All practice questions

Question 1901Question →

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 1902Question →

At a local sports club, the initiation fee is $15$ dollars less than three times the monthly membership fee, $m$ . If the initiation fee is $75$ dollars, what is the monthly membership fee, in dollars?

20

25

30

40

90

Show answer & explanation

Answer: 30

Answer

30

The statement '15 less than three times the monthly membership fee,

m

' translates to the algebraic expression

3m - 15

. Since the initiation fee is

75

dollars, we can set up the equation

3m - 15 = 75

. Adding

15

to both sides yields

3m = 90

. Dividing both sides by

3

gives

m = 30

. Therefore, the monthly membership fee is

30

dollars.

Step-by-Step Solution

1

Translate the verbal description into an algebraic equation.

The expression '15 less than three times the monthly membership fee,

m

' is written as

3m - 15

. Setting this equal to the initiation fee of

75

gives the equation

3m - 15 = 75

.

To solve a word problem, you must first convert the written relationship into a mathematical statement.

2

Isolate the variable term by adding

15

to both sides of the equation.

3m = 90

Performing the inverse operation of subtraction (addition) simplifies the equation to isolate the term with the variable.

3

Solve for

m

by dividing both sides of the equation by

3

.

m = 30

Performing the inverse operation of multiplication (division) isolates the variable

m

completely.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:45s

Question 1903Question →

For each quadratic equation on the left, solve for $x$ by factoring and match it to its correct solution set on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The equation

2x^2 + 5x = 3

matches the solution set

\{-3, \frac{1}{2}\}

; the equation

3x^2 - 10x = -8

matches the solution set

\{\frac{4}{3}, 2\}

; and the equation

x(x - 4) = 12

matches the solution set

\{-2, 6\}

.

Each of the quadratic equations can be solved by first rearranging the terms to set the equation equal to zero. After rewriting them in the standard form

ax^2 + bx + c = 0

, they can be factored into a product of linear binomials. Setting each factor equal to zero and solving for

x

yields the solutions. Specifically:

2x^2 + 5x = 3

simplifies to

2x^2 + 5x - 3 = 0

, which factors as

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

and gives the solution set

\{-3, \frac{1}{2}\}

.

3x^2 - 10x = -8

simplifies to

3x^2 - 10x + 8 = 0

, which factors as

(3x - 4)(x - 2) = 0

and gives the solution set

\{\frac{4}{3}, 2\}

.

x(x - 4) = 12

simplifies to

x^2 - 4x - 12 = 0

, which factors as

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

and gives the solution set

\{-2, 6\}

.

Step-by-Step Solution

1

Rearrange the equation

2x^2 + 5x = 3

into the standard form

ax^2 + bx + c = 0

.

2x^2 + 5x - 3 = 0

To apply factoring and the zero product property, the quadratic expression must equal zero.

2

Factor the trinomial

2x^2 + 5x - 3

.

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

Since the constant term is negative and the middle coefficient is positive, the factors must have opposite signs.

3

Apply the zero product property to find the solutions for the first equation.

x = \frac{1}{2}

and

x = -3

, yielding the set

\{-3, \frac{1}{2}\}

Setting the individual linear factors

2x - 1

and

x + 3

to zero gives the solutions.

4

Rearrange the equation

3x^2 - 10x = -8

into standard form.

3x^2 - 10x + 8 = 0

Add

8

to both sides to set the right side to zero.

5

Factor the trinomial

3x^2 - 10x + 8

.

(3x - 4)(x - 2) = 0

The constant term is positive and the middle coefficient is negative, meaning both constant terms in the binomial factors must be negative.

6

Solve for

x

using the zero product property.

x = \frac{4}{3}

and

x = 2

, yielding the set

\{\frac{4}{3}, 2\}

Setting

3x - 4 = 0

gives

x = \frac{4}{3}

, and setting

x - 2 = 0

gives

x = 2

.

7

Expand and rearrange the equation

x(x - 4) = 12

into standard form.

x^2 - 4x - 12 = 0

Distribute the

x

on the left side to get

x^2 - 4x

and subtract

12

from both sides to set the equation to zero.

8

Factor the trinomial

x^2 - 4x - 12

.

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

Find two numbers that multiply to

-12

and add to

-4

. Those numbers are

-6

and

+2

.

9

Solve for

x

using the zero product property.

x = 6

and

x = -2

, yielding the set

\{-2, 6\}

Setting

x - 6 = 0

gives

x = 6

, and setting

x + 2 = 0

gives

x = -2

.

Key Concept

Solving quadratic equations by rewriting them in standard form, factoring the trinomials, and using the zero product property.

Question 1904Question →

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 1905Question →

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 1906Question →

For all real values of $w$ that satisfy the inequality $4 - |3 - 2w| \leq -1$ , which of the following expressions represents the complete set of possible values of $w$ ?

w \leq -1

or

w \geq 4

-1 \leq w \leq 4

w \leq -4

or

w \geq 1

w \geq 4

w \leq -1

Show answer & explanation

Answer: $w \leq -1$ or $w \geq 4$

Answer

The complete set of possible values is represented by the inequality

w \leq -1

or

w \geq 4

.

The correct answer is the solution set representing

w \leq -1

or

w \geq 4

. Isolating the absolute value expression yields

|3 - 2w| \geq 5

. This splits into two cases:

3 - 2w \geq 5

(which solves to

w \leq -1

after dividing by

-2

and reversing the inequality sign) and

3 - 2w \leq -5

(which solves to

w \geq 4

after dividing by

-2

and reversing the inequality sign). Combining these two cases gives the union

w \leq -1

or

w \geq 4

.

Step-by-Step Solution

1

Isolate the absolute value expression on one side of the inequality.

-|3 - 2w| \leq -5

, which simplifies to

|3 - 2w| \geq 5

after multiplying by

-1

and reversing the inequality sign.

Before splitting an absolute value inequality, the absolute value term must be isolated.

2

Split the absolute value inequality

|3 - 2w| \geq 5

into two separate compound inequalities.

3 - 2w \geq 5

or

3 - 2w \leq -5

An absolute value inequality of the form

|u| \geq c

(where

c > 0

) is equivalent to

u \geq c

or

u \leq -c

.

3

Solve the first inequality:

3 - 2w \geq 5

.

Subtracting

3

from both sides gives

-2w \geq 2

. Dividing both sides by

-2

and reversing the inequality sign gives

w \leq -1

.

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

4

Solve the second inequality:

3 - 2w \leq -5

.

Subtracting

3

from both sides gives

-2w \leq -8

. Dividing both sides by

-2

and reversing the inequality sign gives

w \geq 4

.

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

5

Combine the individual solutions to find the total solution set.

w \leq -1

or

w \geq 4

The solution to a 'greater than or equal to' absolute value inequality is the union of the solutions of the two split cases.

Key Concept

Solving absolute value inequalities with negative variable coefficients by isolating the absolute value term, splitting into cases, and reversing inequality signs when multiplying/dividing by a negative number.

Estimated Time:2m 0s

Question 1907Question →

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 1908Question →

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

Question 1909Question →

The table below shows some values for the functions $f$ and $g$ .

$x$	$f(x)$	$g(x)$
$-1$	$2$	$3$
$0$	$1$	$-1$
$1$	$-3$	$2$
$2$	$0$	$1$
$3$	$-2$	$0$

What is the value of $g(f(2))$ ?

0

-3

1

-1

2

Show answer & explanation

Answer: $-1$

Answer

The correct answer is

-1

.

To find the value of

g(f(2))

, we begin with the inner function

f(2)

. Looking at the table, when

x = 2

, the value of

f(x)

is

0

. This means

f(2) = 0

. We then substitute this value into the outer function, giving us

g(0)

. Looking at the table again, when

x = 0

, the value of

g(x)

is

-1

. Therefore,

g(f(2)) = -1

.

Step-by-Step Solution

1

Find the value of the inner function

f(2)

from the table.

f(2) = 0

For the composed function

g(f(2))

, the inner function

f

must be evaluated first at the input

2

.

2

Substitute this result into the outer function to evaluate

g(0)

using the table.

g(0) = -1

The output of the inner function becomes the input for the outer function

g

.

Key Concept

Evaluating composite functions using a table of values

Estimated Time:45s

Question 1910Question →

If $a$ and $b$ are non-zero real numbers, which of the following expressions is equivalent to $\frac{a^4 b^{-2}}{(a^2 b^{-3})^2}$ ?

a^2 b^4

b^{-1}

b^{-8}

b^4

b^{-11}

Show answer & explanation

Answer: $b^4$

Answer

b^4

To simplify the expression, first apply the power of a product rule to the denominator:

(a^2 b^{-3})^2 = (a^2)^2 (b^{-3})^2 = a^4 b^{-6}

. Next, divide the numerator by the simplified denominator by subtracting the exponents of the corresponding bases:

\frac{a^4 b^{-2}}{a^4 b^{-6}} = a^{4-4} b^{-2 - (-6)} = a^0 b^4

. Since

a^0 = 1

for any non-zero real number

a

, the expression simplifies to

b^4

.

Step-by-Step Solution

1

Simplify the denominator using the power of a product and power of a power rules.

(a^2 b^{-3})^2 = (a^2)^2 \cdot (b^{-3})^2 = a^4 b^{-6}

When raising a product to a power, raise each factor to that power. When raising a power to a power, multiply the exponents.

2

Substitute the simplified denominator back into the original fraction.

\frac{a^4 b^{-2}}{a^4 b^{-6}}

To prepare the expression for division.

3

Divide the numerator by the denominator by subtracting the exponents of like bases.

a^{4-4} b^{-2 - (-6)} = a^0 b^4 = 1 \cdot b^4 = b^4

When dividing terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.

Key Concept

Properties of exponents, specifically the power of a product rule, power of a power rule, and quotient rule.

Question 1911Question →

For positive real values of $u$ and $v$ , the expression $\left( \frac{9u^{-1}v^2}{4u^3v^{-4}} \right)^{-1/2}$ can be simplified to which of the following?

\frac{2u^2v}{3}

\frac{2u^2}{3v^3}

\frac{3u^2}{2v^3}

\frac{2}{3uv^3}

\frac{4u^2}{9v^3}

Show answer & explanation

Answer: $\frac{2u^2}{3v^3}$

Answer

The expression is equivalent to

\frac{2u^2}{3v^3}

.

The correct answer is obtained by first simplifying the quotient inside the parenthesis to get

\frac{9}{4}u^{-4}v^6

. Then, raising each factor to the

-1/2

power yields

\left(\frac{9}{4}\right)^{-1/2} = \frac{2}{3}

,

(u^{-4})^{-1/2} = u^2

, and

(v^6)^{-1/2} = v^{-3}

. Combining these results and rewriting with positive exponents gives the simplified expression.

Step-by-Step Solution

1

Simplify the

u

terms inside the parenthesis using the quotient rule for exponents.

\frac{u^{-1}}{u^3} = u^{-1 - 3} = u^{-4}

The quotient rule states that

\frac{x^a}{x^b} = x^{a-b}

.

2

Simplify the

v

terms inside the parenthesis using the quotient rule for exponents.

\frac{v^2}{v^{-4}} = v^{2 - (-4)} = v^6

Subtracting a negative exponent is equivalent to adding its absolute value.

3

Apply the outer exponent of

-1/2

to the coefficient.

\left(\frac{9}{4}\right)^{-1/2} = \left(\frac{4}{9}\right)^{1/2} = \frac{2}{3}

A negative exponent represents taking the reciprocal of the base, and a fractional exponent of

1/2

represents the square root.

4

Apply the outer exponent of

-1/2

to the simplified variable terms using the power of a power rule.

(u^{-4})^{-1/2} = u^2

and

(v^6)^{-1/2} = v^{-3}

The power of a power rule states that

(x^a)^b = x^{ab}

.

5

Combine the simplified parts and rewrite the expression with positive exponents.

\frac{2}{3} u^2 v^{-3} = \frac{2u^2}{3v^3}

An expression with a negative exponent in the numerator can be moved to the denominator with a positive exponent.

Key Concept

Applying product, quotient, and power rules of exponents to algebraic expressions with negative and rational exponents.

Question 1912Question →

For what value of $k$ does the equation $\frac{1}{5}(kx - 3) - \frac{1}{3}(2x - 5) = 2$ have a solution of $x = 7$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

Substituting

x = 7

reduces the equation to

\frac{1}{5}(7k - 3) - 3 = 2

. Adding 3 to both sides yields

\frac{1}{5}(7k - 3) = 5

. Multiplying by 5 gives

7k - 3 = 25

. Adding 3 gives

7k = 28

, which results in

k = 4

.

Step-by-Step Solution

1

Substitute

x = 7

into the equation.

\frac{1}{5}(7k - 3) - \frac{1}{3}(2(7) - 5) = 2

Since

x = 7

is given as a solution, substituting it into the equation must make the equality true.

2

Evaluate and simplify the expression in the second term.

\frac{1}{3}(14 - 5) = \frac{1}{3}(9) = 3

Follow the order of operations by simplifying the expression inside the parentheses first.

3

Isolate the fractional term containing the variable

k

.

\frac{1}{5}(7k - 3) - 3 = 2 \implies \frac{1}{5}(7k - 3) = 5

Add 3 to both sides of the equation to eliminate the subtraction of 3.

4

Clear the fraction and solve the remaining linear equation for

k

.

7k - 3 = 25 \implies 7k = 28 \implies k = 4

Multiply both sides by 5 to eliminate the denominator, add 3 to isolate the term with

k

, and divide by 7.

Key Concept

Solving multi-step linear equations containing parameters and fractions.

Estimated Time:1m 30s

Question 1913Question →

If $m = -3$ , $n = \frac{1}{8}$ , and $p = -2$ , what is the value of the algebraic expression $m^{-2} - n^{-1/3} \cdot p^{-3}$ ? Express your answer as a simplified fraction.

Fill in the blanks below

The value of the expression is .

Show answer & explanation

Answer

The correct answer is 13/36.

Evaluating each term individually gives

m^{-2} = \frac{1}{9}

,

n^{-1/3} = 2

, and

p^{-3} = -\frac{1}{8}

. Substituting these values into the expression yields

\frac{1}{9} - 2 \cdot \left(-\frac{1}{8}\right)

. Following the order of operations, we first perform the multiplication:

2 \cdot \left(-\frac{1}{8}\right) = -\frac{1}{4}

. We then subtract this result from the first term:

\frac{1}{9} - \left(-\frac{1}{4}\right) = \frac{1}{9} + \frac{1}{4} = \frac{13}{36}

.

Step-by-Step Solution

1

Evaluate

m^{-2}

when

m = -3

.

m^{-2} = (-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}

A negative exponent represents the reciprocal of the base raised to the positive power, and a negative base raised to an even power yields a positive result.

2

Evaluate

n^{-1/3}

when

n = \frac{1}{8}

.

n^{-1/3} = \left(\frac{1}{8}\right)^{-1/3} = \left(8^{-1}\right)^{-1/3} = 8^{1/3} = 2

Apply the negative exponent rule to find the reciprocal of the fraction, then find the cube root of 8.

3

Evaluate

p^{-3}

when

p = -2

.

p^{-3} = (-2)^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8}

A negative exponent represents the reciprocal, and a negative base raised to an odd power yields a negative result.

4

Substitute the evaluated terms back into the original algebraic expression.

\frac{1}{9} - 2 \cdot \left(-\frac{1}{8}\right)

Replace each variable expression with its calculated numerical value.

5

Perform the multiplication before subtraction following the order of operations.

2 \cdot \left(-\frac{1}{8}\right) = -\frac{2}{8} = -\frac{1}{4}

The order of operations (PEMDAS/GEMS) dictates that multiplication must be performed before subtraction.

6

Subtract the product from the first term.

\frac{1}{9} - \left(-\frac{1}{4}\right) = \frac{1}{9} + \frac{1}{4} = \frac{4}{36} + \frac{9}{36} = \frac{13}{36}

Subtracting a negative value is equivalent to addition. Find a common denominator to add the fractions.

Key Concept

Evaluating algebraic expressions involving negative bases, negative exponents, fractional exponents, and the order of operations.

Question 1914Question →

A high school debate club consists of 15 members: 6 freshmen, 5 sophomores, and 4 juniors. If a committee of 3 members is to be selected at random from the club, what is the probability that the committee will contain at least 1 freshman and at least 1 sophomore?

\frac{24}{91}

\frac{247}{455}

\frac{51}{91}

\frac{251}{455}

\frac{6}{7}

Show answer & explanation

Answer: $\frac{51}{91}$

Answer

The correct probability is

\frac{51}{91}

.

To find the probability that a randomly chosen 3-member committee from a club of 15 members (6 freshmen, 5 sophomores, 4 juniors) has at least 1 freshman and at least 1 sophomore, we can find the total number of possible committees and subtract the number of unfavorable committees. The total number of committees is

\binom{15}{3} = 455

. The unfavorable committees are those with no freshmen (chosen from the 9 sophomores and juniors:

\binom{9}{3} = 84

) or no sophomores (chosen from the 10 freshmen and juniors:

\binom{10}{3} = 120

). Because these two groups overlap when only juniors are chosen (

\binom{4}{3} = 4

), the total number of unfavorable committees is

84 + 120 - 4 = 200

by the Principle of Inclusion-Exclusion. Thus, there are

455 - 200 = 255

favorable committees, yielding a probability of

\frac{255}{455} = \frac{51}{91}

.

Step-by-Step Solution

1

Calculate the total number of possible 3-member committees that can be formed from the 15 club members.

\binom{15}{3} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = 455

total committees.

To find the probability, we first need to determine the size of the entire sample space (the total number of possible outcomes).

2

Determine the number of committees that do not meet the requirement of containing at least one freshman and at least one sophomore. This occurs if a committee has no freshmen, no sophomores, or neither.

Let

A

be the event of choosing a committee with no freshmen (only sophomores and juniors):

\binom{9}{3} = 84

ways. Let

B

be the event of choosing a committee with no sophomores (only freshmen and juniors):

\binom{10}{3} = 120

ways. Let

A \cap B

be the event of choosing a committee with neither freshmen nor sophomores (only juniors):

\binom{4}{3} = 4

ways.

It is easier to count the complement (the unfavorable outcomes) and subtract it from the total.

3

Use the Principle of Inclusion-Exclusion to find the total number of unfavorable committees (no freshmen or no sophomores).

|A \cup B| = |A| + |B| - |A \cap B| = 84 + 120 - 4 = 200

unfavorable committees.

Since the events of having no freshmen and having no sophomores overlap when only juniors are selected, we must subtract the intersection to avoid double-counting.

4

Subtract the number of unfavorable committees from the total number of committees to find the number of favorable committees, and then compute the probability.

Favorable committees:

455 - 200 = 255

ways. Probability:

\frac{255}{455} = \frac{51}{91}

.

Subtracting the complement from the total gives the number of valid outcomes, and dividing this by the total outcomes yields the desired probability.

Key Concept

Using combinations and the Principle of Inclusion-Exclusion to calculate probabilities of compound events.

Estimated Time:3m 0s

Question 1915Question →

Pipe A can fill a water tank in $x$ hours, where $x > 0$ . Pipe B can fill the same water tank in $x + 3$ hours. If both pipes are opened at the same time, they can fill the tank together in exactly 2 hours. What is the value of $x$ ?

0.5

1

2

3

6

Show answer & explanation

Answer: 3

Answer

The value of

x

is 3.

The correct value is 3. The work rate equation for the two pipes is

\frac{1}{x} + \frac{1}{x+3} = \frac{1}{2}

. Combining the fractions on the left gives

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

. Cross-multiplying yields

4x + 6 = x^2 + 3x

, which simplifies to the quadratic equation

x^2 - x - 6 = 0

. Factoring this quadratic equation gives

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

, which has the solutions

x = 3

and

x = -2

. Since

x

represents a positive number of hours, the only valid solution is 3.

Step-by-Step Solution

1

Set up the work rate equation based on the individual and combined rates.

\frac{1}{x} + \frac{1}{x+3} = \frac{1}{2}

The sum of the individual rates (in tanks per hour) equals the combined rate of the two pipes.

2

Find a common denominator to combine the fractions on the left side of the equation.

\frac{(x+3) + x}{x(x+3)} = \frac{1}{2} \implies \frac{2x+3}{x^2+3x} = \frac{1}{2}

To add fractions, they must share a common denominator, which is

x(x+3)

.

3

Cross-multiply to eliminate the denominators and form a quadratic equation.

2(2x+3) = x^2+3x \implies 4x + 6 = x^2 + 3x \implies x^2 - x - 6 = 0

Cross-multiplication converts the rational equation into a polynomial equation.

4

Factor the quadratic equation to find the solutions for

x

.

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

or

x = -2

Since

x

represents time, it must be positive (

x > 0

), so the only valid solution is

x = 3

.

Key Concept

Solving rational equations in the context of work-rate problems

Question 1916Question →

A manufacturer of precision components produces cylindrical rods. The target diameter of the rods is $1.20\text{ centimeters}$ . A rod is classified as Grade A if its actual diameter, $d$ centimeters, satisfies the inequality $|3 - 2.5d| \leq 0.15$ . To be used in a specific high-stress assembly, the rod's diameter must also satisfy the tolerance inequality $|d - 1.22| < 0.04$ . Which of the following inequality expressions represents the complete set of all possible diameters, in centimeters, of rods that qualify as Grade A and are suitable for the assembly?

1.14 \le d \le 1.26

1.14 \le d < 1.26

1.18 < d < 1.26

1.18 < d \le 1.26

1.14 < d < 1.18

Show answer & explanation

Answer: $1.18 < d < 1.26$

Answer

The set of diameters satisfying both conditions is

1.18 < d < 1.26

.

To satisfy both conditions, a rod's diameter must meet the Grade A requirement of

1.14 \le d \le 1.26

and the assembly requirement of

1.18 < d < 1.26

. The intersection of these two intervals is the more restrictive range, which is

1.18 < d < 1.26

. This ensures both inequalities are simultaneously true.

Step-by-Step Solution

1

Solve the first inequality representing Grade A rods:

|3 - 2.5d| \le 0.15

.

1.14 \le d \le 1.26

Rewrite the absolute value inequality as a compound inequality:

-0.15 \le 3 - 2.5d \le 0.15

. Subtract

3

from all parts to get

-3.15 \le -2.5d \le -2.85

. Divide all parts by

-2.5

, reversing the direction of the inequality signs:

1.26 \ge d \ge 1.14

, which simplifies to

1.14 \le d \le 1.26

.

2

Solve the second inequality representing suitability for the assembly:

|d - 1.22| < 0.04

.

1.18 < d < 1.26

Rewrite the absolute value inequality as a compound inequality:

-0.04 < d - 1.22 < 0.04

. Add

1.22

to all parts to isolate

d

:

1.18 < d < 1.26

.

3

Find the intersection of the two solution sets:

[1.14, 1.26] \cap (1.18, 1.26)

.

1.18 < d < 1.26

For a rod to qualify for Grade A and be suitable for the assembly, its diameter must satisfy both conditions. The overlapping range is bounded below by the stricter lower bound of

1.18

(exclusive) and above by the stricter upper bound of

1.26

(exclusive).

Key Concept

Absolute Value Equations and Inequalities

Question 1917Question →

For the imaginary unit $i$ , where $i^2 = -1$ , and any integer $n$ , what is the value of the expression $\frac{i^{4n+3} (1 + i)^8}{(1 - i)^6}$ ?

-2

2

-i

-2i

2i

Show answer & explanation

Answer: -2

Answer

-2

To find the value of the expression, we simplify each part. First,

i^{4n+3}

simplifies to

-i

because

i^{4n} = 1

and

i^3 = -i

. Second,

(1+i)^8

is simplified by first squaring the base to get

(1+i)^2 = 2i

, and then raising the result to the fourth power:

(2i)^4 = 16

. Third,

(1-i)^6

is simplified by first squaring the base to get

(1-i)^2 = -2i

, and then cubing the result:

(-2i)^3 = -8i^3 = 8i

. Substituting these back into the expression yields

\frac{-i \cdot 16}{8i} = -2

. Thus, the expression simplifies to

-2

.

Step-by-Step Solution

1

Simplify the term

i^{4n+3}

using the properties of powers of

i

.

i^{4n+3} = -i

Since

i^4 = 1

, we can rewrite

i^{4n+3}

as

(i^4)^n \cdot i^3 = 1^n \cdot (-i) = -i

.

2

Simplify the numerator term

(1+i)^8

.

(1+i)^8 = 16

We can rewrite

(1+i)^8

as

((1+i)^2)^4

. Since

(1+i)^2 = 1 + 2i + i^2 = 2i

, we have

(2i)^4 = 2^4 \cdot i^4 = 16 \cdot 1 = 16

.

3

Simplify the denominator term

(1-i)^6

.

(1-i)^6 = 8i

We can rewrite

(1-i)^6

as

((1-i)^2)^3

. Since

(1-i)^2 = 1 - 2i + i^2 = -2i

, we have

(-2i)^3 = (-2)^3 \cdot i^3 = -8 \cdot (-i) = 8i

.

4

Substitute the simplified components back into the original expression and divide.

\frac{-i \cdot 16}{8i} = -2

Substituting the terms gives

\frac{-i \cdot 16}{8i}

. The common factor of

i

in the numerator and denominator cancels out, and dividing

-16

by

8

yields

-2

.

Key Concept

Simplifying complex expressions involving powers of the imaginary unit and powers of complex binomials.

Estimated Time:1m 30s

Question 1918Question →

An online retailer sells two types of subscription boxes: a Basic Box and a Premium Box. Last month, the retailer sold a total of $250$ boxes. The number of Basic Boxes sold was $10$ more than $3$ times the number of Premium Boxes sold. If the retailer made a total profit of $4,350$ dollars, and the profit from each Premium Box is $5$ dollars less than twice the profit from each Basic Box, what is the profit, in dollars, for a single Premium Box?

15

35

31

25

28

Show answer & explanation

Answer: 25

Answer

The profit for a single Premium Box is

25

dollars.

Solving the system of equations for the box quantities gives

190

Basic Boxes and

60

Premium Boxes. Setting up the profit relation

p = 2b - 5

and the total profit equation

190b + 60p = 4350

, we substitute

b = \frac{p+5}{2}

to get

155p = 3875

, which simplifies to

p = 25

dollars.

Step-by-Step Solution

1

Define variables for the quantities of boxes sold and set up a system of equations.

Let

B

be the number of Basic Boxes sold and

P

be the number of Premium Boxes sold. The given relationships are:

B + P = 250

and

B = 3P + 10

.

This translates the word problem statements about the number of boxes into solvable linear equations.

2

Solve for the quantities of each box sold.

Substitute

B = 3P + 10

into the first equation:

(3P + 10) + P = 250 \implies 4P + 10 = 250 \implies 4P = 240 \implies P = 60

. Then,

B = 3(60) + 10 = 190

.

Finding the exact number of each type of box sold is required to formulate the profit equation.

3

Define variables for the profit of each box and set up the profit equations.

Let

b

be the profit of a Basic Box and

p

be the profit of a Premium Box. The total profit equation is

190b + 60p = 4350

. The relationship between the profits is

p = 2b - 5

, which can be rearranged to

b = \frac{p+5}{2}

.

This sets up the system of equations representing the profit values.

4

Substitute and solve for the profit of a Premium Box (

p

).

Substitute

b = \frac{p+5}{2}

into the profit equation:

190\left(\frac{p+5}{2}\right) + 60p = 4350 \implies 95(p+5) + 60p = 4350 \implies 95p + 475 + 60p = 4350 \implies 155p = 3875 \implies p = 25

.

Solving this single-variable equation gives the final required value for the profit of a single Premium Box.

Key Concept

Translating and Solving Multi-Step Algebraic Word Problems

Question 1919Question →

Let matrix $A = \begin{pmatrix} 2 & 5 \\ -1 & 3 \end{pmatrix}$ and matrix $B = \begin{pmatrix} 4 & -2 \\ 3 & 1 \end{pmatrix}$ . If matrix $C = 2A + 3B$ , what is the value of the entry in the second row and first column of matrix $C$ ?

Show answer & explanation

Answer: 7

Answer

The entry in the second row and first column of matrix

C

is 7.

The entry in the second row, first column of matrix

C

, denoted as

c_{2,1}

, is calculated by multiplying the corresponding entry in matrix

A

(

a_{2,1} = -1

) by 2 and the corresponding entry in matrix

B

(

b_{2,1} = 3

) by 3, and then adding the results:

2(-1) + 3(3) = -2 + 9 = 7

.

Step-by-Step Solution

1

Locate the element in the second row, first column of both matrices

A

and

B

.

The element in the second row, first column of

A

(designated

a_{2,1}

) is

-1

, and the element in the second row, first column of

B

(designated

b_{2,1}

) is

3

.

To find a specific entry in the resulting matrix

C = 2A + 3B

, we perform the scalar multiplication and addition operations on the corresponding entries in the same position of matrices

A

and

B

.

2

Multiply the corresponding elements by their respective scalar factors.

For matrix

A

,

2 \times (-1) = -2

. For matrix

B

,

3 \times 3 = 9

.

Scalar multiplication requires multiplying each individual entry of a matrix by the given scalar factor.

3

Add the two resulting values to find the entry for matrix

C

.

-2 + 9 = 7

Matrix addition is performed by adding the corresponding elements of the matrices.

Key Concept

Matrix scalar multiplication and addition of corresponding elements

Estimated Time:45s

Question 1920Question →

The algebraic expression $\frac{(x^2)^a}{x^{-4}}$ is equivalent to $x^{10}$ for all non-zero real numbers $x$ . What is the value of $a$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

The correct value for

a

is

3

. Applying the power of a power rule to

(x^2)^a

yields

x^{2a}

. Then, applying the quotient rule to

\frac{x^{2a}}{x^{-4}}

yields

x^{2a - (-4)} = x^{2a+4}

. Equating the exponents gives

2a + 4 = 10

, which solves to

a = 3

.

Step-by-Step Solution

1

Apply the power of a power property to the numerator.

(x^2)^a = x^{2a}

When raising a power to another power, multiply the exponents:

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

.

2

Apply the quotient property of exponents to simplify the fraction.

\frac{x^{2a}}{x^{-4}} = x^{2a - (-4)} = x^{2a + 4}

When dividing exponential expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator:

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

.

3

Set the resulting exponent equal to the exponent of the equivalent expression and solve for

a

.

2a + 4 = 10 \implies 2a = 6 \implies a = 3

Since the bases are identical and the expressions are equivalent, their exponents must be equal.

Key Concept

Properties of Exponents in Algebraic Expressions

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Alternative Method

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Common Mistakes

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Common Mistakes

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