All practice questions

Question 1801Question →

Let the functions $f$ and $g$ be defined by $f(x) = |2x - 5|$ and $g(x) = \sqrt{x + 7}$ for all real numbers in their respective domains. If $a$ is a real number such that $(f \circ g)(a) = 3$ , what is the sum of all possible values of $a$ ?

Show answer & explanation

Answer: 3

Answer

The sum of all possible values of

a

is

3

.

The correct answer is

3

. The composition

(f \circ g)(a) = 3

translates to

f(g(a)) = |2\sqrt{a + 7} - 5| = 3

. This absolute value relation splits into two equations:

2\sqrt{a + 7} - 5 = 3

and

2\sqrt{a + 7} - 5 = -3

. Solving the first equation yields

\sqrt{a + 7} = 4 \Rightarrow a = 9

. Solving the second equation yields

\sqrt{a + 7} = 1 \Rightarrow a = -6

. Since both

9

and

-6

are greater than or equal to

-7

, they are within the domain of the radical function. The sum of these values is

9 + (-6) = 3

.

Step-by-Step Solution

1

Express the composition

(f \circ g)(a)

in terms of

a

.

|2\sqrt{a + 7} - 5| = 3

By definition of function composition,

(f \circ g)(a) = f(g(a))

. Substituting

g(a) = \sqrt{a + 7}

into the expression for

f(x)

yields

f(g(a)) = |2g(a) - 5| = |2\sqrt{a + 7} - 5|

.

2

Set up the two algebraic cases to eliminate the absolute value.

2\sqrt{a + 7} - 5 = 3

or

2\sqrt{a + 7} - 5 = -3

An absolute value equation of the form

|u| = c

(where

c \geq 0

) has two possible cases:

u = c

or

u = -c

.

3

Solve the first equation case for

a

.

a = 9

Adding

5

to both sides of

2\sqrt{a + 7} - 5 = 3

gives

2\sqrt{a + 7} = 8

. Dividing by

2

gives

\sqrt{a + 7} = 4

. Squaring both sides yields

a + 7 = 16

, which gives

a = 9

.

4

Solve the second equation case for

a

.

a = -6

Adding

5

to both sides of

2\sqrt{a + 7} - 5 = -3

gives

2\sqrt{a + 7} = 2

. Dividing by

2

gives

\sqrt{a + 7} = 1

. Squaring both sides yields

a + 7 = 1

, which gives

a = -6

.

5

Verify domain constraints and sum the valid solutions.

3

Both

a = 9

and

a = -6

satisfy the domain requirement for

g(x) = \sqrt{x+7}

, which is

x \geq -7

. The sum of these two valid values is

9 + (-6) = 3

.

Key Concept

Evaluating and solving equations containing composite functions, absolute values, and radical functions.

Alternative Method

Instead of expanding the composition immediately, substitute a temporary variable

u = g(a) = \sqrt{a+7}

. The equation becomes

f(u) = 3 \Rightarrow |2u - 5| = 3

. Solve this simplified absolute value equation to get

2u - 5 = 3 \Rightarrow u = 4

, and

2u - 5 = -3 \Rightarrow u = 1

. Next, substitute back

\sqrt{a+7}

for

u

: solving

\sqrt{a+7} = 4

yields

a = 9

, and solving

\sqrt{a+7} = 1

yields

a = -6

. Summing these two solutions gives

9 + (-6) = 3

.

Estimated Time:2m 0s

Question 1802Question →

Match each algebraic expression on the left with its fully simplified equivalent expression on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The expression

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

simplifies to

-ab

; the expression

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

simplifies to

ab

; and the expression

a^2(a - b) - a(a^2 - ab)

simplifies to

0

.

Each expression is correctly simplified by distributing the external factors across parenthetical terms and combining the resulting like terms.

Step-by-Step Solution

1

Simplify the first expression by distributing coefficients and combining like terms.

--a^2 + 2ab + a^2 - 3b = -ab

Distributing the negative sign yields

--a^2 + 2ab

and distributing

a

yields

a^2 - 3ab

. Adding them eliminates the

a^2

terms, leaving

-ab

.

2

Simplify the second expression by distributing coefficients and combining like terms.

2a^2b - 2ab - 2a^2b + 3ab = ab

Distributing

2

yields

2a^2b - 2ab

and distributing

-ab

yields

-2a^2b + 3ab

. Adding them eliminates the

2a^2b

terms, leaving

ab

.

3

Simplify the third expression by distributing coefficients and combining like terms.

a^3 - a^2b - a^3 + a^2b = 0

Distributing

a^2

yields

a^3 - a^2b

and distributing

-a

yields

-a^3 + a^2b

. Adding them eliminates all terms, leaving

0

.

Key Concept

Simplifying algebraic expressions containing multiple variables and higher powers by applying the distributive property and combining like terms.

Question 1803Question →

A specialty coffee shop blends three types of coffee beans: Colombian ( $8.00$ dollars per pound), Ethiopian ( $11.00$ dollars per pound), and Sumatran ( $14.00$ dollars per pound). The shop manager wants to create a $50$ -pound blend that costs exactly $11.60$ dollars per pound. Due to supply constraints, the weight of the Ethiopian beans in the blend must be exactly $2$ pounds more than $\frac{1}{5}$ of the combined weight of the Colombian and Sumatran beans. What is the value of $2s - c$ , where $s$ is the amount of Sumatran beans, in pounds, and $c$ is the amount of Colombian beans, in pounds, in the blend?

Show answer & explanation

Answer: 35

Answer

The correct answer is

35

.

The correct answer of

35

is found by translating the given constraints into three linear equations, solving for

e = 10

using substitution of the sum

c+s = 50 - e

, and then solving the resulting two-variable system to find

s = 25

and

c = 15

. Evaluating

2s - c

yields

2(25) - 15 = 35

.

Step-by-Step Solution

1

Define variables and write the system of equations representing the constraints.

Let

c

,

e

, and

s

represent the weight in pounds of Colombian, Ethiopian, and Sumatran beans, respectively. The system is:
1)

c + e + s = 50

(total weight)
2)

8c + 11e + 14s = 50 \times 11.60 = 580

(total cost)
3)

e = \frac{1}{5}(c + s) + 2

(Ethiopian bean weight constraint)

This translates the word problem's conditions into algebraic expressions.

2

Solve for the variable

e

using substitution.

From equation (1), we have

c + s = 50 - e

. Substituting this expression into equation (3) yields:

e = \frac{1}{5}(50 - e) + 2

Multiply both sides by

5

:

5e = 50 - e + 10

6e = 60 \implies e = 10

Grouping

c + s

allows us to solve for

e

directly without dealing with three separate variable eliminations.

3

Substitute

e = 10

back into the first two equations to simplify the system to two variables.

Equation (1) becomes:

c + s = 40 \implies c = 40 - s

Equation (2) becomes:

8c + 11(10) + 14s = 580 \implies 8c + 14s = 470

This reduces the remaining problem to a standard system of two linear equations.

4

Solve for

c

and

s

by substitution.

Substitute

c = 40 - s

into the simplified cost equation:

8(40 - s) + 14s = 470

320 - 8s + 14s = 470

320 + 6s = 470 \implies 6s = 150 \implies s = 25

Then, find

c

:

c = 40 - 25 = 15

This gives the exact weight of both the Sumatran and Colombian beans.

5

Evaluate the expression

2s - c

requested by the prompt.

2s - c = 2(25) - 15 = 50 - 15 = 35

This satisfies the specific algebraic quantity requested in the question.

Key Concept

Translating word problems into systems of linear equations and solving them using algebraic substitution.

Question 1804Question →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = 2x - 3$ and $g(x) = x + 5$ . What is the value of $f(g(4))$ ?

Show answer & explanation

Answer: 15

Answer

The value of

f(g(4))

is

15

.

To evaluate the composite function

f(g(4))

, the inner function must be evaluated first. Substituting

4

into

g(x) = x + 5

gives

g(4) = 9

. Next, substitute this output of

9

into the outer function

f(x) = 2x - 3

, which gives

f(9) = 2(9) - 3 = 15

.

Step-by-Step Solution

1

Evaluate the inner function

g(4)

g(4) = 9

Substitute

4

for

x

in the definition of

g(x) = x + 5

.

2

Evaluate the outer function

f(x)

at the output of the inner function

f(9) = 15

Substitute

9

(the value of

g(4)

) for

x

in the definition of

f(x) = 2x - 3

.

Key Concept

Function Composition and Evaluation

Estimated Time:45s

Question 1805Question →

Which of the following is the completely factored form of the expression $x^3 - 9x$ ?

x(x - 3)^2

x(x^3 - 9)

x(x - 3)(x + 3)

9x(x - 1)(x + 1)

x(x + 3)^2

Show answer & explanation

Answer: $x(x - 3)(x + 3)$

Answer

x(x - 3)(x + 3)

The expression

x^3 - 9x

can be factored by first finding the greatest common factor of the terms. Since both terms share a factor of

x

, factoring out

x

yields

x(x^2 - 9)

. The binomial

x^2 - 9

is a difference of squares because it can be written as

x^2 - 3^2

. Applying the difference of squares formula,

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

, gives

(x - 3)(x + 3)

. Therefore, the completely factored form is

x(x - 3)(x + 3)

.

Step-by-Step Solution

1

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

x(x^2 - 9)

Both

x^3

and

9x

share a common factor of

x

.

2

Factor the remaining binomial expression inside the parentheses using the difference of squares formula.

x(x - 3)(x + 3)

The expression

x^2 - 9

is a difference of squares, which factors as

(a - b)(a + b)

where

a = x

and

b = 3

.

Key Concept

Factoring out the greatest common factor and factoring a difference of squares.

Question 1806Question →

For all real numbers $x$ such that $x \neq 0$ , the expression $\left(\frac{1}{x} + 1\right)^2 - \frac{1}{x^2}$ is equivalent to which of the following?

\frac{x+2}{x}

1

\frac{3}{x^2}

\frac{x+2}{2x}

2x

Show answer & explanation

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

Answer

\frac{x+2}{x}

Expanding the squared term yields

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

. Subtracting

\frac{1}{x^2}

from this leaves

\frac{2}{x} + 1

. Finding a common denominator of

x

to add these terms results in

\frac{x+2}{x}

.

Step-by-Step Solution

1

Expand the binomial term

\left(\frac{1}{x} + 1\right)^2

.

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

Using the algebraic identity

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

where

a = \frac{1}{x}

and

b = 1

.

2

Substitute this expansion back into the original expression and subtract

\frac{1}{x^2}

.

\frac{2}{x} + 1

The positive and negative

\frac{1}{x^2}

terms cancel each other out.

3

Write the expression as a single fraction over the common denominator

x

.

\frac{x+2}{x}

Convert

1

to

\frac{x}{x}

and add the numerators.

Key Concept

Simplifying rational expressions by expanding binomials and finding common denominators.

Alternative Method

Use the difference of squares factorization:

A^2 - B^2 = (A-B)(A+B)

. Let

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

and

B = \frac{1}{x}

. Then the expression simplifies to

\left(\frac{1}{x} + 1 - \frac{1}{x}\right)\left(\frac{1}{x} + 1 + \frac{1}{x}\right) = 1 \cdot \left(\frac{2}{x} + 1\right) = \frac{x+2}{x}

.

Estimated Time:1m 0s

Question 1807Question →

If $x(x - 6) = 7$ , what is the positive difference between the two solutions to this equation?

6

8

10

14

0

Show answer & explanation

Answer: 8

Answer

The positive difference between the two solutions is

8

.

First, expand the left side of the equation to get

x^2 - 6x = 7

. Next, subtract

7

from both sides to write the equation in standard form:

x^2 - 6x - 7 = 0

. Factoring this quadratic expression gives

(x - 7)(x + 1) = 0

. Setting each factor to zero yields the solutions

x = 7

and

x = -1

. The positive difference between these two solutions is

7 - (-1) = 8

.

Step-by-Step Solution

1

Distribute the

x

on the left side of the equation.

x^2 - 6x = 7

Multiplying

x

by

(x - 6)

expands the left side to prepare for standard form.

2

Subtract

7

from both sides to set the quadratic equation to zero.

x^2 - 6x - 7 = 0

Setting the quadratic equation to standard form

ax^2 + bx + c = 0

is necessary before factoring.

3

Factor the quadratic trinomial.

(x - 7)(x + 1) = 0

We need two numbers that multiply to

-7

and add to

-6

, which are

-7

and

1

.

4

Set each factor to zero to find the solutions.

x = 7

and

x = -1

The zero product property states that if a product is zero, at least one factor must be zero.

5

Calculate the positive difference between the two solutions.

7 - (-1) = 8

The positive difference is found by subtracting the smaller solution from the larger solution.

Key Concept

Solving Quadratic Equations by Factoring

Estimated Time:1m 0s

Question 1808Question →

The area of a rectangle is represented by the expression $18x^3 - 50x$ square inches. If the width of the rectangle is $2x$ inches, which of the following expressions represents the length of the rectangle, in inches?

(3x - 5)^2

9x - 25

9x^2 - 50x

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

18x^3 - 52x

Show answer & explanation

Answer: $(3x - 5)(3x + 5)$

Answer

The length of the rectangle is represented by the expression

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

.

The correct answer is the expression

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

because the length is found by dividing the area,

18x^3 - 50x

, by the width,

2x

. Simplifying this quotient yields

9x^2 - 25

. The expression

9x^2 - 25

is a difference of squares and factors completely into

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

.

Step-by-Step Solution

1

Set up the equation for the length of the rectangle by dividing the area by the width.

\text{Length} = \frac{18x^3 - 50x}{2x}

Since the area of a rectangle is the product of its length and width, the length can be found by dividing the area by the width.

2

Perform the polynomial division by dividing each term of the numerator by the denominator

2x

.

\frac{18x^3 - 50x}{2x} = \frac{18x^3}{2x} - \frac{50x}{2x} = 9x^2 - 25

Dividing a polynomial by a monomial requires dividing every term of the polynomial by that monomial.

3

Factor the resulting expression

9x^2 - 25

using the difference of squares identity,

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

.

9x^2 - 25 = (3x)^2 - 5^2 = (3x - 5)(3x + 5)

The quadratic expression is a difference of two perfect squares, which can be factored into a product of binomial conjugates.

Key Concept

Factoring a difference of squares after dividing a polynomial by a monomial

Estimated Time:1m 30s

Question 1809Question →

The table below gives values for the functions $f$ and $g$ at selected values of $x$ :

$x$	$f(x)$	$g(x)$
1	3	5
2	5	4
3	2	1
4	1	2
5	4	3

Based on the table, what is the value of $f(g(3))$ ?

1

2

3

4

5

Show answer & explanation

Answer: 3

Answer

3

To evaluate the composite function

f(g(3))

, we first evaluate the inner function,

g(3)

. Looking at the table, when

x = 3

,

g(3) = 1

. We then substitute this output into the outer function to evaluate

f(1)

. According to the table, when

x = 1

,

f(1) = 3

. Therefore, the value of the composite function is 3.

Step-by-Step Solution

1

Locate the input value of 3 in the table to evaluate the inner function

g(3)

.

g(3) = 1

We must evaluate the inner function first in the composition

f(g(x))

.

2

Substitute the result of

g(3)

into the outer function, giving

f(1)

, and find its value in the table.

f(1) = 3

Evaluating

f(x)

at

x = 1

completes the composition

f(g(3))

.

Key Concept

Evaluating a composite function from a table of values by finding the output of the inner function and using it as the input for the outer function.

Estimated Time:45s

Question 1810Question →

For all non-zero real numbers $x$ and $y$ , the algebraic expression $\left(\frac{x^{a/2} y^{-1}}{x^{-2/3} y^{b/3}}\right)^{-6}$ is equivalent to $\frac{y^{12}}{x^{16}}$ , where $a$ and $b$ are integers. What is the value of $a + b$ ?

-14

1

7

9

11

Show answer & explanation

Answer: 7

Answer

7

The correct answer is 7. Simplifying the expression inside the parentheses using the quotient rule yields

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

. Raising this expression to the power of

-6

results in

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

. Equating this to

x^{-16} y^{12}

gives

a = 4

and

b = 3

, so

a + b = 7

.

Step-by-Step Solution

1

Simplify the expression inside the parentheses using the quotient rule for exponents,

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

.

x^{\frac{a}{2} - \left(-\frac{2}{3}\right)} y^{-1 - \frac{b}{3}} = x^{\frac{a}{2} + \frac{2}{3}} y^{-1 - \frac{b}{3}}

To combine the base

x

and base

y

terms inside the parentheses before applying the outer power.

2

Apply the outer exponent of

-6

to the simplified expression using the power of a power rule,

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

.

x^{-6\left(\frac{a}{2} + \frac{2}{3}\right)} y^{-6\left(-1 - \frac{b}{3}\right)} = x^{-3a - 4} y^{6 + 2b}

To distribute the negative power of

-6

to each factor in the product.

3

Equate the resulting exponents to the exponents of the given equivalent expression,

\frac{y^{12}}{x^{16}} = x^{-16} y^{12}

, and solve the equations for

a

and

b

.

For

x

:

-3a - 4 = -16 \implies -3a = -12 \implies a = 4

. For

y

:

6 + 2b = 12 \implies 2b = 6 \implies b = 3

.

Since the expressions are equivalent for all non-zero values of

x

and

y

, their respective exponents must be equal.

4

Calculate the sum of

a

and

b

.

a + b = 4 + 3 = 7

To find the final requested value.

Key Concept

Properties of exponents in algebraic expressions including the quotient rule, power rule, and operations with fractional/negative exponents.

Question 1811Question →

For the functions $f(x) = (x - 3)^2 - 4$ and $g(x) = |x + 1|$ , what is the sum of all real values of $x$ for which $f(g(x)) = 5$ ?

-3

-2

4

5

6

Show answer & explanation

Answer: $-3$

Answer

The sum of all real values of

x

for which

f(g(x)) = 5

is

-3

.

To find the sum of all real values of

x

for which

f(g(x)) = 5

, we set up the composite function:

(g(x) - 3)^2 - 4 = 5

. Adding

4

to both sides gives

(g(x) - 3)^2 = 9

. Taking the square root yields two possible cases:

g(x) - 3 = 3

(which means

g(x) = 6

) or

g(x) - 3 = -3

(which means

g(x) = 0

). Next, we substitute

g(x) = |x + 1|

. For the first case,

|x + 1| = 6

gives

x + 1 = 6 \implies x = 5

or

x + 1 = -6 \implies x = -7

. For the second case,

|x + 1| = 0

gives

x + 1 = 0 \implies x = -1

. Summing these three real solutions gives

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

.

Step-by-Step Solution

1

Substitute

g(x)

into

f(x)

and set the expression equal to

5

.

f(g(x)) = (g(x) - 3)^2 - 4 = 5

This sets up the composition equation to be solved.

2

Isolate the squared term and solve for

g(x)

by taking the square root of both sides.

(g(x) - 3)^2 = 9

, which gives two cases:

g(x) - 3 = 3 \implies g(x) = 6

or

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

Solving a quadratic equation of the form

u^2 = k

yields two possibilities:

u = \pm\sqrt{k}

.

3

Substitute the expression for

g(x) = |x + 1|

into both cases and solve for

x

.

For

g(x) = 6

,

|x + 1| = 6 \implies x + 1 = 6

or

x + 1 = -6

, yielding

x = 5

or

x = -7

. For

g(x) = 0

,

|x + 1| = 0 \implies x + 1 = 0

, yielding

x = -1

.

Absolute value equations of the form

|v| = c

yield

v = \pm c

if

c > 0

, and

v = 0

if

c = 0

.

4

Calculate the sum of all the real solutions.

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

The question asks for the sum of all real values of

x

that satisfy the equation.

Key Concept

Function composition involves substituting one function into another, and solving the resulting equation requires accounting for multiple cases when dealing with quadratic and absolute value expressions.

Question 1812Question →

What is the positive difference between the two real solutions to the equation $(2x - 1)^2 = x(3x - 5) + 7$ ?

1

3

5

6

7

Show answer & explanation

Answer: 5

Answer

The positive difference between the two real solutions is 5.

The correct answer is 5. By expanding the equation, we get

4x^2 - 4x + 1 = 3x^2 - 5x + 7

. Rearranging the terms to set the equation to zero yields

x^2 + x - 6 = 0

. Factoring this equation gives

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

, which has the solutions

x = -3

and

x = 2

. The positive difference between these two solutions is

|2 - (-3)| = 5

.

Step-by-Step Solution

1

Expand both sides of the equation.

4x^2 - 4x + 1 = 3x^2 - 5x + 7

Expanding the squared term on the left side and distributing the

x

on the right side allows us to write the equation in polynomial form.

2

Rearrange the equation by moving all terms to the left side to set the right side to zero.

x^2 + x - 6 = 0

Subtracting

3x^2

, adding

5x

, and subtracting

7

from both sides simplifies the equation into standard quadratic form,

ax^2 + bx + c = 0

.

3

Factor the quadratic trinomial.

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

Finding two integers that multiply to

-6

and add to

1

gives

3

and

-2

, allowing us to factor the equation over the integers.

4

Solve for

x

by setting each linear factor to zero.

x = -3

and

x = 2

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

5

Calculate the positive difference between the two solutions.

|2 - (-3)| = 5

Subtracting the smaller solution from the larger solution gives the positive distance between them on the number line.

Key Concept

Solving quadratic equations by expanding, rearranging into standard form, and factoring over the integers.

Estimated Time:2m 0s

Question 1813Question →

What is the positive difference between the two real solutions to the equation $(x - 1)^2 = 5x - 5$ ?

1

5

7

6

4

Show answer & explanation

Answer: 5

Answer

The positive difference between the two real solutions is 5.

Expanding the left side of

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

yields

x^2 - 2x + 1 = 5x - 5

. Moving all terms to the left side by subtracting

5x

and adding

5

gives the standard quadratic equation

x^2 - 7x + 6 = 0

. Factoring this expression gives

(x - 1)(x - 6) = 0

, which yields the solutions

1

and

6

. The positive difference between these solutions is

6 - 1 = 5

.

Step-by-Step Solution

1

Expand the squared binomial on the left side of the equation.

x^2 - 2x + 1 = 5x - 5

Before factoring a quadratic equation, all terms must be expanded and moved to one side to set the equation equal to zero.

2

Subtract

5x

and add

5

to both sides to rearrange the equation into standard quadratic form.

x^2 - 7x + 6 = 0

This sets the equation equal to zero, which is a prerequisite for using the zero product property.

3

Factor the quadratic expression by finding two numbers that multiply to

6

and add to

-7

.

(x - 1)(x - 6) = 0

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

4

Set each factor to zero to solve for

x

.

x = 1

and

x = 6

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

5

Subtract the smaller solution from the larger solution to find the positive difference.

6 - 1 = 5

The question asks for the positive difference between the two solutions.

Key Concept

Solving quadratic equations by expanding, rearranging into standard form, and factoring over the integers.

Question 1814Question →

A rental car company charges a flat fee of $30$ dollars plus $0.20$ dollars per mile driven. If a customer's total rental bill is $56$ dollars, how many miles did they drive?

Show answer & explanation

Answer: 130

Answer

130 miles

Subtracting the flat fee of

30

dollars from the total bill of

56

dollars leaves

26

dollars representing the cost of the miles driven. Dividing this remaining cost of

26

dollars by the rate of

0.20

dollars per mile yields a total of

130

miles driven.

Step-by-Step Solution

1

Set up a linear equation for the total cost

30 + 0.20m = 56

The total cost of

56

dollars is the sum of the flat fee of

30

dollars and the variable cost of

0.20

dollars per mile multiplied by the number of miles

m

.

2

Subtract the flat fee from both sides of the equation

0.20m = 26

Isolating the variable term shows that the total amount spent on the mileage portion of the trip is

26

dollars.

3

Divide by the rate per mile to find the total miles

m = 130

Dividing the mileage portion of the cost by the cost per mile gives the total distance driven.

Key Concept

Translating and Solving Linear Word Problems

Question 1815Question →

Determine the values of $x$ for which the given rational expression is undefined.

Fill in the blanks below

The rational expression

\frac{x + 5}{x^2 - 4x - 12}

is undefined for two real values of

x

. The smaller of these values is and the larger of these values is .

Show answer & explanation

Answer

The rational expression is undefined for

x = -2

and

x = 6

.

The expression is undefined when the denominator is zero. Solving

x^2 - 4x - 12 = 0

yields

(x-6)(x+2) = 0

, which gives

x = -2

and

x = 6

. The smaller value is

-2

and the larger value is

6

.

Step-by-Step Solution

1

Identify the condition that makes a rational expression undefined.

The denominator must equal zero:

x^2 - 4x - 12 = 0

Division by zero is undefined in the set of real numbers.

2

Factor the quadratic equation

x^2 - 4x - 12 = 0

.

(x - 6)(x + 2) = 0

We need to find two numbers that multiply to

-12

and add to

-4

. These numbers are

-6

and

2

.

3

Solve for

x

by setting each factor to zero.

x = 6

or

x = -2

By the zero product property, if

(x - 6)(x + 2) = 0

, then

x - 6 = 0

or

x + 2 = 0

.

4

Determine the smaller and larger values to place in the blanks.

Smaller value is

-2

and larger value is

6

.

Comparing the two numbers,

-2

is less than

6

.

Key Concept

A rational expression is undefined when its denominator equals zero.

Question 1816Question →

A local community theater group sells tickets for their upcoming play. A student ticket costs $8$ dollars, which is $4$ dollars less than half the price of an adult ticket. If $a$ represents the cost of an adult ticket in dollars, what is the value of $a$ ?

6

8

12

20

24

Show answer & explanation

Answer: $24$

Answer

The cost of the adult ticket is

24

dollars.

The correct answer is

24

. According to the problem, the student ticket price of

8

dollars is equal to

4

dollars less than half the price of the adult ticket (

a

). This relationship can be expressed algebraically as

8 = \frac{1}{2}a - 4

. Adding

4

to both sides of the equation results in

12 = \frac{1}{2}a

. Multiplying both sides by

2

yields

a = 24

.

Step-by-Step Solution

1

Translate the word problem into a linear equation.

8 = \frac{1}{2}a - 4

The student ticket price (

8

dollars) is equal to

4

dollars subtracted from half the price of the adult ticket (

a

).

2

Isolate the variable term by adding

4

to both sides of the equation.

12 = \frac{1}{2}a

To solve for

a

, we first eliminate the constant subtraction on the variable's side.

3

Solve for

a

by multiplying both sides of the equation by

2

.

a = 24

Multiplying by the reciprocal of

\frac{1}{2}

isolates the variable

a

.

Key Concept

Translating verbal statements into two-step linear equations and solving for the unknown variable.

Question 1817Question →

If the expression $6x^2 - 11x - 10$ is factored completely into the form $(ax + b)(cx - d)$ , where $a, b, c,$ and $d$ are positive integers, what is the value of $a + b + c + d$ ?

Show answer & explanation

Answer: 12

Answer

The value of the sum of the coefficients and constants

a + b + c + d

is

12

.

The factored form of

6x^2 - 11x - 10

is

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

. Matching this with

(ax + b)(cx - d)

where

a, b, c,

and

d

are positive integers results in

a = 3

,

b = 2

,

c = 2

, and

d = 5

. The sum of these values is

3 + 2 + 2 + 5 = 12

.

Step-by-Step Solution

1

Factor the quadratic trinomial

6x^2 - 11x - 10

using the grouping method.

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

Factoring splits the quadratic expression into its constituent linear binomial factors.

2

Equate the factored expression to the given form

(ax + b)(cx - d)

to find the values of

a, b, c,

and

d

.

a = 3

,

b = 2

,

c = 2

, and

d = 5

Since the variables represent positive integers, we match the positive constant term to

b

and the negative constant term to

-d

.

3

Sum the values of

a, b, c,

and

d

.

12

Calculating the final sum answers the target mathematical question.

Key Concept

Factoring quadratic polynomials with a leading coefficient greater than 1 using the grouping method.

Question 1818Question →

A scientist is mixing two solutions in a laboratory. Solution A contains $\frac{1}{4}$ active ingredient by volume, and Solution B contains $\frac{3}{4}$ active ingredient by volume. The scientist needs to mix these to create a $20$ -liter solution containing exactly $\frac{11}{20}$ active ingredient by volume.

Let

x

represent the volume, in liters, of Solution A used. The correct relationship is represented by the linear equation:

\frac{1}{4}x + \frac{3}{4}(20 - x) = 11

An assistant translates the problem description incorrectly, writing the equation as:

\frac{1}{4}x + \frac{3}{4}(x - 20) = 11

What is the absolute difference, in liters, between the value of $x$ obtained from the assistant's incorrect equation and the value of $x$ obtained from the correct equation?

8

30

18

26

44

Show answer & explanation

Answer: 18

Answer

The correct answer is 18, which is the absolute difference between the two solutions.

The correct answer is 18. Solving the correct equation

\frac{1}{4}x + \frac{3}{4}(20 - x) = 11

by multiplying by 4 gives

x + 60 - 3x = 44

, which simplifies to

-2x = -16

and yields

x = 8

. Solving the assistant's incorrect equation

\frac{1}{4}x + \frac{3}{4}(x - 20) = 11

in a similar manner gives

x + 3x - 60 = 44

, which simplifies to

4x = 104

and yields

x = 26

. The absolute difference between these two values is

|26 - 8| = 18

.

Step-by-Step Solution

1

Solve the correct equation for

x

.

Multiply the entire equation by 4 to clear the denominators:

x + 3(20 - x) = 44

. Distribute the 3:

x + 60 - 3x = 44

. Combine like terms:

-2x + 60 = 44

. Subtract 60 from both sides:

-2x = -16

. Divide by -2:

x = 8

.

This determines the correct volume of Solution A needed.

2

Solve the assistant's incorrect equation for

x

.

Multiply the entire equation by 4 to clear the denominators:

x + 3(x - 20) = 44

. Distribute the 3:

x + 3x - 60 = 44

. Combine like terms:

4x - 60 = 44

. Add 60 to both sides:

4x = 104

. Divide by 4:

x = 26

.

This determines the incorrect volume of Solution A obtained by the assistant.

3

Calculate the absolute difference between the two solutions.

|26 - 8| = 18

.

The question asks for the absolute difference between the two values of

x

.

Key Concept

Solving linear equations involving fractions, parentheses, and algebraic manipulation.

Question 1819Question →

Match each algebraic expression on the left with its fully simplified equivalent expression on the right for all real values of $m$ and $n$ .

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches are: (1)

m(m^2 - 3mn) - 2n(m^2 - n^2) - (m^3 - mn^2)

matches with

-5m^2n + mn^2 + 2n^3

; (2)

(m - n)^3 - m(m^2 - 3n^2) + n^3

matches with

-3m^2n + 6mn^2

; (3)

3mn(m - n) - 2m(n^2 - mn) - (m^2n - 5mn^2)

matches with

4m^2n

; and (4)

m^2(2m - n) - n(m^2 - n^2) - (m^3 + n^3)

matches with

m^3 - 2m^2n

.

Each expression is simplified by systematically expanding parenthetical groups and collecting like terms with identical variable powers.

Step-by-Step Solution

1

Simplify the expression

m(m^2 - 3mn) - 2n(m^2 - n^2) - (m^3 - mn^2)

-5m^2n + mn^2 + 2n^3

Distribute each term across the parenthetical expressions:

m^3 - 3m^2n - 2m^2n + 2n^3 - m^3 + mn^2

. Group the like terms:

(m^3 - m^3) + (-3m^2n - 2m^2n) + mn^2 + 2n^3

, which simplifies to

-5m^2n + mn^2 + 2n^3

.

2

Simplify the expression

(m - n)^3 - m(m^2 - 3n^2) + n^3

-3m^2n + 6mn^2

Use the binomial expansion formula to expand

(m - n)^3 = m^3 - 3m^2n + 3mn^2 - n^3

. Distribute the

-m

term to obtain

-m^3 + 3mn^2

. Sum all the expressions and combine like terms:

(m^3 - m^3) - 3m^2n + (3mn^2 + 3mn^2) + (-n^3 + n^3) = -3m^2n + 6mn^2

.

3

Simplify the expression

3mn(m - n) - 2m(n^2 - mn) - (m^2n - 5mn^2)

4m^2n

Expand the terms by distributing the outer coefficients to get

3m^2n - 3mn^2 - 2mn^2 + 2m^2n - m^2n + 5mn^2

. Grouping similar variables yields

(3 + 2 - 1)m^2n + (-3 - 2 + 5)mn^2 = 4m^2n + 0 = 4m^2n

.

4

Simplify the expression

m^2(2m - n) - n(m^2 - n^2) - (m^3 + n^3)

m^3 - 2m^2n

Expand the terms by distributing the multiplication:

2m^3 - m^2n - m^2n + n^3 - m^3 - n^3

. Grouping like terms yields

(2m^3 - m^3) + (-m^2n - m^2n) + (n^3 - n^3) = m^3 - 2m^2n

.

Key Concept

Simplifying multivariable expressions by distributing terms (including negative signs) and combining like terms.

Question 1820Question →

For all real values of $x$ and $y$ , the expression $5x^3y - 2x^2(3xy - y) - 4x^2y$ can be simplified. Which of the following is equivalent to this simplified expression?

-x^3y - 6x^2y

5x^3y - 8x^2y

-x^3y - 2x^2y

-3x^5y^2

-x^3y + 6x^2y

Show answer & explanation

Answer: $-x^3y - 2x^2y$

Answer

-x^3y - 2x^2y

Distributing

-2x^2

across the parentheses

(3xy - y)

yields

-6x^3y + 2x^2y

. Substituting this back gives

5x^3y - 6x^3y + 2x^2y - 4x^2y

. Combining the

x^3y

terms (

5x^3y - 6x^3y = -x^3y

) and the

x^2y

terms (

2x^2y - 4x^2y = -2x^2y

) yields the simplified expression

-x^3y - 2x^2y

.

Step-by-Step Solution

1

Distribute the term

-2x^2

to both terms inside the parentheses

(3xy - y)

.

-6x^3y + 2x^2y

Multiply

-2x^2

by

3xy

(yielding

-6x^3y

) and multiply

-2x^2

by

-y

(yielding

+2x^2y

).

2

Substitute the expanded terms back into the original expression.

5x^3y - 6x^3y + 2x^2y - 4x^2y

Replace

-2x^2(3xy - y)

with the expanded terms

-6x^3y + 2x^2y

.

3

Combine like terms by grouping the

x^3y

terms and the

x^2y

terms.

-x^3y - 2x^2y

Combine

5x^3y - 6x^3y = -x^3y

and

2x^2y - 4x^2y = -2x^2y

.

Key Concept

Simplifying expressions by distributing terms and combining like terms.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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Answer

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Alternative Method

Answer

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