All practice questions

Question 2301Question →

A straight path on a coordinate map is modeled by the equation $2x - y = 8$ . A second path, which is perpendicular to the first path, passes through the point $(3, 5)$ . What is the $y$ -intercept of the second path?

-1

\frac{7}{2}

\frac{13}{2}

\frac{33}{2}

\frac{8}{3}

Show answer & explanation

Answer: $\frac{13}{2}$

Answer

The y-intercept of the second path is 13/2

The y-intercept is 13/2. First, the slope of the given line is found to be 2. The perpendicular line must have a slope of -1/2. Substituting this slope and the point (3, 5) into the slope-intercept form gives the y-intercept of 13/2.

Step-by-Step Solution

1

Find the slope of the first path by rewriting the equation in slope-intercept form (

y = mx + b

).

The equation

2x - y = 8

becomes

y = 2x - 8

, indicating the slope

m_1 = 2

.

Knowing the slope of the first path allows us to determine the slope of any perpendicular path.

2

Calculate the slope of the perpendicular path by taking the negative reciprocal of the first path's slope.

The perpendicular slope

m_2 = -\frac{1}{2}

.

Perpendicular lines have slopes that are negative reciprocals of one another.

3

Use the point-slope formula with the point

(3, 5)

and the perpendicular slope

-\frac{1}{2}

to write the equation of the second path.

The equation is

y - 5 = -\frac{1}{2}(x - 3)

, which simplifies to

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

.

This establishes the linear relationship of the second path containing the point (3, 5).

4

Find the y-intercept by setting

x = 0

and simplifying.

Setting

x = 0

gives

y = \frac{3}{2} + 5 = \frac{13}{2}

.

The y-intercept of a line is the y-coordinate where the line crosses the y-axis (when x = 0).

Key Concept

Perpendicular lines in a coordinate plane have slopes that are negative reciprocals of each other, and their equations can be solved using point-slope form.

Estimated Time:1m 30s

Question 2302Question →

In the standard $(x, y)$ coordinate plane, a line $L$ passes through the point $(2, 3)$ and has a positive $y$ -intercept $(0, b)$ . The area of the triangular region in the first quadrant bounded by line $L$ , the $x$ -axis, and the $y$ -axis is $12$ square units. What is the value of $b$ ?

3

4

6

8

12

Show answer & explanation

Answer: $6$

Answer

The correct value of

b

is

6

.

To find the value of

b

, we can use the intercept form of a linear equation:

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

, where

a

is the

x

-intercept and

b

is the

y

-intercept. The area of the right triangle formed by the axes and the line is given by

\frac{1}{2}ab = 12

, which means

ab = 24

, or

a = \frac{24}{b}

. Since the line passes through the point

(2, 3)

, we substitute these coordinates into the intercept equation to get

\frac{2}{a} + \frac{3}{b} = 1

. Substituting

a = \frac{24}{b}

into this equation gives

\frac{2b}{24} + \frac{3}{b} = 1

, which simplifies to

\frac{b}{12} + \frac{3}{b} = 1

. Multiplying the entire equation by

12b

to clear the denominators results in

b^2 + 36 = 12b

. Rearranging this quadratic equation gives

b^2 - 12b + 36 = 0

, which factors as

(b - 6)^2 = 0

. Solving for

b

yields

b = 6

.

Step-by-Step Solution

1

Express the area of the right triangle in the first quadrant in terms of the

x

-intercept

(a, 0)

and

y

-intercept

(0, b)

.

The area is

\frac{1}{2}ab = 12

, which simplifies to

ab = 24

, or

a = \frac{24}{b}

.

The triangular region is a right triangle with base

a

and height

b

along the coordinate axes.

2

Set up the equation of the line using the intercept form

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

and substitute the given point

(2, 3)

.

Substituting

x = 2

and

y = 3

gives the equation

\frac{2}{a} + \frac{3}{b} = 1

.

Since the line passes through the point

(2, 3)

, this point must satisfy the equation of the line.

3

Substitute

a = \frac{24}{b}

into the intercept equation and solve the resulting quadratic equation for

b

.

Substituting

a

gives

\frac{2}{\frac{24}{b}} + \frac{3}{b} = 1 \implies \frac{b}{12} + \frac{3}{b} = 1

. Multiplying by

12b

yields

b^2 + 36 = 12b \implies b^2 - 12b + 36 = 0 \implies (b - 6)^2 = 0 \implies b = 6

.

Solving the quadratic equation yields the value of the

y

-intercept

b

.

Key Concept

Linear equations and graphing using intercept form and triangle area relations

Alternative Method

An alternative approach is to use the slope formula. The line passes through

(a, 0)

,

(2, 3)

, and

(0, b)

. The slope between

(0, b)

and

(2, 3)

is

\frac{3 - b}{2}

, and the slope between

(0, b)

and

(a, 0)

is

-\frac{b}{a}

. Equating these gives

\frac{3 - b}{2} = -\frac{b}{a} \implies 3a - ab = -2b

. Since the area is

12

, we know

ab = 24

. Substituting

ab = 24

gives

3a - 24 = -2b \implies 3a + 2b = 24

. Since

a = \frac{24}{b}

, we get

3\left(\frac{24}{b}\right) + 2b = 24 \implies \frac{72}{b} + 2b = 24 \implies b^2 - 12b + 36 = 0 \implies b = 6

.

Estimated Time:1m 30s

Question 2303Question →

A closed cardboard box has a height of $x$ inches, a width of $3x - 2$ inches, and a length of $2x + 5$ inches. When the volume of the box, in cubic inches, is written as a polynomial in standard form, what is the coefficient of the $x^2$ term?

Show answer & explanation

Answer: 11

Answer

The coefficient of the

x^2

term is 11.

Expanding the volume expression

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

yields

6x^3 + 11x^2 - 10x

. The coefficient of the

x^2

term is the numerical value associated with

x^2

, which is 11.

Step-by-Step Solution

1

Set up the polynomial expression for the volume.

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

The volume of a rectangular prism is the product of its length, width, and height.

2

Multiply the binomials

(3x - 2)

and

(2x + 5)

by distributing terms.

(3x - 2)(2x + 5) = 6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10

To find the product of two binomials, multiply each term of the first binomial by each term of the second binomial and combine like terms.

3

Distribute the monomial

x

to each term in the simplified trinomial.

x(6x^2 + 11x - 10) = 6x^3 + 11x^2 - 10x

The height

x

must scale the entire base area polynomial.

4

Identify the coefficient of the quadratic term

x^2

.

11

The coefficient of a term is the numerical factor multiplied by the variable part.

Key Concept

Multiplying polynomials and identifying coefficients of specific terms in the resulting standard form polynomial.

Question 2304Question →

In the standard $(x,y)$ coordinate plane, a parallelogram has vertices $A(1, 2)$ , $B(6, 2)$ , and $C(8, 6)$ . If the fourth vertex, $D$ , has a $y$ -coordinate of $6$ and lies to the left of $C$ , what is the $x$ -coordinate of $D$ ?

-7

3

5

9

13

Show answer & explanation

Answer: $3$

Answer

The correct answer is

3

, which is the

x

-coordinate of vertex

D

.

The correct answer is the coordinate value

3

. Since vertices

A(1, 2)

and

B(6, 2)

share the same

y

-coordinate, the side

AB

is horizontal with a length of

6 - 1 = 5

units. The opposite side

CD

must also be horizontal and have the same length of

5

units. Since vertex

C

is at

(8, 6)

and vertex

D

lies to its left, the

x

-coordinate of

D

is found by subtracting

5

from the

x

-coordinate of

C

, which gives

8 - 5 = 3

.

Step-by-Step Solution

1

Identify the relationship between the vertices of the parallelogram.

Sides

AB

and

CD

are parallel and equal in length.

By definition, opposite sides of a parallelogram are equal in length and parallel.

2

Calculate the horizontal length of the bottom side

AB

.

Length of

AB = 6 - 1 = 5

units.

Since both

A(1,2)

and

B(6,2)

share a

y

-coordinate of

2

, the segment is horizontal, and its length is the difference in their

x

-coordinates.

3

Apply the horizontal translation to find the

x

-coordinate of

D

.

x = 8 - 5 = 3

.

Since

D

lies to the left of

C(8,6)

on the horizontal line

y=6

, we subtract the side length of

5

units from the

x

-coordinate of

C

to get the

x

-coordinate of

D

.

Key Concept

Using the properties of parallelograms and coordinates on a 2D plane to find a missing vertex.

Estimated Time:45s

Question 2305Question →

In the standard $(x, y)$ coordinate plane, a point is first dilated by a scale factor of $2$ with the center of dilation at $(1, 1)$ . The resulting intermediate point is then reflected across the line $y = -x$ to yield a final image at $(-5, -9)$ . What were the coordinates of the original point before these two transformations were applied?

(17, 9)

(4, 2)

(5, 3)

(-4, -2)

(-2, -4)

Show answer & explanation

Answer: $(5, 3)$

Answer

The original point before the transformations was

(5, 3)

.

To find the coordinates of the original point, we reverse the transformations in reverse order. First, we reverse the reflection across

y = -x

. Since reflecting

(x', y')

across

y = -x

results in

(-y', -x') = (-5, -9)

, we have

y' = 5

and

x' = 9

, which makes the intermediate point

(9, 5)

. Next, we reverse the dilation centered at

(1, 1)

with a scale factor of

2

. Under this dilation, the original point

(x, y)

maps to

(1 + 2(x - 1), 1 + 2(y - 1)) = (9, 5)

. Solving the equation

1 + 2(x - 1) = 9

gives

x = 5

, and solving

1 + 2(y - 1) = 5

gives

y = 3

. Thus, the original point is

(5, 3)

.

Step-by-Step Solution

1

Determine the intermediate point by reversing the reflection across the line

y = -x

.

The final image is at

(-5, -9)

. The rule for reflection across

y = -x

is

(x', y') \rightarrow (-y', -x')

. To reverse this, we apply the same rule to the final image:

x' = -(-9) = 9

and

y' = -(-5) = 5

. Thus, the intermediate point is

(9, 5)

.

We must work backward from the final result, reversing the second transformation first.

2

Set up the equation to reverse the dilation centered at

(1, 1)

with a scale factor of

2

.

The dilation formula for a point

(x, y)

centered at

(h, k)

with scale factor

c

is

(x', y') = (h + c(x - h), k + c(y - k))

. Substituting

h=1

,

k=1

,

c=2

,

x'=9

, and

y'=5

yields the equations:

9 = 1 + 2(x - 1)

and

5 = 1 + 2(y - 1)

.

The intermediate point is the result of dilating the original point, so we solve for the original coordinates.

3

Solve the coordinate equations to find the original values of

x

and

y

.

For

x

:

8 = 2(x - 1) \Rightarrow 4 = x - 1 \Rightarrow x = 5

. For

y

:

4 = 2(y - 1) \Rightarrow 2 = y - 1 \Rightarrow y = 3

. The original point is

(5, 3)

.

Solving these algebraic equations gives the horizontal and vertical coordinates of the pre-image.

Key Concept

Transformations in the Coordinate Plane

Question 2306Question →

A non-degenerate triangle has side lengths of $5$ , $12$ , and $x$ . A second non-degenerate triangle has side lengths of $x$ , $10$ , and $y$ . If $x$ and $y$ must be integers, and the perimeter of the second triangle is the minimum possible integer value, what is the sum of all possible values of $y$ ?

1

3

6

10

27

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. By applying the Triangle Inequality Theorem, the shared side length

x

of the first triangle must satisfy

12 - 5 < x < 12 + 5

, which simplifies to

7 < x < 17

. Since

x

must be an integer, its possible values are

\{8, 9, 10, 11, 12, 13, 14, 15, 16\}

. For the second triangle with side lengths

x

,

10

, and

y

, the Triangle Inequality Theorem requires

|x - 10| < y < x + 10

. To minimize the perimeter

P = x + 10 + y

, we minimize

x + y

. Checking the possible values of

x

, we find that when

x = 8

, the minimum integer value for

y

is

3

(giving

P = 21

); when

x = 9

, the minimum integer value for

y

is

2

(giving

P = 21

); and when

x = 10

, the minimum integer value for

y

is

1

(giving

P = 21

). For any

x \ge 11

, the minimum value of

y

is

x - 9

, resulting in a perimeter of at least

23

. Therefore, the minimum perimeter of the second triangle is

21

, which is achieved when

y

is

3

,

2

, or

1

. The sum of these values of

y

is

1 + 2 + 3 = 6

.

Step-by-Step Solution

1

Apply the Triangle Inequality Theorem to the first triangle to find the range of the shared side length

x

.

7 < x < 17

The sum of any two side lengths of a triangle must be strictly greater than the third side length. Thus,

12 - 5 < x < 12 + 5

.

2

List the possible integer values of

x

based on the range.

x \in \{8, 9, 10, 11, 12, 13, 14, 15, 16\}

The problem states that

x

must be an integer.

3

Apply the Triangle Inequality Theorem to the second triangle with sides

x

,

10

, and

y

to express the range of

y

in terms of

x

.

|x - 10| < y < x + 10

The third side length

y

must be strictly between the difference and the sum of the other two sides (

x

and

10

).

4

Determine the minimum integer value of the perimeter of the second triangle,

P = x + 10 + y

, by testing the possible values of

x

and finding the minimum integer

y

for each.

The minimum perimeter is

21

, achieved when

(x, y) = (8, 3)

,

(9, 2)

, or

(10, 1)

.

Minimizing the perimeter

P = x + y + 10

is equivalent to minimizing the sum

x + y

for integer values of

y > |x - 10|

.

5

Sum the possible integer values of

y

that yield the minimum perimeter.

3 + 2 + 1 = 6

We need to find the sum of all possible values of

y

that result in the minimum perimeter of

21

.

Key Concept

Triangle Inequality Theorem and Perimeter Optimization

Estimated Time:3m 0s

Question 2307Question →

Let the functions $f$ and $g$ be defined by $f(x) = |x - 4|$ and $g(x) = 2x + 1$ for all real numbers. For what values of $x$ does the composition $f(g(x)) = 5$ ?

x = -1

and

x = 4

x = 4

only

x = 2

and

x = 6

x = -2

and

x = 2

x = -4

and

x = 4

Show answer & explanation

Answer: $x = -1$ and $x = 4$

Answer

x = -1

and

x = 4

To find the values of

x

for which

f(g(x)) = 5

, we substitute the expression for

g(x)

into

f(x)

, yielding

|(2x + 1) - 4| = 5

, which simplifies to

|2x - 3| = 5

. Setting up the two possible cases for the absolute value gives

2x - 3 = 5

(which solves to

x = 4

) and

2x - 3 = -5

(which solves to

x = -1

). Both values of

x

satisfy the original composition equation.

Step-by-Step Solution

1

Substitute the inner function

g(x) = 2x + 1

into the outer function

f(x) = |x - 4|

to find the composite function

f(g(x))

.

f(g(x)) = |(2x + 1) - 4| = |2x - 3|

Function composition requires evaluating the outer function at the output of the inner function.

2

Set the composite expression equal to

5

.

|2x - 3| = 5

We are solving for the values of

x

where the output of the composite function is

5

.

3

Split the absolute value equation into its positive and negative cases and solve each linear equation for

x

.

Case 1:

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

. Case 2:

2x - 3 = -5 \implies 2x = -2 \implies x = -1

.

An absolute value equation

|u| = c

(with

c > 0

) is equivalent to

u = c

or

u = -c

.

Key Concept

Evaluating a composite function with an absolute value and solving the resulting equations by considering both positive and negative cases.

Estimated Time:1m 30s

Question 2308Question →

The measures of the interior angles of a triangle are in the ratio $2:3:7$ . If the measure of the largest angle is decreased by $15^\circ$ and the measure of the smallest angle is increased by $15^\circ$ , what is the ratio of the interior angles of the new triangle, ordered from smallest to largest?

1:2:3

1:3:8

1:1:2

2:3:6

5:6:13

Show answer & explanation

Answer: $1:1:2$

Answer

The ratio of the interior angles of the new triangle is

1:1:2

.

The correct answer is the ratio

1:1:2

. The sum of the interior angles in a triangle is always

180^\circ

. Given the ratio

2:3:7

, the sum of the parts is

12

, which means each part represents

180^\circ / 12 = 15^\circ

. The original angles are therefore

30^\circ

,

45^\circ

, and

105^\circ

. Increasing the smallest angle by

15^\circ

gives

45^\circ

, and decreasing the largest by

15^\circ

gives

90^\circ

. The new angle measures are

45^\circ

,

45^\circ

, and

90^\circ

, which simplifies to

1:1:2

.

Step-by-Step Solution

1

Set up the equation for the sum of the interior angles of a triangle.

2x + 3x + 7x = 180^\circ

The sum of the interior angles of any triangle is always

180^\circ

.

2

Solve for the value of

x

.

12x = 180^\circ \implies x = 15^\circ

Combining like terms simplifies the equation to find the value of one ratio unit.

3

Calculate the original measures of the three angles.

Smallest:

30^\circ

, Middle:

45^\circ

, Largest:

105^\circ

Multiply each part of the ratio by

x = 15^\circ

.

4

Apply the modifications to the smallest and largest angles.

New smallest:

30^\circ + 15^\circ = 45^\circ

; New largest:

105^\circ - 15^\circ = 90^\circ

; Middle:

45^\circ

(unchanged).

Perform the operations described in the problem statement.

5

Order the new angle measures from smallest to largest and simplify the ratio.

45^\circ : 45^\circ : 90^\circ \implies 1 : 1 : 2

Divide each term in the ratio by the greatest common divisor, which is

45

.

Key Concept

Angle sum theorem of a triangle and ratio partition applications

Alternative Method

Instead of calculating the actual angle values, note that the sum of the ratio parts is

2+3+7 = 12

, and the sum of the interior angles of a triangle is

180^\circ

. This means

1

ratio unit is equal to

180^\circ / 12 = 15^\circ

. Since the smallest angle is increased by

15^\circ

(exactly

1

ratio unit) and the largest is decreased by

15^\circ

(exactly

1

ratio unit), we can apply these modifications directly to the ratio terms. The new ratio terms are

2+1 = 3

,

3

(unchanged), and

7-1 = 6

. This gives a ratio of

3:3:6

, which simplifies to

1:1:2

.

Estimated Time:1m 30s

Question 2309Question →

In the standard $(x, y)$ coordinate plane, two circles are defined by the equations $x^2 + y^2 - 4x + 6y = 3$ and $x^2 + y^2 + 8x - 2y = 8$ . Is the statement that the line passing through the centers of these two circles has a slope of $-\frac{2}{3}$ true or false?

True

False

Show answer & explanation

Answer: True

Answer

true

The correct answer is true because completing the square for the given circle equations determines the centers to be

(2, -3)

and

(-4, 1)

. Calculating the slope between these two coordinates using the slope formula results in

-\frac{2}{3}

, which matches the statement.

Step-by-Step Solution

1

Complete the square for the first circle's equation to find its center.

The equation

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

is rewritten as

(x - 2)^2 + (y + 3)^2 = 16

, identifying the center of the first circle as

(2, -3)

.

Converting the circle's equation to the standard form

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

reveals the center coordinates

(h, k)

.

2

Complete the square for the second circle's equation to find its center.

The equation

x^2 + y^2 + 8x - 2y = 8

is rewritten as

(x + 4)^2 + (y - 1)^2 = 25

, identifying the center of the second circle as

(-4, 1)

.

Converting the second circle's equation to standard form provides the coordinates of the second point needed to determine the slope.

3

Apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

to the two center points.

Using the coordinates

(2, -3)

and

(-4, 1)

, we calculate

m = \frac{1 - (-3)}{-4 - 2} = \frac{4}{-6} = -\frac{2}{3}

.

This calculation determines the slope of the line passing through both centers to evaluate the statement.

Key Concept

Slope of a Line

Question 2310Question →

If $x$ is a real number that satisfies the equation $\frac{3}{5}(2x - 7) + 0.4 = 0.2(x + 3)$ , what is the value of $5x - 2$ ?

Show answer & explanation

Answer: 20

Answer

The value of the expression

5x - 2

is

20

.

First, convert the fraction

\frac{3}{5}

to the decimal

0.6

. The equation becomes

0.6(2x - 7) + 0.4 = 0.2(x + 3)

. Distribute on both sides to get

1.2x - 4.2 + 0.4 = 0.2x + 0.6

. Combine constant terms on the left side to get

1.2x - 3.8 = 0.2x + 0.6

. Subtract

0.2x

and add

3.8

to both sides to isolate the variable, resulting in

x = 4.4

. Finally, substitute

x = 4.4

into the expression

5x - 2

to get

5(4.4) - 2 = 22 - 2 = 20

.

Step-by-Step Solution

1

Convert the fraction and distribute the coefficients

1.2x - 4.2 + 0.4 = 0.2x + 0.6

To clear parentheses and align terms using decimals.

2

Combine constants on the left side

1.2x - 3.8 = 0.2x + 0.6

To simplify the left-hand side of the linear equation.

3

Isolate the variable

x

x = 4.4

To determine the value of the unknown variable.

4

Evaluate the target expression

20

To compute the final value of the expression

5x - 2

.

Key Concept

Solving linear equations with fractional and decimal coefficients

Question 2311Question →

A circle in the standard $(x,y)$ coordinate plane is defined by the equation $(x - 8)^2 + (y + 5)^2 = 121$ . What is the diameter of this circle?

Show answer & explanation

Answer: 22

Answer

The diameter of the circle is 22.

By comparing the given equation

(x - 8)^2 + (y + 5)^2 = 121

to the standard circle equation

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

, we find that

r^2 = 121

. Taking the square root of

121

gives a radius of

r = 11

. Since the diameter of a circle is twice the radius, the diameter is

2 \times 11 = 22

.

Step-by-Step Solution

1

Identify the relationship between the circle's equation and its radius squared.

r^2 = 121

The standard equation of a circle is

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

, meaning the constant on the right side of the equation represents the square of the radius.

2

Determine the radius of the circle.

r = 11

Taking the square root of

121

yields the radius of the circle, which must be a positive length.

3

Calculate the diameter of the circle.

d = 22

The diameter of a circle is defined as twice the length of its radius (

d = 2r

).

Key Concept

The standard equation of a circle is

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

, where

r

is the radius and the diameter is

2r

.

Question 2312Question →

A smartphone's battery is fully charged to $100\%$ . When the phone is in sleep mode, the battery drains at a constant rate of $1\%$ per hour. When the phone is in active use, the battery drains at a constant rate of $6\%$ per hour. Over a $20$ -hour period, the phone is either in sleep mode or in active use, and the battery level decreases to $45\%$ . For how many hours was the phone in active use during this period?

5

7

9

11

13

Show answer & explanation

Answer: 7

Answer

The phone was in active use for 7 hours.

To find the active hours, first calculate the total percentage of the battery consumed:

100\% - 45\% = 55\%

. Let

x

represent the number of hours in active use. The remaining time,

20 - x

, represents the hours in sleep mode. Set up the equation representing the total battery drain:

6x + 1(20 - x) = 55

. Simplifying this equation gives

5x + 20 = 55

, which reduces to

5x = 35

, resulting in

x = 7

hours.

Step-by-Step Solution

1

Calculate the total percentage of the battery that drained during the period.

100\% - 45\% = 55\%

To determine the exact battery percentage consumed by both active use and sleep mode combined.

2

Set up a system of equations using

x

for hours in active use and

y

for hours in sleep mode.

x + y = 20

and

6x + 1y = 55

To represent the constraints on the total duration (20 hours) and the total battery drain (55%).

3

Substitute

y = 20 - x

into the drain equation and solve for

x

.

6x + (20 - x) = 55 \implies 5x + 20 = 55 \implies 5x = 35 \implies x = 7

To find the number of hours the phone was actively used.

Key Concept

Translating and solving a system of linear equations from a real-world word problem.

Question 2313Question →

Which of the following expressions is equivalent to $\frac{a^{-2} - b^{-2}}{a^{-1} - b^{-1}}$ for all non-zero real numbers $a$ and $b$ where $a \neq b$ ?

a^2 - b^2

\frac{1}{a+b}

\frac{a+b}{ab}

\frac{1}{a-b}

a+b

Show answer & explanation

Answer: $\frac{a+b}{ab}$

Answer

The correct answer is

\frac{a+b}{ab}

because simplifying the original expression using the difference of squares or by rewriting negative exponents as fractions leads directly to this value.

By applying the difference of squares factorization to the numerator, the expression simplifies to

a^{-1} + b^{-1}

. Converting these to standard fractions and finding a common denominator yields

\frac{a+b}{ab}

.

Step-by-Step Solution

1

Factor the numerator as a difference of squares.

a^{-2} - b^{-2} = (a^{-1})^2 - (b^{-1})^2 = (a^{-1} - b^{-1})(a^{-1} + b^{-1})

Since the terms are squared with negative exponents, we can use the difference of squares identity

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

where

x = a^{-1}

and

y = b^{-1}

.

2

Substitute the factored numerator back into the expression and cancel the common factor.

\frac{(a^{-1} - b^{-1})(a^{-1} + b^{-1})}{a^{-1} - b^{-1}} = a^{-1} + b^{-1}

Since

a \neq b

, we know

a^{-1} - b^{-1} \neq 0

, allowing us to divide out the common factor in the numerator and denominator.

3

Rewrite the expression using positive exponents and find a common denominator.

a^{-1} + b^{-1} = \frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a+b}{ab}

A negative exponent

x^{-n}

represents the reciprocal

\frac{1}{x^n}

. Finding the common denominator

ab

allows us to combine the two fractions into a single expression.

Key Concept

Properties of negative exponents and difference of squares factorization

Alternative Method

Convert all terms to fractions with positive exponents first: write the numerator as

\frac{1}{a^2} - \frac{1}{b^2} = \frac{b^2 - a^2}{a^2b^2}

and the denominator as

\frac{1}{a} - \frac{1}{b} = \frac{b-a}{ab}

. Then, divide the two complex fractions by multiplying the numerator by the reciprocal of the denominator:

\frac{(b-a)(b+a)}{a^2b^2} \times \frac{ab}{b-a}

. After canceling the common factors

(b-a)

and

ab

, the expression simplifies directly to

\frac{a+b}{ab}

.

Estimated Time:1m 30s

Question 2314Question →

In the standard $(x, y)$ coordinate plane, a line passes through the points $(2, 5)$ and $(6, k)$ . If the $y$ -intercept of this line is $-1$ , what is the value of $k$ ?

Show answer & explanation

Answer: 17

Answer

The value of

k

is 17.

The correct answer is 17. The y-intercept of

-1

gives the point

(0, -1)

. The slope between

(0, -1)

and

(2, 5)

is

\frac{5 - (-1)}{2 - 0} = 3

. The equation of the line is

y = 3x - 1

. Substituting the point

(6, k)

into the equation gives

k = 3(6) - 1 = 17

.

Step-by-Step Solution

1

Identify the coordinates of the y-intercept.

The point is

(0, -1)

.

The y-intercept is the point where the line crosses the y-axis, which occurs when the x-coordinate is 0.

2

Calculate the slope (

m

) of the line using the points

(0, -1)

and

(2, 5)

.

m = 3

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

, we find

m = \frac{5 - (-1)}{2 - 0} = \frac{6}{2} = 3

.

3

Write the equation of the line.

y = 3x - 1

Using the slope-intercept form

y = mx + b

, where the slope

m = 3

and the y-intercept

b = -1

.

4

Substitute the point

(6, k)

into the line equation to solve for

k

.

k = 17

Since the point

(6, k)

lies on the line, substituting

x = 6

and

y = k

yields

k = 3(6) - 1 = 18 - 1 = 17

.

Key Concept

Linear Equations and Graphing

Alternative Method

Since the slope of a straight line is constant, the slope between

(0, -1)

and

(2, 5)

must equal the slope between

(2, 5)

and

(6, k)

. Setting the slopes equal:

\frac{5 - (-1)}{2 - 0} = \frac{k - 5}{6 - 2} \implies 3 = \frac{k - 5}{4} \implies 12 = k - 5 \implies k = 17

.

Estimated Time:1m 0s

Question 2315Question →

For all positive real numbers $w$ , the expression $\left(\frac{w^4 \cdot \sqrt{w}}{w^{-1.5}}\right)^{2/3}$ is equivalent to $w^k$ , where $k$ is a constant. What is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

Applying the exponent rules in sequence: first, rewrite the square root as a fractional exponent to get

w^{0.5}

. Next, multiply the terms in the numerator by adding their exponents:

4 + 0.5 = 4.5

. Then, divide the numerator by the denominator by subtracting the denominator's exponent from the numerator's exponent:

4.5 - (-1.5) = 6

. Finally, raise this result to the

2/3

power by multiplying the exponents:

6 \times (2/3) = 4

. This yields

w^4

, so the constant exponent is

4

.

Step-by-Step Solution

1

Convert the radical expression to an exponential expression.

\sqrt{w} = w^{0.5}

Converting all terms to base

w

with decimal or fractional exponents makes it easier to apply exponent properties.

2

Apply the product rule of exponents to the numerator.

w^4 \cdot w^{0.5} = w^{4 + 0.5} = w^{4.5}

When multiplying exponential terms with the same base, add their exponents:

w^a \cdot w^b = w^{a+b}

.

3

Apply the quotient rule of exponents to the fraction.

\frac{w^{4.5}}{w^{-1.5}} = w^{4.5 - (-1.5)} = w^6

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

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

.

4

Apply the power rule of exponents to the simplified term.

(w^6)^{2/3} = w^{6 \cdot \frac{2}{3}} = w^4

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

(w^a)^b = w^{a \cdot b}

.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 2316Question →

In the standard $(x, y)$ coordinate plane, line $L_1$ is perpendicular to the line represented by the equation $3x - 2y = 6$ . Line $L_2$ is parallel to $L_1$ and passes through the midpoint of the line segment with endpoints $(-1, 5)$ and $(3, -1)$ . If the equation of $L_2$ is written in the form $Ax + By = 8$ , where $A$ and $B$ are integers, what is the value of $A + B$ ?

Show answer & explanation

Answer: 5

Answer

The value of

A + B

is 5.

The given line

3x - 2y = 6

has a slope of

\frac{3}{2}

. A perpendicular line

L_1

has a slope of

-\frac{2}{3}

. A line

L_2

parallel to

L_1

also has a slope of

-\frac{2}{3}

. The midpoint of the segment with endpoints

(-1, 5)

and

(3, -1)

is calculated as

\left(\frac{-1+3}{2}, \frac{5-1}{2}\right) = (1, 2)

. Using the point-slope form with the slope

-\frac{2}{3}

and point

(1, 2)

yields

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

, which simplifies to

2x + 3y = 8

. Thus,

A = 2

,

B = 3

, and

A + B = 5

.

Step-by-Step Solution

1

Find the slope of the given line

3x - 2y = 6

.

Slope is

\frac{3}{2}

Convert the equation to slope-intercept form

y = mx + b

to find the slope.

2

Determine the slope of line

L_1

.

Slope is

-\frac{2}{3}

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Determine the slope of line

L_2

.

Slope is

-\frac{2}{3}

Parallel lines have the same slope.

4

Calculate the midpoint of the segment with endpoints

(-1, 5)

and

(3, -1)

.

Midpoint is

(1, 2)

Use the midpoint formula:

\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

.

5

Write the equation of line

L_2

and convert it to standard form.

Equation is

2x + 3y = 8

Use the point-slope form with slope

-\frac{2}{3}

and midpoint

(1, 2)

, then rearrange to match the target constant 8 on the right-hand side.

6

Identify

A

and

B

, and compute their sum.

A = 2

,

B = 3

, and

A + B = 5

Match the coefficients of

2x + 3y = 8

to

Ax + By = 8

and add them.

Key Concept

Determining equations of parallel and perpendicular lines on the coordinate plane using slopes and midpoints.

Question 2317Question →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $(x - 2)^2 + y^2 = 20$ , and a line is defined by the equation $y = 2x - 4$ . If the circle and the line intersect at two points, $(x_1, y_1)$ and $(x_2, y_2)$ , what is the value of $x_1 + x_2 + y_1 + y_2$ ?

Show answer & explanation

Answer: 4

Answer

The value of

x_1 + x_2 + y_1 + y_2

is 4.

Substituting the equation of the line into the circle's equation gives

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

. Since

(2x-4) = 2(x-2)

, we can write this as

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

, which simplifies to

5(x-2)^2 = 20

, and further to

(x-2)^2 = 4

. Solving for

x

yields

x - 2 = 2

or

x - 2 = -2

, so

x = 4

or

x = 0

. Substituting these values into the linear equation gives the points

(4, 4)

and

(0, -4)

. The sum of all coordinates is

4 + 0 + 4 + (-4) = 4

.

Step-by-Step Solution

1

Substitute the linear equation into the circle equation.

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

To find the coordinates of the intersection points by reducing the system to a single variable equation.

2

Simplify the equation using algebraic substitution.

5(x - 2)^2 = 20

Factoring 2 out of the term

(2x-4)

yields

2(x-2)

. Squaring it results in

4(x-2)^2

. Combining this with the first term

(x-2)^2

simplifies the expression.

3

Solve for the two possible x-coordinates.

x = 0

and

x = 4

Dividing both sides by 5 yields

(x-2)^2 = 4

, which means

x-2 = \pm 2

.

4

Substitute the x-coordinates back into the linear equation to find the corresponding y-coordinates.

For

x = 0

,

y = -4

, giving the point

(0, -4)

. For

x = 4

,

y = 4

, giving the point

(4, 4)

.

To determine the coordinates of both intersection points.

5

Calculate the sum of all coordinates:

x_1 + x_2 + y_1 + y_2

.

0 + 4 + (-4) + 4 = 4

To find the final value requested by the question.

Key Concept

Solving systems of linear and quadratic (circular) equations by substitution and factoring.

Question 2318Question →

In $\triangle ABC$ , the side lengths are $AB = 12$ , $BC = 15$ , and $AC = 18$ . A point $P$ lies strictly inside $\triangle ABC$ . If the lengths of the segments $BP$ and $CP$ are both integers, what is the maximum possible value of the sum of these two lengths?

26

27

28

29

30

Show answer & explanation

Answer: 28

Answer

28

The correct answer is 28. According to the properties of triangles, for any point

P

strictly inside

\triangle ABC

, the sum of the interior segments is strictly less than the sum of the other two sides:

BP + CP < AB + AC = 12 + 18 = 30

. Since

BP

and

CP

are integers, we check the boundary where

P

lies on the side

AC

at an integer distance

CP = z

from

C

. Applying Stewart's Theorem, the boundary length

BP = y_{bound}

is

\sqrt{z^2 - 22.5z + 225}

. For the largest possible integer value

z = 17

, the boundary value is approximately

11.47

. Because the point must lie strictly inside the triangle,

BP

must be strictly less than this boundary, so the maximum integer value for

BP

is 11. This yields a maximum sum of

11 + 17 = 28

. Lower integer values of

z

yield smaller maximum sums (for example, if

z = 16

, the boundary is exactly 11, so

BP

can be at most 10, giving a sum of 26).

Step-by-Step Solution

1

Apply the interior point triangle inequality theorem.

For any point

P

strictly inside

\triangle ABC

, the sum of the distances to two vertices is strictly less than the sum of the other two sides:

BP + CP < AB + AC

.

This establishes the theoretical upper bound for the sum of the two segment lengths.

2

Calculate the theoretical upper bound.

Since

AB = 12

and

AC = 18

, we have

BP + CP < 12 + 18 = 30

. Since

BP

and

CP

must be integers, the sum

BP + CP

can be at most 29.

This sets the initial integer limit before evaluating if it is geometrically possible.

3

Analyze the boundary conditions for integer lengths using Stewart's Theorem.

Let

CP = z

and

BP = y

, where

y

and

z

are integers. As

P

approaches the side

AC

, the boundary value

y_{bound}

represents the distance from

B

to a point on

AC

at distance

z

from

C

. Using Stewart's Theorem, this boundary satisfies:

y_{bound}^2 = z^2 - 22.5z + 225

. Since

P

is strictly inside the triangle,

y

must be strictly less than

y_{bound}

.

This provides the mathematical relationship determining whether a point is inside the triangle for any given integer length of one segment.

4

Test the maximum possible integer value for

z

to maximize the sum

y + z

.

Since

P

is strictly inside,

z

must be strictly less than

AC = 18

, so the maximum integer for

z

is 17. For

z = 17

, the boundary value is

y_{bound} = \sqrt{17^2 - 22.5(17) + 225} = \sqrt{131.5} \approx 11.47

. Since

y < y_{bound}

, the maximum integer value for

y

is 11. This yields a maximum sum of

11 + 17 = 28

.

This determines the actual maximum integer sum that can be geometrically realized within the triangle.

Key Concept

Triangle Inequality Theorem and Interior Point Properties

Question 2319Question →

In the standard $(x, y)$ coordinate plane, a rectangle has vertices at $(1, 2)$ , $(7, 2)$ , $(7, 10)$ , and $(1, 10)$ . What is the length of a diagonal of this rectangle?

Show answer & explanation

Answer: 10

Answer

The length of a diagonal of the rectangle is 10.

The width of the rectangle is

7 - 1 = 6

, and the height is

10 - 2 = 8

. The diagonal forms the hypotenuse of a right triangle with legs of 6 and 8. By the Pythagorean theorem, the length of the diagonal is

\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10

.

Step-by-Step Solution

1

Calculate the horizontal width of the rectangle.

Width =

7 - 1 = 6

The horizontal distance is found by subtracting the x-coordinates of the horizontal vertices.

2

Calculate the vertical height of the rectangle.

Height =

10 - 2 = 8

The vertical distance is found by subtracting the y-coordinates of the vertical vertices.

3

Apply the Pythagorean theorem to find the diagonal length.

Diagonal length =

\sqrt{6^2 + 8^2} = 10

The diagonal of a rectangle forms the hypotenuse of a right triangle with legs equal to the width and the height.

Key Concept

Using coordinate differences to find dimensions of a figure and using the Pythagorean theorem to find its diagonal length.

Question 2320Question →

A food truck selling gourmet grilled cheese sandwiches has a daily fixed operating cost of $120$ . Each sandwich costs $2.50$ to make and is sold for $6.50$ . What is the minimum number of sandwiches the food truck must sell in one day to make a net profit of at least $180$ ?

Show answer & explanation

Answer: 75

Answer

The food truck must sell a minimum of 75 sandwiches to make a net profit of at least $180.

Representing the number of sandwiches sold as

x

, the total revenue is

6.50x

and the total cost is

120 + 2.50x

. The net profit is the difference between revenue and cost:

6.50x - (120 + 2.50x)

, which simplifies to

4x - 120

. Setting up the inequality for a profit of at least

180

gives

4x - 120 \geq 180

. Solving for

x

gives

4x \geq 300

, which simplifies to

x \geq 75

. Therefore, the minimum number of sandwiches that must be sold is

75

.

Step-by-Step Solution

1

Define the variable and write expressions for revenue and cost.

Let

x

represent the number of sandwiches sold. Total Revenue =

6.50x

and Total Cost =

120 + 2.50x

.

Defining the variable and translating the verbal descriptions of revenue and cost into algebraic expressions is necessary to model the profit.

2

Formulate the net profit expression.

Net Profit = Total Revenue - Total Cost =

6.50x - (120 + 2.50x) = 4x - 120

.

Net profit is calculated by subtracting all fixed and variable costs from the total revenue.

3

Set up and solve the linear inequality.

4x - 120 \geq 180 \Rightarrow 4x \geq 300 \Rightarrow x \geq 75

.

To find the minimum number of sandwiches needed to reach a target profit of at least

180

, we solve the inequality

4x - 120 \geq 180

for

x

.

Key Concept

Translating a real-world scenario into a linear inequality and solving for the unknown variable.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Common Mistakes

Alternative Method

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Common Mistakes

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