All practice questions

Question 2041Question →

In the standard $(x,y)$ coordinate plane, a line has the equation $3x + 2y = 12$ . What is the $x$ -intercept of this line?

Show answer & explanation

Answer: 4

Answer

The

x

-intercept of the line is

4

.

Step-by-Step Solution

1

Substitute

y = 0

into the equation to find the point where the line intersects the

x

-axis.

3x + 2(0) = 12

By definition, the

x

-intercept occurs where the

y

-coordinate is equal to

0

.

2

Simplify the equation and solve for the variable

x

.

3x = 12 \Rightarrow x = 4

Simplifying

2(0)

to

0

leaves

3x = 12

. Dividing both sides of the equation by

3

isolates

x

, giving

x = 4

.

Key Concept

Determining the

x

-intercept of a linear equation by evaluating it at

y = 0

.

Estimated Time:45s

Question 2042Question →

Consider the system of equations below:

y = \sqrt{x + 37}

y = x - 5

If

(x, y)

is a real solution to this system, what is the value of

xy

?

-12

-4

6

24

84

Show answer & explanation

Answer: 84

Answer

84

The correct answer is 84 because solving the system yields the quadratic equation

x^2 - 11x - 12 = 0

, which has roots

x = 12

and

x = -1

. Substituting these back into the original equations shows that only

x = 12

is valid (giving

y = 7

), while

x = -1

is extraneous (since

\sqrt{36} = 6 \neq -6

). The product of the coordinates of the valid solution is

12 \times 7 = 84

.

Step-by-Step Solution

1

Set the two expressions for

y

equal to each other to solve for

x

.

\sqrt{x + 37} = x - 5

Since both equations are solved for

y

, their right-hand sides must be equal at the point of intersection.

2

Square both sides of the equation to eliminate the radical.

x + 37 = (x - 5)^2 \Rightarrow x + 37 = x^2 - 10x + 25

Squaring is the inverse operation of the square root. The binomial on the right must be expanded fully using

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

.

3

Rearrange the quadratic equation into standard form.

x^2 - 11x - 12 = 0

Subtracting

x

and

37

from both sides sets the quadratic equation to zero so it can be solved.

4

Factor the quadratic equation to find the algebraic solutions.

(x - 12)(x + 1) = 0 \Rightarrow x = 12 \text{ or } x = -1

Factoring the trinomial allows us to find the roots by setting each linear factor to zero.

5

Substitute each solution back into the original system to check for extraneous solutions.

For

x = 12

:

y = 12 - 5 = 7

and

y = \sqrt{12 + 37} = 7

(Valid). For

x = -1

:

y = -1 - 5 = -6

, but

y = \sqrt{-1 + 37} = 6 \neq -6

(Extraneous).

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original radical relationship because the principal square root must be non-negative.

6

Calculate the product

xy

of the coordinates of the valid solution.

xy = 12 \times 7 = 84

The question asks for the product of

x

and

y

for the real solution

(12, 7)

.

Key Concept

Solving systems containing radical equations and verifying for extraneous solutions.

Question 2043Question →

If $\frac{2}{3}(3x - 4) - \frac{1}{4}(x + 2) = \frac{22}{3}$ , what is the value of $3x + 2$ ?

Show answer & explanation

Answer: 20

Answer

The correct answer is 20, which is the value of the expression

3x + 2

when

x = 6

.

The value of the expression

3x + 2

is 20. Solving the linear equation

\frac{2}{3}(3x - 4) - \frac{1}{4}(x + 2) = \frac{22}{3}

yields

x = 6

. Substituting

x = 6

into

3x + 2

gives

3(6) + 2 = 20

.

Step-by-Step Solution

1

Multiply the entire equation by the least common multiple of the denominators (12).

8(3x - 4) - 3(x + 2) = 88

This clears the fractions to make solving the equation simpler.

2

Distribute the coefficients to remove parentheses.

24x - 32 - 3x - 6 = 88

This allows like terms to be grouped together.

3

Combine like terms on the left side of the equation.

21x - 38 = 88

Simplifies the equation to prepare for isolating the variable.

4

Add 38 to both sides of the equation.

21x = 126

Isolates the variable term on one side of the equation.

5

Divide both sides by 21 to solve for

x

.

x = 6

Finds the value of the variable

x

.

6

Substitute

x = 6

into the target expression

3x + 2

.

3(6) + 2 = 20

Calculates the final requested value.

Key Concept

Solving multi-step linear equations with fractions and evaluating algebraic expressions.

Question 2044Question →

For a certain real number $x$ , the equation $\frac{3}{4}(x - 3) - \frac{1}{3}(2x + 5) = -2$ is true. What is the value of $2x - 5$ ?

Show answer & explanation

Answer: 41

Answer

The value of the expression

2x - 5

is 41.

Solving the linear equation by multiplying both sides by the least common denominator of 12 yields the simplified equation

9(x - 3) - 4(2x + 5) = -24

. Expanding the terms gives

9x - 27 - 8x - 20 = -24

, which simplifies to

x - 47 = -24

. Adding 47 to both sides gives

x = 23

. Finally, evaluating the expression

2x - 5

for

x = 23

results in

2(23) - 5 = 41

.

Step-by-Step Solution

1

Multiply both sides of the equation by the least common denominator of 12 to eliminate fractions.

9(x - 3) - 4(2x + 5) = -24

Multiplying by the LCD clears all fractional coefficients, making the equation easier to solve.

2

Distribute the constants and expand the terms on the left side of the equation.

9x - 27 - 8x - 20 = -24

Applying the distributive property removes the parentheses.

3

Combine like terms on the left side of the equation.

x - 47 = -24

Simplifying the equation makes it easier to isolate the variable

x

.

4

Isolate the variable

x

by adding 47 to both sides of the equation.

x = 23

This determines the value of the unknown variable

x

.

5

Substitute

x = 23

into the target expression

2x - 5

.

41

The question asks for the value of the expression

2x - 5

, not just the value of

x

.

Key Concept

Solving multi-step linear equations with fractional coefficients by clearing the denominators and then evaluating algebraic expressions.

Alternative Method

Instead of multiplying by the LCD first, distribute the fractions directly:

\frac{3}{4}x - \frac{9}{4} - \frac{2}{3}x - \frac{5}{3} = -2

. Combine the

x

terms:

(\frac{3}{4} - \frac{2}{3})x = \frac{1}{12}x

. Combine the constant terms:

-\frac{9}{4} - \frac{5}{3} = -\frac{27}{12} - \frac{20}{12} = -\frac{47}{12}

. This gives the equation

\frac{1}{12}x - \frac{47}{12} = -2

. Add

\frac{47}{12}

to both sides:

\frac{1}{12}x = -2 + \frac{47}{12} = -\frac{24}{12} + \frac{47}{12} = \frac{23}{12}

. Multiply by 12 to get

x = 23

, then evaluate

2x - 5 = 41

.

Estimated Time:1m 30s

Question 2045Question →

Matrices $A$ and $B$ are defined as:

A = \begin{bmatrix} x & 4 \\ -3 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -2 \\ 5 & 3 \end{bmatrix}

Let $C$ represent the product matrix $AB$ , where:

C = \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{bmatrix}

If the element $c_{12}$ is equal to $2$ , what is the value of $x$ ?

Show answer & explanation

Answer: 5

Answer

5

To find the entry

c_{12}

in the first row and second column of the product matrix

C = AB

, we calculate the dot product of the first row of matrix

A

and the second column of matrix

B

. The first row of

A

is

[x, 4]

and the second column of

B

is

[-2, 3]^T

. Thus,

c_{12} = x(-2) + 4(3) = -2x + 12

. Setting this equal to the given value of

2

gives the equation

-2x + 12 = 2

. Subtracting

12

from both sides results in

-2x = -10

. Dividing by

-2

yields

x = 5

.

Step-by-Step Solution

1

Identify the formula for the element

c_{12}

in the product matrix

C = AB

.

c_{12} = -2x + 12

The element

c_{12}

is located in the first row and second column of the product matrix, so it is the product of the first row of

A

, which is

[x, 4]

, and the second column of

B

, which is

[-2, 3]^T

.

2

Set the expression for

c_{12}

equal to the given value of

2

and solve the linear equation for

x

.

x = 5

Setting

-2x + 12 = 2

leads to

-2x = -10

, and dividing both sides by

-2

yields

x = 5

.

Key Concept

Matrix Multiplication and Element-wise Operations

Question 2046Question →

If $x$ and $y$ are positive real numbers greater than $1$ such that $\log_y(x) + 6\log_x(y) = 5$ and $xy = 64$ , what is the sum of all possible values of $x$ ?

16 + 16\sqrt{2}

4 + 2\sqrt{2}

4\sqrt{2}

80

528

Show answer & explanation

Answer: $16 + 16\sqrt{2}$

Answer

16 + 16\sqrt{2}

Using the change of base property

\log_x(y) = \frac{1}{\log_y(x)}

, the equation can be written in terms of

u = \log_y(x)

as

u + \frac{6}{u} = 5

. Solving this quadratic equation gives

u = 2

or

u = 3

. If

\log_y(x) = 2

, then

x = y^2

. Substituting into

xy = 64

yields

y^3 = 64 \implies y = 4

, which gives

x = 16

. If

\log_y(x) = 3

, then

x = y^3

. Substituting into

xy = 64

yields

y^4 = 64 \implies y = 2\sqrt{2}

, which gives

x = 16\sqrt{2}

. The sum of these values is

16 + 16\sqrt{2}

.

Step-by-Step Solution

1

Apply the change of base formula to express the equation in terms of a single logarithmic base.

Since

\log_x(y) = \frac{1}{\log_y(x)}

, the equation

\log_y(x) + 6\log_x(y) = 5

becomes

\log_y(x) + \frac{6}{\log_y(x)} = 5

.

This allows us to solve for the log expression using substitution.

2

Substitute

u = \log_y(x)

and solve the resulting quadratic equation for

u

.

u + \frac{6}{u} = 5 \implies u^2 - 5u + 6 = 0 \implies (u-2)(u-3) = 0

, so

u = 2

or

u = 3

.

Solving the quadratic equation gives the possible relationships between

x

and

y

.

3

Analyze the first case where

u = 2

and solve for

x

.

If

\log_y(x) = 2

, then

x = y^2

. Substituting this into

xy = 64

gives

y^3 = 64

, which yields

y = 4

. Thus,

x = 4^2 = 16

.

This determines the first possible value of

x

.

4

Analyze the second case where

u = 3

and solve for

x

.

If

\log_y(x) = 3

, then

x = y^3

. Substituting this into

xy = 64

gives

y^4 = 64

, which yields

y = 64^{1/4} = (2^6)^{1/4} = 2^{3/2} = 2\sqrt{2}

. Thus,

x = (2\sqrt{2})^3 = 16\sqrt{2}

.

This determines the second possible value of

x

.

5

Sum the possible values of

x

.

16 + 16\sqrt{2}

The question asks for the sum of all possible values of

x

.

Key Concept

Solving systems of exponential and logarithmic equations using base-change properties and substitution

Alternative Method

Instead of using substitution directly, you can write both equations in terms of base 2 or natural logs: let

\log_y(x) = k

, which means

x = y^k

. We then have

k + 6/k = 5

giving

k = 2

or

k = 3

. This leads directly to

x = y^2

and

x = y^3

, which can then be substituted into the second equation.

Estimated Time:3m 0s

Question 2047Question →

A community theater group sells student tickets and adult tickets for their weekend plays. The number of tickets sold for Friday and Saturday night shows is represented by the matrix $T$ :

T = \begin{bmatrix} 80 & 120 \\ 100 & 150 \end{bmatrix}

where the first and second rows represent Friday and Saturday, respectively, and the first and second columns represent student and adult tickets, respectively. The price of each ticket is represented by the matrix $P$ :

P = \begin{bmatrix} 6 \\ 10 \end{bmatrix}

where the first row represents the price of a student ticket ( $6) and the second row represents the price of an adult ticket ($ 10). Which of the following matrices represents the total ticket sales revenue, in dollars, for Friday and Saturday, respectively?

\begin{bmatrix} 480 & 1,200 \\ 600 & 1,500 \end{bmatrix}

\begin{bmatrix} 2,100 \\ 1,680 \end{bmatrix}

\begin{bmatrix} 1,680 \\ 2,100 \end{bmatrix}

\begin{bmatrix} 1,580 \\ 2,100 \end{bmatrix}

\begin{bmatrix} 1,520 \\ 1,900 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 1,680 \\ 2,100 \end{bmatrix}$

Answer

The matrix with entries 1,680 and 2,100

The correct answer is the matrix with entries 1,680 and 2,100. This is found by multiplying the sales matrix by the price matrix. The revenue for Friday is calculated as

80 \times 6 + 120 \times 10 = 480 + 1,200 = 1,680

. The revenue for Saturday is

100 \times 6 + 150 \times 10 = 600 + 1,500 = 2,100

. Placing these in a column matrix matching the order of the days yields the correct matrix.

Step-by-Step Solution

1

Set up the matrix multiplication of the ticket sales matrix

T

and the ticket price matrix

P

.

TP = \begin{bmatrix} 80 & 120 \\ 100 & 150 \end{bmatrix} \begin{bmatrix} 6 \\ 10 \end{bmatrix}

To find the total revenue for Friday and Saturday, we must multiply the ticket sales by their respective prices.

2

Perform the multiplication by taking the dot product of each row of

T

with the column of

P

.

\begin{bmatrix} 80(6) + 120(10) \\ 100(6) + 150(10) \end{bmatrix} = \begin{bmatrix} 480 + 1200 \\ 600 + 1500 \end{bmatrix}

The matrix product of a

2 \times 2

matrix and a

2 \times 1

matrix results in a

2 \times 1

matrix where the entries are the sums of the products of corresponding elements.

3

Simplify the sums in the resulting matrix.

\begin{bmatrix} 1,680 \\ 2,100 \end{bmatrix}

Adding the products gives the final total revenues for Friday and Saturday, respectively.

Key Concept

Matrix Multiplication in Word Problems

Alternative Method

Instead of formal matrix multiplication, you can calculate the scalar totals for each day directly (Friday:

80 \times 6 + 120 \times 10 = 1,680

; Saturday:

100 \times 6 + 150 \times 10 = 2,100

) and match them with the corresponding rows of the resulting

2 \times 1

matrix.

Estimated Time:1m 30s

Question 2048Question →

The slope of a line is $2$ , and the line passes through the point $(3, 1)$ . What is the $y$ -intercept of this line in the standard $(x,y)$ coordinate plane?

-5

-4

1

2.5

5

Show answer & explanation

Answer: $-5$

Answer

The

y

-intercept of the line is

-5

.

The correct answer is found by substituting the slope

m = 2

and the point

(3, 1)

into the slope-intercept equation

y = mx + b

. This gives

1 = 2(3) + b

, which simplifies to

1 = 6 + b

. Subtracting

6

from both sides isolates

b

, giving

b = -5

.

Step-by-Step Solution

1

Recall the slope-intercept form of a linear equation.

y = mx + b

, where

m

is the slope and

b

is the

y

-intercept.

This formula allows us to solve for the

y

-intercept using the given slope and a point on the line.

2

Substitute the given slope

m = 2

and the coordinates of the point

(3, 1)

into the slope-intercept equation.

1 = 2(3) + b

The point

(3, 1)

lies on the line, so its coordinates must satisfy the equation of the line.

3

Simplify the equation and solve for

b

.

1 = 6 + b \Rightarrow b = -5

Subtracting

6

from both sides isolates

b

, which represents the

y

-intercept of the line.

Key Concept

Linear Equations and Graphing

Question 2049Question →

If $x$ is a positive real number such that $\log_2(x) + \log_4(x) + \log_{16}(x) = 7$ , what is the value of $x$ ?

Show answer & explanation

Answer: 16

Answer

The value of

x

is

16

.

By converting all logarithms to base 2, we write

\log_4(x)

as

\frac{1}{2}\log_2(x)

and

\log_{16}(x)

as

\frac{1}{4}\log_2(x)

. Summing these gives

\frac{7}{4}\log_2(x) = 7

, which simplifies to

\log_2(x) = 4

. Converting to exponential form, we find

x = 2^4 = 16

.

Step-by-Step Solution

1

Apply the change of base formula to express the logarithms with a common base of 2.

\log_4(x) = \frac{1}{2}\log_2(x)

and

\log_{16}(x) = \frac{1}{4}\log_2(x)

, giving the equation

\log_2(x) + \frac{1}{2}\log_2(x) + \frac{1}{4}\log_2(x) = 7

.

Rewriting the terms with a common base allows them to be combined algebraically.

2

Combine the coefficients of the like terms on the left-hand side.

\frac{7}{4}\log_2(x) = 7

.

The sum of the coefficients is

1 + \frac{1}{2} + \frac{1}{4} = \frac{4}{4} + \frac{2}{4} + \frac{1}{4} = \frac{7}{4}

.

3

Isolate the logarithm term by dividing or multiplying by the reciprocal coefficient.

\log_2(x) = 4

.

Multiplying both sides by

\frac{4}{7}

solves for the value of

\log_2(x)

.

4

Convert the equation from logarithmic form to its equivalent exponential form.

x = 2^4 = 16

.

By definition,

\log_b(a) = c

is equivalent to

b^c = a

.

Key Concept

Change of Base Formula for Logarithms

Question 2050Question →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = |2x - 6| - 16$ and $g(x) = (x - 3)^2 - 5$ . What is the sum of all real values of $x$ that satisfy the equation $f(g(x)) = 0$ ?

0

6

9

10

12

Show answer & explanation

Answer: 9

Answer

9

The correct answer is 9. First, substitute the inner function into the outer function to get

f(g(x)) = |2g(x) - 6| - 16 = 0

. This simplifies to the absolute value equation

|2g(x) - 6| = 16

. Splitting this into its two possible cases gives

2g(x) - 6 = 16 \implies g(x) = 11

, and

2g(x) - 6 = -16 \implies g(x) = -5

. Solving the first case,

(x-3)^2 - 5 = 11 \implies (x-3)^2 = 16

. Taking both square roots yields

x - 3 = 4 \implies x = 7

and

x - 3 = -4 \implies x = -1

. Solving the second case,

(x-3)^2 - 5 = -5 \implies (x-3)^2 = 0 \implies x - 3 = 0 \implies x = 3

. The sum of all unique real values of

x

that satisfy the original equation is

7 + (-1) + 3 = 9

.

Step-by-Step Solution

1

Set up the composite function equation using the outer function definition.

|2g(x) - 6| - 16 = 0

Substitute the expression of

g(x)

as the input variable into

f(x)

.

2

Isolate the absolute value term and split the equation into two cases.

|2g(x) - 6| = 16 \implies 2g(x) - 6 = 16

or

2g(x) - 6 = -16

An absolute value equation of the form

|A| = B

where

B > 0

has two solutions:

A = B

and

A = -B

.

3

Solve for the target values of

g(x)

in each case.

g(x) = 11

and

g(x) = -5

Isolate the function

g(x)

by adding 6 and dividing by 2 on both sides of each equation.

4

Solve the first case

g(x) = 11

for the variable

x

.

(x-3)^2 - 5 = 11 \implies (x-3)^2 = 16 \implies x - 3 = \pm 4

, yielding

x = 7

and

x = -1

Substitute the algebraic rule for

g(x)

, isolate the squared expression, and extract the square roots.

5

Solve the second case

g(x) = -5

for the variable

x

.

(x-3)^2 - 5 = -5 \implies (x-3)^2 = 0 \implies x - 3 = 0

, yielding

x = 3

Substitute the algebraic rule for

g(x)

, isolate the squared expression, and solve for

x

.

6

Sum all unique real solutions found.

7 + (-1) + 3 = 9

Add the distinct solutions

x = 7

,

x = -1

, and

x = 3

to find the total sum.

Key Concept

Evaluating and solving equations involving composite functions, absolute values, and quadratic expressions.

Question 2051Question →

If $\log_2(a) + \log_2(b) = 5$ and $\log_2(a^2) - \log_2(b) = 4$ for positive real numbers $a$ and $b$ , what is the value of $a + b$ ?

5

10

12

16

32

Show answer & explanation

Answer: 12

Answer

12

The correct answer is 12. We can simplify the system of equations by using the power rule of logarithms, which allows us to rewrite

\log_2(a^2)

as

2\log_2(a)

. Letting

x = \log_2(a)

and

y = \log_2(b)

gives us the system

x + y = 5

and

2x - y = 4

. Adding these equations gives

3x = 9

, which means

x = 3

. Substituting this back gives

y = 2

. Converting back from logarithmic form to exponential form, we get

a = 2^3 = 8

and

b = 2^2 = 4

. Therefore,

a + b = 8 + 4 = 12

.

Step-by-Step Solution

1

Use the power property of logarithms,

\log_b(x^k) = k \log_b(x)

, to rewrite the second equation.

The equation

\log_2(a^2) - \log_2(b) = 4

becomes

2\log_2(a) - \log_2(b) = 4

.

This simplifies the term

\log_2(a^2)

so that it is linear in terms of

\log_2(a)

.

2

Substitute variables to simplify solving the system of equations. Let

x = \log_2(a)

and

y = \log_2(b)

.

The system of equations becomes:
1)

x + y = 5

2)

2x - y = 4

Variable substitution reduces the logarithmic system to a standard system of linear equations.

3

Solve the linear system by adding the two equations together.

Adding the equations yields

(x + y) + (2x - y) = 5 + 4

, which simplifies to

3x = 9

, so

x = 3

. Substituting

x = 3

back into the first equation gives

3 + y = 5

, so

y = 2

.

Addition eliminates the variable

y

, allowing us to solve for

x

and then find

y

.

4

Convert the solved values of

x

and

y

back into

a

and

b

using the exponential form definition of a logarithm.

Since

x = \log_2(a) = 3

, we have

a = 2^3 = 8

. Since

y = \log_2(b) = 2

, we have

b = 2^2 = 4

.

The definition of a logarithm

\log_b(z) = w

is equivalent to

z = b^w

.

5

Calculate the sum of

a

and

b

.

a + b = 8 + 4 = 12

.

To find the final value requested by the question stem.

Key Concept

Solving systems of logarithmic equations using logarithm properties and exponential conversions

Question 2052Question →

A system of equations consists of the circle defined by $x^2 + (y - 4)^2 = 10$ and the line defined by $y = 2x - 1$ . The two points of intersection of this system and the origin, $(0, 0)$ , form the vertices of a triangle in the standard $(x, y)$ coordinate plane. What is the area of this triangle?

2

\sqrt{7}

1

4

5

Show answer & explanation

Answer: 1

Answer

The correct area of the triangle is 1.

The correct answer of 1 is found by substituting the linear equation into the circle's equation, solving the resulting quadratic equation to find the coordinates of the two intersection points,

(1, 1)

and

(3, 5)

, and then applying the coordinate area formula for a triangle with a vertex at the origin.

Step-by-Step Solution

1

Substitute the linear equation into the circle's equation to form a quadratic equation in terms of

x

.

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

This allows us to solve for the x-coordinates of the intersection points by replacing

y

with

2x - 1

in

(y - 4)^2

to get

(2x - 5)^2

.

2

Expand the squared binomial and simplify the quadratic equation.

5x^2 - 20x + 15 = 0

Expanding

(2x - 5)^2

yields

4x^2 - 20x + 25

. Adding

x^2

and subtracting

10

from both sides simplifies the equation to standard quadratic form.

3

Divide the quadratic equation by 5 and solve for

x

by factoring.

x = 1

and

x = 3

Dividing the equation by 5 yields

x^2 - 4x + 3 = 0

, which factors as

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

.

4

Find the corresponding y-coordinates by substituting the

x

-values back into the linear equation

y = 2x - 1

.

The intersection points are

(1, 1)

and

(3, 5)

.

For

x = 1

,

y = 2(1) - 1 = 1

. For

x = 3

,

y = 2(3) - 1 = 5

.

5

Calculate the area of the triangle with vertices at the origin

(0, 0)

and the two intersection points

(1, 1)

and

(3, 5)

.

Area = 1

Using the coordinate area formula for a triangle with one vertex at the origin,

\text{Area} = \frac{1}{2} |x_1 y_2 - x_2 y_1| = \frac{1}{2} |1(5) - 3(1)| = \frac{1}{2} |2| = 1

.

Key Concept

Solving a system of linear and non-linear equations and using coordinate geometry to find the area of a triangle.

Alternative Method

Instead of using the coordinate area formula, we can find the distance between the two intersection points to serve as the base of the triangle (

b = \sqrt{(3-1)^2 + (5-1)^2} = 2\sqrt{5}

), and find the perpendicular distance from the origin to the line

2x - y - 1 = 0

to serve as the height (

h = \frac{|-1|}{\sqrt{2^2 + (-1)^2}} = \frac{1}{\sqrt{5}}

). The area is then

\frac{1}{2} \times 2\sqrt{5} \times \frac{1}{\sqrt{5}} = 1

.

Estimated Time:2m 30s

Question 2053Question →

In the inequality $7 - 3|2z + 1| \ge -8$ , which of the following inequality expressions represents the complete solution set for $z$ ?

z \le -3

or

z \ge 2

z \le 2

-3 \le z \le 2

-1.5 \le z \le 0.5

z \ge 2

Show answer & explanation

Answer: $-3 \le z \le 2$

Answer

-3 \le z \le 2

Subtracting 7 from both sides of

7 - 3|2z + 1| \ge -8

yields

-3|2z + 1| \ge -15

. Dividing both sides by

-3

and reversing the inequality sign results in

|2z + 1| \le 5

. Writing this as the compound inequality

-5 \le 2z + 1 \le 5

, then subtracting 1 and dividing by 2 yields the correct solution interval

-3 \le z \le 2

.

Step-by-Step Solution

1

Subtract 7 from both sides to begin isolating the absolute value term.

-3|2z + 1| \ge -15

To solve an absolute value inequality, we must first isolate the absolute value term on one side.

2

Divide both sides by -3 and reverse the direction of the inequality sign.

|2z + 1| \le 5

Dividing an inequality by a negative number reverses the direction of the inequality sign.

3

Rewrite the absolute value inequality as a compound inequality.

-5 \le 2z + 1 \le 5

An inequality of the form

|u| \le a

(where

a \ge 0

) is equivalent to the compound inequality

-a \le u \le a

.

4

Subtract 1 from all three parts of the compound inequality.

-6 \le 2z \le 4

This is the first step to isolate the variable

z

in the middle.

5

Divide all three parts by 2.

-3 \le z \le 2

This fully isolates

z

, giving the final solution interval.

Key Concept

Solving multi-step absolute value inequalities, including isolating the absolute value expression, reversing the inequality sign when dividing by a negative number, and expressing the solution as a compound inequality.

Question 2054Question →

The length of a rectangle is $3$ inches greater than its width. If the area of the rectangle is $10$ square inches, what is the width of the rectangle, in inches?

1

2

3.5

5

7

Show answer & explanation

Answer: $2$

Answer

The width of the rectangle is

2

inches.

The correct answer is the option stating that the width is

2

inches. This is obtained by expressing the length as

w+3

and setting the area equation to

w(w+3) = 10

. Expanding and setting it to zero gives the quadratic equation

w^2 + 3w - 10 = 0

, which factors to

(w+5)(w-2) = 0

. Since width must be positive, the only valid solution is

2

.

Step-by-Step Solution

1

Define variables for the dimensions of the rectangle based on the problem statement.

Let the width of the rectangle be

w

inches. The length of the rectangle is

w + 3

inches.

The problem states that the length is

3

inches greater than the width.

2

Set up an equation representing the area of the rectangle.

The area is the product of width and length:

w(w + 3) = 10

.

The area of a rectangle is calculated as width times length, and the given area is

10

square inches.

3

Rewrite the equation in standard quadratic form

aw^2 + bw + c = 0

.

Distribute

w

to get

w^2 + 3w = 10

, then subtract

10

from both sides:

w^2 + 3w - 10 = 0

.

Standard quadratic form is required to solve the equation by factoring or using the quadratic formula.

4

Factor the quadratic equation.

(w + 5)(w - 2) = 0

.

Finding factors that multiply to

-10

and add to

3

allows us to find the roots.

5

Solve for

w

and apply real-world constraints.

w = -5

or

w = 2

. Since a physical width must be positive, discard

-5

, leaving

w = 2

.

A dimension of a geometric shape cannot be negative.

Key Concept

Solving quadratic equations derived from geometric word problems by factoring or using the quadratic formula.

Question 2055Question →

A line in the standard $(x,y)$ coordinate plane is defined by the equation $kx + 5y = 20$ , where $k$ is a constant. If the $x$ -intercept of this line is $6$ units greater than its $y$ -intercept, what is the value of $k$ ?

\frac{1}{2}

2

\frac{10}{3}

10

-10

Show answer & explanation

Answer: $2$

Answer

2

To find the

y

-intercept of the line

kx + 5y = 20

, set

x = 0

, which yields

5y = 20

, or

y = 4

. The problem states that the

x

-intercept is

6

units greater than the

y

-intercept, so the

x

-intercept is

4 + 6 = 10

. Substituting the point

(10, 0)

back into the original equation gives

k(10) + 5(0) = 20

. Solving for

k

gives

10k = 20

, which simplifies to

k = 2

.

Step-by-Step Solution

1

Find the

y

-intercept of the line.

The

y

-coordinate of the

y

-intercept is

4

.

To find the

y

-intercept, set

x = 0

in the equation

kx + 5y = 20

, which gives

5y = 20

, so

y = 4

.

2

Determine the

x

-intercept of the line based on the given relationship.

The

x

-coordinate of the

x

-intercept is

10

.

The problem states that the

x

-intercept is

6

units greater than the

y

-intercept. Since the

y

-intercept value is

4

, the

x

-intercept value is

4 + 6 = 10

.

3

Substitute the

x

-intercept coordinates into the equation to solve for

k

.

k = 2

The

x

-intercept is the point

(10, 0)

. Substituting

x = 10

and

y = 0

into the equation

kx + 5y = 20

gives

k(10) + 5(0) = 20

, which simplifies to

10k = 20

. Dividing both sides by

10

yields

k = 2

.

Key Concept

Finding intercepts of a linear equation in standard form and using coordinate substitution to solve for an unknown coefficient.

Question 2056Question →

Which of the following inequalities represents the complete set of real values of $x$ that satisfy the inequality $15 \leq 3 - 4x$ ?

x \geq -3

x \leq -15

x \geq -15

x \leq -3

x \geq 3

Show answer & explanation

Answer: $x \leq -3$

Answer

The inequality

x \leq -3

Subtracting 3 from both sides of the inequality

15 \leq 3 - 4x

yields

12 \leq -4x

. Dividing both sides by

-4

and reversing the inequality sign results in

-3 \geq x

, which is equivalent to

x \leq -3

.

Step-by-Step Solution

1

Subtract 3 from both sides of the inequality.

12 \leq -4x

This isolates the variable term on the right side of the inequality.

2

Divide both sides of the inequality by

-4

and reverse the inequality sign.

-3 \geq x

Reversing the inequality sign is required whenever both sides of an inequality are multiplied or divided by a negative number.

3

Rewrite the inequality to place the variable on the left side.

x \leq -3

Reorganizing the inequality with

x

on the left side is the standard format for representing the solution set.

Key Concept

Solving linear inequalities by isolating the variable and reversing the inequality sign when dividing by a negative number.

Question 2057Question →

A local library has two types of books: fiction and non-fiction. The number of fiction books is $120$ more than twice the number of non-fiction books. If the library has a total of $1\text{}860$ books, how many more fiction books than non-fiction books does the library have?

580

780

700

870

1\text{}280

Show answer & explanation

Answer: $700$

Answer

The library has

700

more fiction books than non-fiction books.

The correct answer is

700

because defining the non-fiction books as

n

gives

2n + 120

fiction books. Summing these to equal

1860

yields

3n + 120 = 1860

, which solves to

n = 580

. The number of fiction books is therefore

2(580) + 120 = 1280

. The difference between the two quantities is

1280 - 580 = 700

.

Step-by-Step Solution

1

Define variables for the two categories of books.

Let

n

be the number of non-fiction books. Then the number of fiction books,

f

, can be written as

f = 2n + 120

.

This translates the statement 'the number of fiction books is

120

more than twice the number of non-fiction books' into an algebraic expression.

2

Set up an equation representing the total number of books.

n + f = 1860 \implies n + (2n + 120) = 1860 \implies 3n + 120 = 1860

.

The sum of the fiction and non-fiction books must equal the total library inventory of

1860

books.

3

Solve the equation for the number of non-fiction books,

n

.

3n = 1740 \implies n = 580

.

Subtracting

120

from both sides and then dividing by

3

isolates the variable

n

.

4

Calculate the number of fiction books and find the difference.

Fiction books:

f = 2(580) + 120 = 1280

. Difference:

f - n = 1280 - 580 = 700

.

The question asks for the difference between the number of fiction and non-fiction books.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Question 2058Question →

A certain relationship between a number $x$ and other values is described as follows: one-fourth of the difference when $3x$ is subtracted from $2$ , decreased by one-third of the sum of $x$ and $3$ , is strictly greater than the difference when $x$ is subtracted from $1$ . Which of the following inequalities represents the complete set of all possible values of $x$ ?

x < 6

x > -18

x < -18

x > \frac{6}{25}

x < -\frac{2}{3}

Show answer & explanation

Answer: $x < -18$

Answer

x < -18

To find the correct solution set, translate the word problem into the inequality

\frac{2 - 3x}{4} - \frac{x + 3}{3} > 1 - x

. First, multiply all terms by the least common denominator,

12

, to clear the fractions, giving

3(2 - 3x) - 4(x + 3) > 12(1 - x)

. Distributing the constants yields

6 - 9x - 4x - 12 > 12 - 12x

. Combining like terms on the left side simplifies the expression to

-13x - 6 > 12 - 12x

. Adding

12x

to both sides results in

-x - 6 > 12

. Adding

6

to both sides gives

-x > 18

. Finally, dividing by

-1

and reversing the inequality sign results in the solution set

x < -18

.

Step-by-Step Solution

1

Translate the verbal description into an algebraic inequality.

\frac{2 - 3x}{4} - \frac{x + 3}{3} > 1 - x

To represent the relationships described in the word problem mathematically.

2

Multiply the entire inequality by the least common denominator, 12, to clear the fractions.

3(2 - 3x) - 4(x + 3) > 12(1 - x)

Clearing denominators makes it easier to combine like terms and solve for

x

.

3

Distribute the constants on both sides.

6 - 9x - 4x - 12 > 12 - 12x

To remove parentheses and separate individual terms for simplification.

4

Combine like terms on the left side of the inequality.

-13x - 6 > 12 - 12x

Simplifying the expressions on each side makes the inequality easier to solve.

5

Add

12x

and

6

to both sides of the inequality to isolate variables on one side.

-x > 18

To group variable terms on the left and constant terms on the right.

6

Divide both sides by

-1

and reverse the inequality sign.

x < -18

Dividing or multiplying an inequality by a negative number requires reversing the inequality sign to maintain a true statement.

Key Concept

Solving multi-step linear inequalities with rational terms and variable terms on both sides, including reversing the inequality sign when dividing by a negative number.

Alternative Method

Instead of clearing fractions first, you can distribute the division to each term (e.g.,

\frac{2}{4} - \frac{3x}{4} - \frac{x}{3} - \frac{3}{3} > 1 - x

), group the

x

terms on one side and constants on the other using fraction arithmetic, and then solve for

x

. However, clearing fractions with the LCD is generally faster and less prone to arithmetic errors.

Estimated Time:2m 0s

Question 2059Question →

Two infinite geometric series, Series A and Series B, are defined as follows:

* Series A has a first term of $2^x$ and a common ratio of $\frac{1}{2}$ .
* Series B has a first term of $2^{x+3}$ and a common ratio of $\frac{3}{4}$ .

If the sum of Series A and Series B is 136, what is the value of $x$ ?

0.5

1

2

3

4

Show answer & explanation

Answer: $2$

Answer

2

The correct answer is the value 2. Substituting

x = 2

into the sum of Series A gives

2^{2+1} = 8

. Substituting

x = 2

into the sum of Series B gives

2^{2+5} = 128

. The total sum is

8 + 128 = 136

, which matches the given sum.

Step-by-Step Solution

1

Express the sum of Series A, denoted as

S_A

, using the infinite geometric series sum formula

S = \frac{a_1}{1-r}

.

S_A = \frac{2^x}{1 - 1/2} = \frac{2^x}{1/2} = 2 \cdot 2^x = 2^{x+1}

Since the common ratio

r = 1/2

satisfies

|r| < 1

, the infinite sum exists and can be simplified using exponent properties.

2

Express the sum of Series B, denoted as

S_B

, using the infinite geometric series sum formula.

S_B = \frac{2^{x+3}}{1 - 3/4} = \frac{2^{x+3}}{1/4} = 4 \cdot 2^{x+3} = 2^2 \cdot 2^{x+3} = 2^{x+5}

Since the common ratio

r = 3/4

satisfies

|r| < 1

, the infinite sum exists and can be simplified using exponent rules.

3

Set up the equation for the sum of both series and solve for

x

.

2^{x+1} + 2^{x+5} = 136 \implies 2 \cdot 2^x + 32 \cdot 2^x = 136 \implies 34 \cdot 2^x = 136 \implies 2^x = 4 \implies x = 2

Factoring out

2^x

from the terms allows us to isolate the exponential expression and find the value of

x

.

Key Concept

Sum of an Infinite Geometric Series and Exponential Properties

Question 2060Question →

A line in the standard $(x,y)$ coordinate plane passes through the point $(4, 1)$ and has a negative slope $m$ . The line and the coordinate axes bound a region in the first quadrant. If the area of this region is $8$ square units, which of the following is the value of $m$ ?

-4

-\frac{9}{16}

-\frac{1}{4}

-\frac{3}{4}

-\frac{9}{8}

Show answer & explanation

Answer: $-\frac{1}{4}$

Answer

-\frac{1}{4}

The correct answer is

-\frac{1}{4}

. The line passes through

(4, 1)

with slope

m

. Using the point-slope formula, the equation of the line is

y - 1 = m(x - 4)

, which simplifies to

y = mx - 4m + 1

. The

y

-intercept is found by setting

x = 0

, giving

1 - 4m

. The

x

-intercept is found by setting

y = 0

, giving

\frac{4m-1}{m}

. The area of the right-triangular region in the first quadrant bounded by the axes is

\frac{1}{2} \cdot (\text{base}) \cdot (\text{height}) = 8

. Substituting the intercepts, we get

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

. Multiplying by

2m

(which is negative, so we maintain positive side lengths) yields

-(1-4m)^2 = 16m

, which simplifies to

16m^2 + 8m + 1 = 0

. Factoring the quadratic expression gives

(4m+1)^2 = 0

, which has the single solution

m = -\frac{1}{4}

.

Step-by-Step Solution

1

Write the equation of the line using the point-slope form with the point

(4, 1)

and slope

m

.

y - 1 = m(x - 4) \implies y = mx - 4m + 1

The point-slope formula

y - y_1 = m(x - x_1)

defines any line passing through a given point with a specific slope.

2

Find the

x

-intercept and

y

-intercept of the line.

y\text{-intercept} = 1 - 4m

(when

x=0

), and

x\text{-intercept} = \frac{4m-1}{m}

(when

y=0

)

The intercepts represent the vertices of the right triangle formed by the line and the coordinate axes.

3

Set up the area equation for the right triangle in the first quadrant, noting that since

m < 0

, both intercepts are positive.

Area = \frac{1}{2} \cdot \text{base} \cdot \text{height} \implies \frac{1}{2} \cdot \left(\frac{4m-1}{m}\right) \cdot (1-4m) = 8

The area of the region bounded by the axes and the line is a right triangle whose legs are the intercepts.

4

Solve the algebraic equation for

m

.

(4m-1)(1-4m) = 16m \implies 16m^2 + 8m + 1 = 0 \implies (4m+1)^2 = 0 \implies m = -\frac{1}{4}

Multiplying both sides by

2m

and expanding the terms leads to a quadratic equation in terms of

m

, which resolves to a single real root.

Key Concept

Using linear equation forms and coordinate geometry to represent boundary lines and compute bounded areas.

Alternative Method

Instead of using the point-slope form, write the line in intercept form:

\frac{x}{a} + \frac{y}{b} = 1

, where

a

and

b

are the

x

- and

y

-intercepts. The area of the region is

\frac{1}{2}ab = 8 \implies ab = 16 \implies b = \frac{16}{a}

. Substitute this back into the intercept form to get

\frac{x}{a} + \frac{y}{16/a} = 1 \implies \frac{x}{a} + \frac{ay}{16} = 1

. Since the line passes through

(4, 1)

, substitute these coordinates:

\frac{4}{a} + \frac{a(1)}{16} = 1

. Multiply the entire equation by

16a

to clear the denominators:

64 + a^2 = 16a \implies a^2 - 16a + 64 = 0 \implies (a-8)^2 = 0 \implies a = 8

. Since

a = 8

, the

y

-intercept is

b = \frac{16}{8} = 2

. Using the intercepts

(8, 0)

and

(0, 2)

, the slope is

m = \frac{2 - 0}{0 - 8} = -\frac{2}{8} = -\frac{1}{4}

.

Estimated Time:2m 30s

Answer

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