All practice questions

Question 321Question →

A manufacturing plant uses two types of machines, Machine X and Machine Y, to produce a certain component. Machine X produces $15$ components per hour, and Machine Y produces $25$ components per hour. On a certain day, both machines were operated for a combined total of $14$ hours, producing a total of $270$ components. For how many hours was Machine X operated?

Show answer & explanation

Answer: 8

Answer

Machine X was operated for 8 hours.

By setting up the system of linear equations where

x

represents the hours Machine X operated and

y

represents the hours Machine Y operated, we get

x + y = 14

and

15x + 25y = 270

. Substituting

y = 14 - x

into the second equation yields

15x + 25(14 - x) = 270

. Expanding and simplifying results in

-10x + 350 = 270

, which solves to

x = 8

. Thus, Machine X operated for 8 hours.

Step-by-Step Solution

1

Define variables for the hours each machine operated and set up the system of linear equations.

Let

x

be the hours Machine X was operated and

y

be the hours Machine Y was operated. The system is:

x + y = 14

15x + 25y = 270

This translates the verbal descriptions in the problem into algebraic expressions and equations.

2

Express

y

in terms of

x

from the first equation and substitute it into the second equation.

y = 14 - x

15x + 25(14 - x) = 270

Substitution reduces the system from two variables to a single equation in terms of

x

.

3

Solve the linear equation for

x

by distributing the constant and combining like terms.

15x + 350 - 25x = 270

-10x + 350 = 270

-10x = -80

x = 8

Isolating

x

gives the specific number of hours Machine X was operated.

Key Concept

Solving systems of linear equations in two variables using substitution or elimination.

Alternative Method

We can solve the system by multiplying the first equation by 25 to get

25x + 25y = 350

. Subtracting the second equation

15x + 25y = 270

from this yields

10x = 80

, which directly gives

x = 8

.

Estimated Time:1m 30s

Question 322Question →

A digital printing press uses paper from a large roll. The total weight $W$ , in pounds, of the paper remaining on the roll after $m$ minutes of continuous printing is given by the equation $W = 120 - 1.5m$ . According to the equation, after how many minutes of continuous printing will the weight of the remaining paper on the roll be exactly $45$ pounds?

Show answer & explanation

Answer: 50

Answer

The correct answer is 50.

To find the number of minutes,

m

, when the remaining weight is

45

pounds, substitute

W = 45

into the given equation

W = 120 - 1.5m

. This yields the equation

45 = 120 - 1.5m

. Subtracting

120

from both sides gives

-75 = -1.5m

. Dividing both sides by

-1.5

results in

m = 50

. Therefore, after

50

minutes, the remaining weight of the paper on the roll is

45

pounds.

Step-by-Step Solution

1

Substitute the target weight into the linear equation.

45 = 120 - 1.5m

We are given that the weight of the remaining paper,

W

, is

45

pounds, and we need to solve for the time in minutes,

m

.

2

Isolate the variable term by subtracting

120

from both sides of the equation.

-75 = -1.5m

To solve for

m

, we first need to isolate the term containing the variable on one side of the equation.

3

Divide both sides by

-1.5

to find the value of

m

.

m = 50

Dividing by the coefficient of the variable completes the isolation and gives the final value.

Key Concept

Interpreting values and solving linear equations in context

Alternative Method

Alternatively, you can determine the total weight of paper consumed:

120 - 45 = 75

pounds. Since the paper is used at a rate of

1.5

pounds per minute, the time required is

\frac{75}{1.5} = 50

minutes.

Estimated Time:1m 30s

Question 323Question →

If $x$ is the real solution to the equation below, what is the value of $x$ ?

\frac{2x^2 - 30x + 100}{x - 5} + \frac{140}{x + 2} = 10

Show answer & explanation

Answer: 8

Answer

The correct answer is 8.

To solve the rational equation, we first state the domain restrictions as

x \neq 5

and

x \neq -2

because these values make the denominators zero. Next, we factor the numerator of the first term:

2x^2 - 30x + 100 = 2(x^2 - 15x + 50) = 2(x - 5)(x - 10)

. Since

x \neq 5

, we can cancel the common factor of

x - 5

from the numerator and denominator, which simplifies the first term to

2(x - 10) = 2x - 20

. The equation can then be rewritten as

2x - 20 + \frac{140}{x + 2} = 10

. Subtracting 10 from both sides yields

2x - 30 + \frac{140}{x + 2} = 0

. Dividing the entire equation by 2 simplifies the coefficients, giving

x - 15 + \frac{70}{x + 2} = 0

. To clear the fraction, we multiply the entire equation by

x + 2

, leading to

(x - 15)(x + 2) + 70 = 0

. Expanding the product gives

x^2 - 13x - 30 + 70 = 0

, which simplifies to

x^2 - 13x + 40 = 0

. Factoring this quadratic equation yields

(x - 5)(x - 8) = 0

, giving the potential solutions

x = 5

and

x = 8

. However,

x = 5

is extraneous because it is restricted from the domain of the original rational expression. Therefore, the only real solution is 8.

Step-by-Step Solution

1

Find the values of

x

that make any denominator equal to zero to identify domain restrictions.

x \neq 5

and

x \neq -2

Dividing by zero is undefined in real numbers, so these values are excluded from the domain.

2

Factor the numerator of the first term,

2x^2 - 30x + 100

.

2(x - 5)(x - 10)

Factoring allows us to check for common factors in the numerator and denominator.

3

Simplify the first rational term by dividing the numerator and denominator by the common factor

x - 5

, under the assumption that

x \neq 5

.

2(x - 10) = 2x - 20

Reducing the expression simplifies the algebraic structure of the equation.

4

Substitute the simplified expression back into the original equation and subtract 10 from both sides.

2x - 30 + \frac{140}{x + 2} = 0

To consolidate the constant terms and prepare the equation for clearing the remaining fraction.

5

Divide the entire equation by 2.

x - 15 + \frac{70}{x + 2} = 0

Dividing by the common factor simplifies the coefficients before expanding.

6

Multiply the entire equation by

x + 2

to clear the denominator, then expand and combine like terms.

x^2 - 13x + 40 = 0

This transforms the rational equation into a standard quadratic equation.

7

Factor the quadratic equation.

(x - 5)(x - 8) = 0

Factoring is the most direct method to find the roots of the quadratic equation.

8

Verify the candidate solutions against the domain restrictions identified in Step 1.

The candidate solution

x = 5

is extraneous because it is excluded from the domain of the original equation. The candidate solution

x = 8

is valid.

Extraneous solutions introduced during algebraic steps must be excluded from the final solution set.

Key Concept

Solving rational equations by simplifying terms, finding common denominators, and identifying extraneous solutions.

Question 324Question →

In the $xy$ -plane, the system of equations below has exactly one real solution.

y = -x^2 + 6x + 2

y + 2x = k

If $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 18

Answer

18

To find the value of the constant

k

for which the system of equations has exactly one real solution, we can solve the system by substitution. Substituting

y = k - 2x

from the second equation into the first equation gives

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

. Rearranging the terms to write this quadratic equation in standard form,

ax^2 + bx + c = 0

, yields

x^2 - 8x + (k - 2) = 0

. A quadratic equation has exactly one real solution when its discriminant,

b^2 - 4ac

, is equal to zero. Substituting

a = 1

,

b = -8

, and

c = k - 2

into the discriminant formula gives

(-8)^2 - 4(1)(k - 2) = 0

. Simplifying this expression results in

64 - 4k + 8 = 0

, which simplifies further to

72 - 4k = 0

. Solving for

k

yields

k = 18

.

Step-by-Step Solution

1

Rewrite the linear equation to express

y

in terms of

x

and

k

.

y = k - 2x

This allows substituting the expression for

y

into the quadratic equation to eliminate

y

.

2

Substitute the expression for

y

into the quadratic equation.

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

To construct a single quadratic equation in terms of

x

whose solutions correspond to the system's intersection points.

3

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 8x + (k - 2) = 0

To identify the coefficients

a = 1

,

b = -8

, and

c = k - 2

.

4

Set the discriminant of the quadratic equation to zero.

(-8)^2 - 4(1)(k - 2) = 0

A quadratic equation has exactly one real solution if and only if its discriminant is zero.

5

Solve the linear equation for

k

.

k = 18

Simplify the equation:

64 - 4k + 8 = 0 \implies 72 - 4k = 0 \implies 4k = 72 \implies k = 18

.

Key Concept

Nonlinear Systems of Equations

Question 325Question →

If $\frac{\left(\frac{1}{4}\right)^{2x-5}}{8^{3-x}} = \left(\sqrt[3]{2}\right)^{x-5}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

2

By writing all terms with a common base of 2, the equation simplifies to

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

. Equating the exponents yields

-x + 1 = \frac{x-5}{3}

. Solving this linear equation gives

x = 2

.

Step-by-Step Solution

1

Rewrite each base in terms of base 2.

\frac{1}{4} = 2^{-2}

,

8 = 2^3

, and

\sqrt[3]{2} = 2^{\frac{1}{3}}

Expressing all terms with a common base allows the exponents to be directly compared.

2

Substitute the bases back into the left side of the equation and simplify using exponent rules.

\frac{(2^{-2})^{2x-5}}{(2^3)^{3-x}} = \frac{2^{-4x+10}}{2^{9-3x}} = 2^{-x+1}

Applying the power rule

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

and quotient rule

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

simplifies the fraction.

3

Rewrite the right side of the equation using base 2.

(\sqrt[3]{2})^{x-5} = (2^{\frac{1}{3}})^{x-5} = 2^{\frac{x-5}{3}}

Applying the power rule to the fractional exponent converts the radical expression.

4

Equate the simplified left and right sides, then set the exponents equal to each other.

-x + 1 = \frac{x-5}{3}

If two exponential expressions with the same base are equal, their exponents must be equal.

5

Solve the linear equation for

x

.

-3x + 3 = x - 5 \implies 8 = 4x \implies x = 2

Isolating the variable

x

yields the final solution.

Key Concept

Solving exponential equations by expressing all terms with a common base and applying exponent laws.

Question 326Question →

An artist creates custom ceramic tiles in two shapes: square tiles and hexagonal tiles. Each square tile has an area of $12$ square inches, and each hexagonal tile has an area of $28$ square inches. The artist is designing a mosaic with a total area of $460$ square inches using only these two types of tiles. If the artist uses $15$ square tiles, how many hexagonal tiles are used in the mosaic?

Show answer & explanation

Answer: 10

Answer

The correct answer is 10. By setting up a linear equation representing the total area of the mosaic,

12s + 28h = 460

, and substituting the given value of 15 for the number of square tiles, we solve for the number of hexagonal tiles to get 10.

To find the number of hexagonal tiles used in the mosaic, we set up a linear equation in two variables representing the total area. Let

s

represent the number of square tiles, and let

h

represent the number of hexagonal tiles. The total area is the sum of the areas of the square tiles and the hexagonal tiles, which is

12s + 28h = 460

. Given that the artist uses

15

square tiles, we substitute

s = 15

into the equation to get

12(15) + 28h = 460

. Simplifying the equation yields

180 + 28h = 460

. Subtracting

180

from both sides gives

28h = 280

. Dividing both sides by

28

yields

h = 10

. Therefore, the artist uses 10 hexagonal tiles.

Step-by-Step Solution

1

Define variables and set up the linear equation representing the total area of the mosaic.

12s + 28h = 460

, where

s

is the number of square tiles and

h

is the number of hexagonal tiles.

This models the relationship between the quantities of each tile type used and the total surface area of the mosaic.

2

Substitute the given number of square tiles into the equation.

12(15) + 28h = 460

, which simplifies to

180 + 28h = 460

.

We are given that the artist uses exactly 15 square tiles in the design.

3

Isolate the variable term by subtracting the constant from both sides.

28h = 280

.

This isolates the term containing the unknown variable on one side of the equation.

4

Solve for the remaining variable by dividing both sides by its coefficient.

h = 10

.

Dividing both sides by the coefficient yields the final count of hexagonal tiles.

Key Concept

Linear Equations in Two Variables

Estimated Time:1m 30s

Question 327Question →

In 2015, the population of a certain species of owl in a nature reserve was $x$ . By 2020, the population had increased by $35\%$ . Between 2020 and 2025, the population decreased by $20\%$ of its 2020 value. If the population of the owl species in 2025 was $270$ , what was the value of $x$ ?

Show answer & explanation

Answer: 250

Answer

The population in 2015 was 250.

To find the initial population

x

, we apply the percent changes sequentially. A

35\%

increase is represented by multiplying by

1.35

, and a

20\%

decrease is represented by multiplying by

0.80

. The final population in terms of

x

is

x \times 1.35 \times 0.80 = 1.08x

. Given that the population in 2025 is

270

, we set up the equation

1.08x = 270

. Dividing both sides by

1.08

yields

x = 250

.

Step-by-Step Solution

1

Write an expression for the owl population in 2020 in terms of

x

.

The population in 2020 was

1.35x

.

An increase of

35\%

is represented by multiplying the initial amount by

1 + 0.35 = 1.35

.

2

Write an expression for the owl population in 2025 in terms of

x

.

The population in 2025 was

1.08x

.

A decrease of

20\%

from the 2020 population is represented by multiplying the 2020 population by

1 - 0.20 = 0.80

. Thus, the 2025 population is

1.35x \times 0.80 = 1.08x

.

3

Set the expression for the 2025 population equal to the given value of 270 and solve for

x

.

x = 250

Solving the equation

1.08x = 270

by dividing both sides by

1.08

gives

x = \frac{270}{1.08} = 250

.

Key Concept

Consecutive percent changes

Estimated Time:1m 30s

Question 328Question →

A software update reduced the storage space used by an application on a smartphone by $30\%$ . If the application originally used $400$ megabytes ( $\text{MB}$ ) of storage space, how many megabytes of storage space does the application use after the update?

Show answer & explanation

Answer: 280

Answer

The application uses

280

megabytes of storage space after the update.

The correct answer is

280

. The reduction in storage space is

30\%

of the original

400

megabytes, which is calculated as

0.30 \times 400 = 120

megabytes. Subtracting this reduction from the original storage space gives

400 - 120 = 280

megabytes. Alternatively, a reduction of

30\%

means the remaining storage is

70\%

of the original storage, which is

0.70 \times 400 = 280

megabytes.

Step-by-Step Solution

1

Calculate the reduction in storage space.

The reduction is

120

megabytes.

Multiply the original storage space of

400

megabytes by the percent reduction expressed as a decimal (

0.30

).

2

Subtract the reduction from the original storage space.

The new storage space is

280

megabytes.

Subtracting the amount of space saved (

120

megabytes) from the original space (

400

megabytes) gives the remaining space used.

Key Concept

Calculating a decreased amount after a percentage reduction.

Question 329Question →

The function $f$ is defined by $f(x) = x^2 - 6x + c$ , where $c$ is a constant. In the $xy$ -plane, the graph of $f$ has vertex $A$ . The function $g$ is defined by $g(x) = -f(x - 6)$ , and its graph has vertex $B$ . If the distance between points $A$ and $B$ is $10$ , and $c > 10$ , what is the value of $c$ ?

Show answer & explanation

Answer: 13

Answer

The value of

c

is

13

.

Completing the square for

f(x) = x^2 - 6x + c

gives

f(x) = (x - 3)^2 + c - 9

, which shows that vertex

A

is located at

(3, c - 9)

. The transformation

g(x) = -f(x - 6)

translates the graph

6

units to the right and reflects it vertically, giving vertex

B

the coordinates

(3 + 6, -(c - 9)) = (9, 9 - c)

. Using the distance formula, the distance between

A

and

B

is

\sqrt{(9 - 3)^2 + ((9 - c) - (c - 9))^2} = \sqrt{36 + (18 - 2c)^2}

. Setting this distance equal to

10

and squaring both sides gives

36 + (18 - 2c)^2 = 100

, which simplifies to

(18 - 2c)^2 = 64

. Taking the square root of both sides gives

18 - 2c = 8

or

18 - 2c = -8

, yielding solutions of

c = 5

or

c = 13

. Since the question specifies that

c > 10

, the correct value must be

13

.

Step-by-Step Solution

1

Rewrite the function

f(x) = x^2 - 6x + c

in vertex form by completing the square.

f(x) = (x - 3)^2 + c - 9

, which gives the coordinates of vertex

A

as

(3, c - 9)

.

Completing the square reveals the vertex

(h, k)

of a quadratic function in the form

y = a(x - h)^2 + k

.

2

Determine the vertex

B

of the graph of

g(x) = -f(x - 6)

by applying transformations to vertex

A(3, c - 9)

.

The horizontal shift of

f(x - 6)

moves the vertex to

(3 + 6, c - 9) = (9, c - 9)

. The reflection of

-f(x - 6)

negates the

y

-coordinate of the vertex, resulting in

B(9, 9 - c)

.

The transformation

f(x - h)

shifts a graph right by

h

units, and the transformation

-f(x)

reflects it across the

x

-axis.

3

Use the distance formula to set up an equation for the distance between

A(3, c - 9)

and

B(9, 9 - c)

.

\sqrt{(9 - 3)^2 + ((9 - c) - (c - 9))^2} = 10 \implies \sqrt{6^2 + (18 - 2c)^2} = 10

.

The distance

d

between points

(x_1, y_1)

and

(x_2, y_2)

is given by

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

.

4

Solve the radical equation

\sqrt{36 + (18 - 2c)^2} = 10

for

c

.

36 + (18 - 2c)^2 = 100 \implies (18 - 2c)^2 = 64 \implies 18 - 2c = 8

or

18 - 2c = -8

. This yields

c = 5

or

c = 13

.

Squaring both sides eliminates the square root, allowing us to solve the resulting quadratic equation.

5

Apply the constraint

c > 10

to choose the correct value for

c

.

c = 13

.

The question specifies that

c

must be greater than

10

, which excludes

c = 5

.

Key Concept

Using vertex form of quadratic equations to determine vertex coordinates and applying transformations (horizontal shifts and vertical reflections) to find key graphical points.

Question 330Question →

The polynomial function $p$ is defined by $p(x) = x^4 - 8x^3 + 20x^2 - 16x + c$ , where $c$ is a constant. In the $xy$ -plane, the graph of $y = p(x)$ is tangent to the $x$ -axis at two distinct points. What is the value of $c$ ?

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. A quartic polynomial that is tangent to the x-axis at two distinct points must have two roots of multiplicity 2. Thus, it can be factored as p(x) = (x - r_1)^2(x - r_2)^2 = [x^2 - (r_1 + r_2)x + r_1 r_2]^2. Setting S = r_1 + r_2 and P = r_1 r_2, we expand this to p(x) = x^4 - 2Sx^3 + (S^2 + 2P)x^2 - 2SPx + P^2. Matching this with the given equation x^4 - 8x^3 + 20x^2 - 16x + c shows that -2S = -8, so S = 4. Using the quadratic term, S^2 + 2P = 20, which yields 16 + 2P = 20, so P = 2. Finally, the constant term is c = P^2 = 2^2 = 4.

Step-by-Step Solution

1

Express the polynomial using its roots and multiplicities.

p(x) = (x - r_1)^2(x - r_2)^2

Since the graph of the quartic polynomial is tangent to the x-axis at two distinct points, it must have two real roots, each with a multiplicity of 2.

2

Group the terms and introduce variables for the sum and product of the roots.

p(x) = [x^2 - Sx + P]^2, where S = r_1 + r_2 and P = r_1 r_2

This simplifies expansion by using the symmetric properties of roots.

3

Expand the squared quadratic expression.

p(x) = x^4 - 2Sx^3 + (S^2 + 2P)x^2 - 2SPx + P^2

To compare coefficients with the standard form of the given polynomial.

4

Compare the coefficients of the x^3 term.

-2S = -8, which gives S = 4

To determine the sum of the roots from the given cubic coefficient.

5

Compare the coefficients of the x^2 term using the value of S.

S^2 + 2P = 20, which becomes 16 + 2P = 20, leading to P = 2

To find the product of the roots using the quadratic coefficient.

6

Verify the linear coefficient using S and P.

-2SP = -2(4)(2) = -16

To ensure consistency with the given coefficient of x.

7

Determine the constant term c.

c = P^2 = 2^2 = 4

To find the final value of the constant c.

Key Concept

Representing a polynomial function in factored form based on the multiplicity of its roots and using coefficient matching to determine unknown constants.

Question 331Question →

The function $f$ has the property that $f(3) = 11$ . The function $g$ is defined by $g(x) = f(x) - 4$ . What is the value of $g(3)$ ?

Show answer & explanation

Answer: 7

Answer

7

The value of

g(3)

is found by substituting

x = 3

into the equation

g(x) = f(x) - 4

, which results in

g(3) = f(3) - 4

. Substituting the given value of

f(3) = 11

yields

g(3) = 11 - 4

, which simplifies to

7

.

Step-by-Step Solution

1

Substitute

x = 3

into the function definition of

g(x)

.

g(3) = f(3) - 4

To evaluate the function

g

at a specific input, we replace

x

with

3

in the definition

g(x) = f(x) - 4

.

2

Substitute the given value of

f(3) = 11

into the equation.

g(3) = 11 - 4

The problem states that the value of

f(3)

is equal to

11

.

3

Simplify the expression to find the final value.

g(3) = 7

Subtracting

4

from

11

yields

7

.

Key Concept

Applying vertical translations using function notation.

Estimated Time:40s

Question 332Question →

In a certain school, the number of students in the science club was $25\%$ of the number of students in the math club. During a membership drive, the number of students in the science club increased by $30\%$ , and the number of students in the math club increased by $10\%$ . After the drive, the total number of students in both clubs combined was $228$ . If no student is a member of both clubs, what was the total number of students in both clubs combined before the membership drive?

Show answer & explanation

Answer: 200

Answer

The total number of students in both clubs combined before the membership drive was 200.

To find the initial total number of students, we define the initial math club members as

M

. Since the science club has

25\%

of the size of the math club, its initial size is

0.25M

. After a

30\%

increase, the science club size is

1.30(0.25M) = 0.325M

. After a

10\%

increase, the math club size is

1.10M

. The new combined total is

0.325M + 1.10M = 228

. Simplifying gives

1.425M = 228

, which yields

M = 160

. The initial science club size is

0.25(160) = 40

. The total number of students before the drive was

160 + 40 = 200

.

Step-by-Step Solution

1

Define variables for the initial number of students in each club.

Let

M

be the initial number of students in the math club, and

S = 0.25M

be the initial number of students in the science club.

This establishes a relationship between the sizes of the two clubs based on the given percentage.

2

Express the new number of students in each club after the percent increases.

Science club:

1.30 \times S = 1.30(0.25M) = 0.325M

. Math club:

1.10M

.

An increase of

30\%

is represented by multiplying by

1.30

, and an increase of

10\%

is represented by multiplying by

1.10

.

3

Set up and solve an equation using the new combined total.

0.325M + 1.10M = 228 \implies 1.425M = 228 \implies M = 160

.

The sum of the new club sizes equals the new total of

228

students.

4

Calculate the initial science club members and the initial total combined members.

Initial science members:

S = 0.25 \times 160 = 40

. Total initial members:

M + S = 160 + 40 = 200

.

This answers the final question asking for the combined total before the membership drive.

Key Concept

Applying percent changes to individual components to determine initial values in a combined total system.

Alternative Method

Instead of setting up equations using

M

, you can choose an arbitrary starting value for the math club that is easily divisible, such as

100

. If the math club starts with

100

students, the science club starts with

25

students. After the increases, the math club has

110

students and the science club has

25 \times 1.3 = 32.5

students, for a total of

142.5

students. Since the actual final total is

228

, we scale our initial values by a factor of

\frac{228}{142.5} = 1.6

. The initial combined total is therefore

125 \times 1.6 = 200

.

Estimated Time:2m 0s

Question 333Question →

An equation is given as follows:

\frac{x}{x - 3} - \frac{4}{x + 2} = \frac{20}{x^2 - x - 6}

What is the only value of $x$ for which this equation is true?

Show answer & explanation

Answer: 4

Answer

4

To solve the equation, factor the denominator on the right side:

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

. The least common denominator is

(x - 3)(x + 2)

. Multiplying both sides by

(x - 3)(x + 2)

clears the fractions, resulting in

x(x + 2) - 4(x - 3) = 20

. Expanding the terms gives

x^2 + 2x - 4x + 12 = 20

. Simplifying and writing this in standard form yields

x^2 - 2x - 8 = 0

. Factoring the quadratic gives

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

, which gives the candidate solutions

x = 4

and

x = -2

. Substituting

x = -2

back into the original equation results in a denominator of zero, so

x = -2

is an extraneous solution. The only valid solution is

x = 4

.

Step-by-Step Solution

1

Factor the quadratic trinomial in the denominator of the right side of the equation.

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

Identifying the factors of the quadratic trinomial helps find the least common denominator of the rational equation.

2

Multiply the entire equation by the least common denominator

(x - 3)(x + 2)

to clear the fractions.

x(x + 2) - 4(x - 3) = 20

, with the constraints that

x \neq 3

and

x \neq -2

.

This simplifies the rational equation into a standard polynomial equation.

3

Distribute and combine like terms to write the equation in standard quadratic form.

x^2 - 2x - 8 = 0

Rewriting the equation in the form

ax^2 + bx + c = 0

is necessary to solve it by factoring.

4

Factor the quadratic equation.

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

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

5

Check the candidate solutions

x = 4

and

x = -2

against the original equation to identify extraneous solutions.

x = 4

is the only valid solution because

x = -2

makes the denominators in the original equation equal to zero.

Any solution that makes a denominator in the original rational expression equal to zero is extraneous and must be excluded.

Key Concept

Solving rational equations by finding a common denominator, clearing fractions, and checking for extraneous solutions.

Question 334Question →

If $81^{y-2} = 27^{y+1}$ , what is the value of $y$ ?

Show answer & explanation

Answer: 11

Answer

11

The correct answer is 11. By expressing both sides of the equation with the common base of 3, the equation simplifies from

81^{y-2} = 27^{y+1}

to

(3^4)^{y-2} = (3^3)^{y+1}

. Applying the power of a power rule gives

3^{4y-8} = 3^{3y+3}

. Equating the exponents yields

4y - 8 = 3y + 3

, which solves to

y = 11

.

Step-by-Step Solution

1

Express both bases as powers of 3

(3^4)^{y-2} = (3^3)^{y+1}

To solve an exponential equation algebraically, it is helpful to express both sides using a common base.

2

Apply the power of a power exponent rule

3^{4(y-2)} = 3^{3(y+1)}

The rule

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

allows us to simplify the exponent expressions by multiplying the exponents.

3

Equate the exponents

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

Since the bases are equal, the exponents must be equal for the equation to hold true.

4

Distribute the coefficients

4y - 8 = 3y + 3

Expanding the linear expressions prepares the equation for isolation of the variable.

5

Solve the linear equation for

y

y = 11

Subtract

3y

from both sides to get

y - 8 = 3

, then add 8 to both sides to isolate

y

.

Key Concept

Solving exponential equations by expressing bases in terms of a common base and equating exponents.

Alternative Method

Alternatively, substitute the value of 11 back into the original equation to verify that both sides are equal:

81^{11-2} = 81^9 = (3^4)^9 = 3^{36}

and

27^{11+1} = 27^{12} = (3^3)^{12} = 3^{36}

.

Estimated Time:1m 30s

Question 335Question →

What is the sum of all positive real solutions to the equation

\frac{x^2 - 3x}{x^2 - 3x - 2} + \frac{x^2 - 3x - 2}{x^2 - 3x} = \frac{5}{2}

?

Show answer & explanation

Answer: 7

Answer

The sum of all positive real solutions is 7.

By substituting

u = x^2 - 3x

, the original rational equation simplifies to

\frac{u}{u - 2} + \frac{u - 2}{u} = \frac{5}{2}

. Multiplying both sides by the common denominator

2u(u-2)

and simplifying results in the quadratic equation

u^2 - 2u - 8 = 0

. Factoring gives

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

, so

u = 4

or

u = -2

. Substituting back

x^2 - 3x

for

u

leads to two quadratic equations:

x^2 - 3x = 4

(which has solutions

x = 4

and

x = -1

) and

x^2 - 3x = -2

(which has solutions

x = 2

and

x = 1

). Checking the denominators, none of these solutions make the original denominators zero, so they are all valid. The positive solutions are

1

,

2

, and

4

, and their sum is

1 + 2 + 4 = 7

.

Step-by-Step Solution

1

Introduce a substitution variable to simplify the rational equation.

Letting

u = x^2 - 3x

transforms the equation into

\frac{u}{u - 2} + \frac{u - 2}{u} = \frac{5}{2}

.

This reduces the degree of the rational expression and simplifies the algebraic manipulation required to solve it.

2

Eliminate the denominators by multiplying by the least common denominator.

Multiplying by

2u(u-2)

gives

2u^2 + 2(u-2)^2 = 5u(u-2)

, which simplifies to

u^2 - 2u - 8 = 0

.

This converts the rational equation into a standard quadratic equation in terms of

u

.

3

Solve the quadratic equation for

u

by factoring.

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

, which gives

u = 4

or

u = -2

.

Finding the values of

u

allows us to set up equations to solve for the original variable

x

.

4

Substitute back

x^2 - 3x

for

u

and solve the resulting quadratic equations for

x

.

From

x^2 - 3x = 4

, we get

(x-4)(x+1) = 0 \implies x = 4, -1

. From

x^2 - 3x = -2

, we get

(x-2)(x-1) = 0 \implies x = 2, 1

.

This yields all real values of

x

that satisfy the original algebraic structure.

5

Filter for positive real solutions and calculate their sum.

The positive solutions are

1

,

2

, and

4

. Their sum is

1 + 2 + 4 = 7

.

The question specifically asks for the sum of only the positive real solutions.

Key Concept

Solving rational equations using algebraic substitution and factoring quadratic equations.

Alternative Method

Instead of using substitution directly, the equation can be solved by multiplying by the common denominator

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

to get a fourth-degree polynomial:

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

. Letting

z = x^2 - 3x

at this stage simplifies this expression to

2z^2 + 2(z-2)^2 = 5z(z-2)

, which avoids full expansion into a fourth-degree polynomial and leads to the same quadratic in

z

.

Estimated Time:3m 0s

Question 336Question →

Due to rising global temperatures, the thickness $T$ , in meters, of a mountain glacier is decreasing at a constant rate. A researcher models the thickness of the glacier using the equation $T = 48.5 - 0.12w$ , where $w$ represents the number of weeks since the start of a monitoring study. According to the model, by how many meters does the thickness of the glacier decrease every $50$ weeks?

Show answer & explanation

Answer: 6

Answer

The glacier's thickness decreases by 6 meters.

In the linear model

T = 48.5 - 0.12w

, the thickness

T

decreases by

0.12

meters for each increase of

1

in

w

(each week). Therefore, over a period of

50

weeks, the total decrease in thickness is

0.12 \times 50 = 6

meters.

Step-by-Step Solution

1

Identify the weekly rate of decrease from the slope of the equation.

0.12 meters per week

In the linear model

T = 48.5 - 0.12w

, the coefficient of

w

is

-0.12

. This slope represents the rate of change, indicating a decrease of

0.12

meters in thickness per week.

2

Calculate the cumulative decrease over 50 weeks.

6

Multiply the weekly rate of decrease (

0.12

meters per week) by the duration (

50

weeks) to obtain the total decrease:

0.12 \times 50 = 6

.

Key Concept

Interpreting the slope of a linear relationship in context

Estimated Time:1m 30s

Question 337Question →

Last year, a gardener grew $80$ tomato plants. This year, the gardener grew $92$ tomato plants. What is the percent increase in the number of tomato plants from last year to this year?

Show answer & explanation

Answer: 15

Answer

The percent increase in the number of tomato plants is

15

.

To find the percent increase, subtract the original value from the new value to get the absolute increase:

92 - 80 = 12

. Next, divide this increase by the original value:

\frac{12}{80} = 0.15

. Finally, multiply by

100

to convert the decimal to a percentage, resulting in

15

.

Step-by-Step Solution

1

Subtract the original number of tomato plants from the new number of plants

12

plants

To find the absolute change in the quantity of plants.

2

Divide the change by the original number of plants

0.15

To find the ratio of the increase relative to the starting amount.

3

Multiply the ratio by

100

to convert it to a percentage

15

To express the ratio as a percentage value.

Key Concept

Calculating percent change from an initial value to a final value

Question 338Question →

\begin{aligned} 4x - 3y &= 25 \\ -2x + 5y &= -9 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x - y$ ?

Show answer & explanation

Answer: 6

Answer

6

To solve the system of equations, we can use the elimination method. First, multiply the second equation by 2 to align the coefficients of

x

:

2(-2x + 5y) = 2(-9) \implies -4x + 10y = -18

Next, add this new equation to the first equation to eliminate

x

:

(4x - 3y) + (-4x + 10y) = 25 + (-18)

7y = 7

y = 1

Substitute

y = 1

back into the second equation to solve for

x

:

-2x + 5(1) = -9

-2x + 5 = -9

-2x = -14

x = 7

Finally, calculate the value of the requested expression

x - y

:

x - y = 7 - 1 = 6

Thus, the correct response is 6.

Step-by-Step Solution

1

Multiply the second equation by 2 to prepare for the elimination of

x

.

-4x + 10y = -18

This creates coefficients for

x

in both equations that are additive opposites, allowing

x

to be eliminated when the equations are added.

2

Add the first equation and the modified second equation together to solve for

y

.

7y = 7

, which simplifies to

y = 1

Adding the equations eliminates the

x

terms and leaves a single-variable equation in terms of

y

.

3

Substitute

y = 1

back into one of the original equations to solve for

x

.

-2x + 5(1) = -9

, which simplifies to

-2x = -14

, yielding

x = 7

Now that the value of

y

is known, it can be substituted into either equation to find the corresponding value of

x

.

4

Calculate the value of the expression

x - y

.

7 - 1 = 6

The question asks specifically for the value of the difference

x - y

, so we subtract the value of

y

from the value of

x

.

Key Concept

Solving systems of linear equations using the elimination method and evaluating linear combinations of the solutions.

Estimated Time:1m 30s

Question 339Question →

The function $f$ is defined by $f(x) = (x - 4)(x - 2)(x + k)$ , where $k$ is a constant. If the $y$ -intercept of the graph of $y = f(x)$ in the $xy$ -plane is $(0, 24)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

3

The

y

-intercept of the graph of

y = f(x)

is the point where

x = 0

. Since the

y

-intercept is

(0, 24)

, it follows that

f(0) = 24

. Substituting

0

for

x

in the equation for

f(x)

gives

f(0) = (0 - 4)(0 - 2)(0 + k) = (-4)(-2)(k) = 8k

. Setting this equal to the

y

-value of the intercept yields

8k = 24

. Dividing both sides of the equation by

8

gives

k = 3

.

Step-by-Step Solution

1

Use the definition of the

y

-intercept to find the value of

f(0)

f(0) = 24

The

y

-intercept of a graph is the point where the graph crosses the

y

-axis, corresponding to

x = 0

. Given the point

(0, 24)

,

f(0)

must equal

24

.

2

Evaluate the polynomial at

x = 0

in terms of

k

f(0) = 8k

Substituting

0

for

x

in

f(x) = (x - 4)(x - 2)(x + k)

gives

f(0) = (-4)(-2)(k)

, which simplifies to

8k

.

3

Set the evaluated expression equal to the

y

-intercept value and solve for

k

k = 3

Equating

8k

to

24

and dividing both sides by

8

yields

k = 3

.

Key Concept

Evaluating a factored polynomial function at

x = 0

determines its

y

-intercept. Using a given

y

-intercept allows solving for unknown coefficients or constants within the factors of the polynomial.

Question 340Question →

In the $xy$ -plane, the graph of the linear equation $kx - 3y = 18$ , where $k$ is a constant, has a $y$ -intercept of $(0, b)$ and an $x$ -intercept of $(a, 0)$ , where $a$ and $b$ are constants. If $a - b = 15$ , what is the value of $k$ ?

Show answer & explanation

Answer: 2

Answer

2

To find the value of

k

, we first determine the intercepts of the linear equation

kx - 3y = 18

. The

y

-intercept

(0, b)

occurs where

x = 0

. Substituting

x = 0

gives

-3b = 18

, which solves to

b = -6

. Next, we use the given relation

a - b = 15

. Substituting

b = -6

yields

a - (-6) = 15

, or

a + 6 = 15

, which gives

a = 9

. The

x

-intercept is therefore

(9, 0)

. Since this point lies on the line, we substitute

x = 9

and

y = 0

into the original equation:

k(9) - 3(0) = 18

. This simplifies to

9k = 18

, which gives

k = 2

.

Step-by-Step Solution

1

Find the

y

-coordinate of the

y

-intercept (

b

) by setting

x = 0

in the equation

kx - 3y = 18

.

b = -6

The

y

-intercept occurs where the graph crosses the

y

-axis, which corresponds to

x = 0

.

2

Use the equation

a - b = 15

and the value of

b = -6

to solve for

a

.

a = 9

Substituting

b = -6

into

a - b = 15

gives

a - (-6) = 15

, which simplifies to

a + 6 = 15

.

3

Find the value of

k

by substituting the

x

-intercept

(9, 0)

into the equation

kx - 3y = 18

.

k = 2

Since

(a, 0) = (9, 0)

is the

x

-intercept, it must satisfy the equation of the line.

Key Concept

Determining intercepts of a linear equation in two variables and using them to find unknown constants.

Answer

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