^4

terms to find the value of

b

.

b - 3a = -5 \implies b - 6 = -5 \implies b = 1

The

x^4

term in the expanded product comes from

(x^3)(bx) + (-3x^2)(ax^2) = (b - 3a)x^4

, which must equal the

x^4

term on the right side,

-5x^4

.

5

Compute the value of the requested expression

4a + 2b - c

.

4(2) + 2(1) - 4 = 6

Substitute the determined values

a = 2

,

b = 1

, and

c = 4

into the expression.

Key Concept

Operations on polynomials, specifically multiplication, subtraction, and equating corresponding coefficients.

Alternative Method

Alternatively, evaluate the polynomial equation at

x = 2

. Substituting

x = 2

into

(x^3 - 3x^2 + 2x - 4)(ax^2 + bx + c) = 2x^5 - 5x^4 + 5x^3 - 18x^2 + 4x - 16

gives

(8 - 12 + 4 - 4)(4a + 2b + c) = 64 - 80 + 40 - 72 + 8 - 16

, which simplifies to

-4(4a + 2b + c) = -56

. Dividing both sides by

-4

yields

4a + 2b + c = 14

. Since equating the constant terms gives

-4c = -16 \implies c = 4

, we can substitute

c = 4

into

4a + 2b + c = 14

to get

4a + 2b + 4 = 14 \implies 4a + 2b = 10

. Subtracting

c=4

from both sides gives the desired expression value:

4a + 2b - c = 10 - 4 = 6

.

Estimated Time:2m 30s

Question 119Question →

A system of equations consists of the linear equation $y = 2x + 1$ and the quadratic equation $y = x^2 - 2$ . If $(x, y)$ is a solution to this system such that $x > 0$ , what is the value of $y$ ?

Show answer & explanation

Answer: 7

Answer

The correct value of

y

is 7.

Substituting

x = 3

into either equation yields the

y

-value of 7.

Step-by-Step Solution

1

Set the two expressions for

y

equal to each other.

x^2 - 2 = 2x + 1

Since both equations define

y

in terms of

x

, equating them allows us to find the

x

-coordinates of the intersection points.

2

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 2x - 3 = 0

Subtracting

2x

and

1

from both sides collects all terms on one side of the equation.

3

Factor the quadratic equation.

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

Finding two numbers that multiply to

-3

and add to

-2

gives

-3

and

1

, allowing the quadratic to be factored.

4

Solve for

x

and apply the constraint

x > 0

.

x = 3

The factored equation yields solutions of

x = 3

and

x = -1

. The constraint that

x

must be greater than zero means we select

x = 3

.

5

Substitute the

x

-value back into one of the original equations to solve for

y

.

y = 7

Plugging

x = 3

into the linear equation

y = 2x + 1

yields

y = 2(3) + 1 = 7

. Substituting into

y = x^2 - 2

yields

y = 3^2 - 2 = 7

, which confirms the solution.

Key Concept

Solving a system of linear and quadratic equations using substitution.

Question 120Question →

When the expression $2a(a^2 - 3ab) - 3b(a^2 + 2b^2) - (a^3 - 5ab^2)$ is simplified by combining like terms, what is the coefficient of the $a^2b$ term?

Show answer & explanation

Answer: -9

Answer

The coefficient of the

a^2b

term is

-9

.

Expanding the entire expression yields

2a^3 - 6a^2b - 3a^2b - 6b^3 - a^3 + 5ab^2

. Combining the

a^2b

terms gives

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

. Therefore, the coefficient of the

a^2b

term is

-9

.

Step-by-Step Solution

1

Distribute

2a

to the terms inside the first set of parentheses:

2a(a^2 - 3ab)

2a^3 - 6a^2b

To expand the first part of the expression.

2

Distribute

-3b

to the terms inside the second set of parentheses:

-3b(a^2 + 2b^2)

-3a^2b - 6b^3

To expand the second part of the expression, ensuring the negative sign is distributed to all terms inside.

3

Distribute the negative sign to the terms inside the third set of parentheses:

-(a^3 - 5ab^2)

-a^3 + 5ab^2

To expand the third part of the expression, reversing the sign of each term inside.

4

Identify and combine the like terms for the

a^2b

variable combination:

-6a^2b - 3a^2b

-9a^2b

To simplify the expression by combining terms with the same variable components.

Key Concept

Simplifying expressions by distributing terms and combining like terms

Estimated Time:1m 30s

Answer

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