All practice questions

Question 2201Question →

A toy car moves along a straight track. Its position in meters, $s$ , is plotted against time in seconds, $t$ , on a standard coordinate plane. At $t = 2$ seconds, the car is at a position of $-3$ meters. From $t = 2$ to $t = 6$ seconds, the position changes at a constant rate of $m$ meters per second. From $t = 6$ to $t = 8$ seconds, the position changes at a constant rate of $2m + 3$ meters per second. If the average rate of change of the car's position over the entire interval from $t = 2$ to $t = 8$ seconds is $\frac{5}{3}$ meters per second, what is the constant rate of change, in meters per second, from $t = 6$ to $t = 8$ seconds?

1/2

12/5

29/9

4

19/4

Show answer & explanation

Answer: 4

Answer

The correct constant rate of change from

t = 6

to

t = 8

seconds is

4

meters per second.

The correct answer of

4

meters per second is found by setting up the displacement for each interval in terms of

m

. The displacement during the first interval is

4m

, and the displacement during the second interval is

2(2m + 3) = 4m + 6

. Adding these gives a total displacement of

8m + 6

over a total time of

6

seconds. Dividing total displacement by total time yields the average rate of change,

\frac{8m + 6}{6} = \frac{4m + 3}{3}

. Setting this equal to the given average rate of

\frac{5}{3}

yields

m = \frac{1}{2}

. Finally, substituting this back into the rate expression for the second interval,

2m + 3

, gives

2(1/2) + 3 = 4

.

Step-by-Step Solution

1

Express the displacement in each time interval using the rate of change (slope) formula:

\Delta s = \text{slope} \times \Delta t

.

For

t = 2

to

t = 6

:

\Delta s_1 = m \times (6 - 2) = 4m

. For

t = 6

to

t = 8

:

\Delta s_2 = (2m + 3) \times (8 - 6) = 2(2m + 3) = 4m + 6

.

The constant rate of change in a position-time graph is the slope of the line, which relates time intervals to position displacements.

2

Calculate the total displacement over the entire interval from

t = 2

to

t = 8

seconds.

Total displacement

\Delta s_{\text{total}} = \Delta s_1 + \Delta s_2 = 4m + (4m + 6) = 8m + 6

.

The total displacement is the sum of the individual displacements over consecutive sub-intervals.

3

Set up the average rate of change equation using the total displacement and the total time elapsed (

\Delta t_{\text{total}} = 8 - 2 = 6

seconds).

Average rate of change =

\frac{8m + 6}{6} = \frac{4m + 3}{3}

.

The average rate of change is the net displacement divided by the total time elapsed.

4

Equate the expression for the average rate of change to the given value of

\frac{5}{3}

and solve for

m

.

\frac{4m + 3}{3} = \frac{5}{3} \Rightarrow 4m + 3 = 5 \Rightarrow 4m = 2 \Rightarrow m = \frac{1}{2}

.

Solving this linear equation yields the value of the parameter

m

that satisfies the average rate condition.

5

Calculate the rate of change for the second interval from

t = 6

to

t = 8

seconds by substituting

m = \frac{1}{2}

into

2m + 3

.

Rate of change =

2\left(\frac{1}{2}\right) + 3 = 1 + 3 = 4

.

The question asks for the rate of change during the second interval, which is defined in terms of

m

as

2m + 3

.

Key Concept

Slope as a constant rate of change in a piecewise linear model

Estimated Time:3m 0s

Question 2202Question →

In the standard $(x, y)$ coordinate plane, a line $L_1$ passes through the points $(2, 5)$ and $(-1, -4)$ . A second line, $L_2$ , passes through the $y$ -intercept of $L_1$ and has a slope that is twice the slope of $L_1$ . What is the $x$ -intercept of $L_2$ ?

\frac{1}{6}

-\frac{1}{6}

\frac{1}{3}

-\frac{3}{2}

-\frac{13}{2}

Show answer & explanation

Answer: $\frac{1}{6}$

Answer

The

x

-intercept of

L_2

is

\frac{1}{6}

.

To find the

x

-intercept of

L_2

, the slope and

y

-intercept of

L_1

must first be found. The slope of

L_1

is

\frac{5 - (-4)}{2 - (-1)} = 3

. Using the point-slope form with the point

(2, 5)

, the equation of

L_1

is

y - 5 = 3(x - 2) \implies y = 3x - 1

, meaning its

y

-intercept is

(0, -1)

. The second line,

L_2

, has a slope of

2 \times 3 = 6

and passes through the same

y

-intercept

(0, -1)

, giving the equation

y = 6x - 1

. Setting

y = 0

to find the

x

-intercept yields

6x - 1 = 0 \implies x = \frac{1}{6}

.

Step-by-Step Solution

1

Calculate the slope of line

L_1

using the coordinates

(2, 5)

and

(-1, -4)

.

The slope of

L_1

is

m_1 = \frac{5 - (-4)}{2 - (-1)} = \frac{9}{3} = 3

.

The slope of a line containing

(x_1, y_1)

and

(x_2, y_2)

is defined as

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

.

2

Determine the equation and the

y

-intercept of line

L_1

.

The equation of

L_1

in point-slope form is

y - 5 = 3(x - 2) \implies y = 3x - 1

. The

y

-intercept of

L_1

is

(0, -1)

.

Writing the equation in slope-intercept form

y = mx + b

directly identifies the

y

-intercept at

(0, b)

.

3

Determine the equation of line

L_2

.

The slope of

L_2

is

2 \times 3 = 6

. Since

L_2

passes through the

y

-intercept of

L_1

at

(0, -1)

, its equation is

y = 6x - 1

.

Using the relationship for the doubled slope and the shared

y

-intercept, the slope-intercept form of the second line is determined.

4

Find the

x

-intercept of

L_2

by setting

y = 0

.

Setting

y = 0

in the equation

y = 6x - 1

gives

0 = 6x - 1 \implies 6x = 1 \implies x = \frac{1}{6}

.

The

x

-intercept occurs where the line crosses the

x

-axis, which corresponds to setting

y = 0

.

Key Concept

Determining linear equations from coordinates, finding slopes, and calculating coordinate intercepts.

Question 2203Question →

A line segment in the standard $(x, y)$ coordinate plane has endpoints at $(1, a)$ and $(5, a^2)$ . The perpendicular bisector of this segment is parallel to the line defined by the equation $x + 3y = 6$ . What is the sum of all possible values of the constant $a$ ?

Show answer & explanation

Answer: 1

Answer

The sum of all possible values of the constant

a

is 1.

To find the sum of all possible values of the constant

a

, we first find the slope of the line

x + 3y = 6

by writing it in slope-intercept form:

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

. Since the perpendicular bisector is parallel to this line, its slope is also

-\frac{1}{3}

. The line segment is perpendicular to its perpendicular bisector, so the slope of the line segment is the negative reciprocal of

-\frac{1}{3}

, which is

3

. Setting the slope of the segment

\frac{a^2 - a}{5 - 1}

equal to

3

gives the equation

\frac{a^2 - a}{4} = 3

, which simplifies to

a^2 - a - 12 = 0

. Solving this quadratic equation yields

(a - 4)(a + 3) = 0

, giving the values

a = 4

and

a = -3

. The sum of these possible values is

4 + (-3) = 1

.

Step-by-Step Solution

1

Find the slope of the given line

x + 3y = 6

.

The slope is

-\frac{1}{3}

.

Rewriting the equation in slope-intercept form (

y = mx + b

) gives

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

, showing the slope is

-\frac{1}{3}

.

2

Determine the slope of the line segment.

The slope is 3.

The line segment is perpendicular to its perpendicular bisector. Because the perpendicular bisector is parallel to the reference line, its slope is also

-\frac{1}{3}

. The line segment's slope is the negative reciprocal of

-\frac{1}{3}

, which is

3

.

3

Write the slope of the segment in terms of

a

and set it equal to 3.

\frac{a^2 - a}{4} = 3

Using the slope formula with endpoints

(1, a)

and

(5, a^2)

gives the expression

\frac{a^2 - a}{5 - 1}

.

4

Solve the quadratic equation for

a

.

a = 4

or

a = -3

Multiplying both sides by 4 yields

a^2 - a = 12

, which simplifies to the quadratic

a^2 - a - 12 = 0

. Factoring gives

(a - 4)(a + 3) = 0

.

5

Sum all possible values of

a

.

1

The sum of the values is

4 + (-3) = 1

.

Key Concept

Understanding that parallel lines have equal slopes, perpendicular lines have slopes that are negative reciprocals of each other, and applying the slope formula to solve for coordinate variables.

Question 2204Question →

A triangle has two sides of length $7$ centimeters and $10$ centimeters. If the length of the third side, in centimeters, must be an integer, what is the minimum possible length of the third side?

3

17

4

16

8

Show answer & explanation

Answer: 4

Answer

4 centimeters

The Triangle Inequality Theorem states that the length of the third side,

x

, must be strictly greater than the difference of the two known sides (

10 - 7 = 3

) and strictly less than their sum (

10 + 7 = 17

). Therefore,

3 < x < 17

. The smallest integer value that satisfies this inequality is 4.

Step-by-Step Solution

1

Identify the given side lengths of the triangle.

The two given side lengths are 7 centimeters and 10 centimeters.

These are the values needed to apply the Triangle Inequality Theorem.

2

Apply the Triangle Inequality Theorem to find the range of possible lengths for the third side,

x

.

10 - 7 < x < 10 + 7

, which simplifies to

3 < x < 17

.

The theorem states that the length of any side of a triangle must be strictly greater than the difference of the other two sides and strictly less than their sum.

3

Identify the minimum integer value within the valid range.

The smallest integer strictly greater than 3 is 4.

The question specifies that the third side length must be an integer and asks for the minimum possible value.

Key Concept

The Triangle Inequality Theorem states that for any triangle with sides

a

,

b

, and

c

, the length of the third side must satisfy

|a - b| < c < a + b

.

Estimated Time:1m 0s

Question 2205Question →

For all real values of $x$ and $y$ where the expression is defined, simplify the algebraic fraction:

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

Which expression is equivalent to this fraction?

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

\frac{(x+y)^2}{y^3}

\frac{(x+y)^2}{y}

(x+y)^2 y^2

\frac{(x+y)^2 + 2}{y+2}

Show answer & explanation

Answer: $\frac{(x+y)^2}{y}$

Answer

The expression

\frac{(x+y)^2}{y}

The correct expression is derived by first simplifying the denominator

(xy^{-1})^{-2}

to

x^{-2}y^2

. We then group and simplify like terms:

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

,

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

, and

\frac{y}{y^2} = \frac{1}{y}

. Multiplying these yields the simplified expression.

Step-by-Step Solution

1

Simplify the term

(x y^{-1})^{-2}

in the denominator

x^{-2} y^2

Apply the power of a product rule:

(ab)^n = a^n b^n

, which gives

x^{-2} (y^{-1})^{-2}

. Then, use the power of a power rule:

(y^{-1})^{-2} = y^{(-1) \cdot (-2)} = y^2

.

2

Rewrite the original fraction with the simplified denominator

\frac{(x + y)^3 x^{-2} y}{(x+y) x^{-2} y^2}

Substitute the simplified term back into the expression to align similar bases.

3

Simplify the fraction by dividing terms with like bases using the quotient rule

\frac{(x+y)^2}{y}

Divide each component:

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

,

\frac{x^{-2}}{x^{-2}} = x^{-2 - (-2)} = x^0 = 1

, and

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

. Multiplying these results gives

\frac{(x+y)^2}{y}

.

Key Concept

Properties of exponents in algebraic expressions, including power of a product, power of a power, and quotient rules.

Question 2206Question →

In the standard $(x, y)$ coordinate plane, the triangular region $R$ is bounded by the lines $y = 2x$ , $y = -x + 9$ , and the $x$ -axis. A vertical line $x = k$ (where $0 < k < 9$ ) divides region $R$ into two sub-regions. If the area of the sub-region to the right of the line $x = k$ is exactly $8$ , what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

5

The boundary lines intersect to form a triangle with vertices at

(0, 0)

,

(9, 0)

, and

(3, 6)

. Since the area to the right of

x = k

is

8

, and the total area of the triangle is

27

,

k

must be greater than

3

. The region to the right of

x = k

is a right triangle with a base of

9 - k

and a height of

9 - k

. Setting its area

\frac{1}{2}(9 - k)^2

equal to

8

yields

(9 - k)^2 = 16

, which gives

9 - k = 4

(since

k < 9

), and thus

k = 5

.

Step-by-Step Solution

1

Find the vertices of the triangular region

R

by finding the intersection points of the boundary lines

y = 2x

,

y = -x + 9

, and

y = 0

.

The vertices of the triangle are

A(0, 0)

,

B(9, 0)

, and

C(3, 6)

.

This establishes the boundaries and shape of the triangular region.

2

Determine which side of the peak

x = 3

the vertical line

x = k

must lie. Calculate the total area and the area of the left portion.

The total area of the triangle is

27

. The area to the left of the peak

x = 3

is

9

. Since the area of the region to the right of

x = k

is

8

, which is less than

18

,

k

must be greater than or equal to

3

.

This determines the geometric shape of the sub-region to the right of

x = k

as a right triangle.

3

Set up the area formula for the right-hand triangle with vertices

(k, 0)

,

(9, 0)

, and

(k, -k + 9)

, and set it equal to

8

.

The area is

\frac{1}{2}(9 - k)^2 = 8

, which simplifies to

(9 - k)^2 = 16

.

This relates the given area to the unknown coordinate

k

.

4

Solve the equation

(9 - k)^2 = 16

for

k

, keeping in mind that

k < 9

.

Taking the square root gives

9 - k = 4 \implies k = 5

.

This yields the final value of

k

.

Key Concept

Finding the area of a region defined by linear boundary equations and dividing it with a vertical line.

Alternative Method

Using similar triangles: The right-hand triangle formed by the line

x = 3

, the line

y = -x + 9

, and the

x

-axis has vertices at

(3,0)

,

(9,0)

, and

(3,6)

, with an area of

\frac{1}{2} \times 6 \times 6 = 18

. The smaller triangle to the right of

x = k

has an area of

8

and is similar to the larger triangle. The ratio of their areas is

\frac{8}{18} = \frac{4}{9}

, which means the ratio of their linear dimensions is

\sqrt{\frac{4}{9}} = \frac{2}{3}

. The base of the larger triangle is

9 - 3 = 6

, so the base of the smaller triangle must be

6 \times \frac{2}{3} = 4

. This gives

9 - k = 4 \implies k = 5

.

Estimated Time:2m 30s

Question 2207Question →

A circle in the standard $(x, y)$ coordinate plane has center $(3, -4)$ and passes through the point $(6, 0)$ . Which of the following is an equation of this circle?

x^2 + y^2 - 6x + 8y = 0

x^2 + y^2 + 6x - 8y = 0

x^2 + y^2 - 6x + 8y + 20 = 0

x^2 + y^2 - 18 = 0

x^2 + y^2 - 6x + 8y - 24 = 0

Show answer & explanation

Answer: $x^2 + y^2 - 6x + 8y = 0$

Answer

x^2 + y^2 - 6x + 8y = 0

The standard equation of a circle is

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

, where

(h, k)

is the center and

r

is the radius. Given the center is

(3, -4)

, the equation becomes

(x - 3)^2 + (y + 4)^2 = r^2

. Since the circle passes through

(6, 0)

, we substitute these coordinates to find

r^2

:

(6 - 3)^2 + (0 + 4)^2 = 3^2 + 4^2 = 9 + 16 = 25

. The equation is therefore

(x - 3)^2 + (y + 4)^2 = 25

. Expanding this equation gives

x^2 - 6x + 9 + y^2 + 8y + 16 = 25

. Combining constant terms yields

x^2 + y^2 - 6x + 8y + 25 = 25

. Subtracting 25 from both sides results in the general form equation

x^2 + y^2 - 6x + 8y = 0

.

Step-by-Step Solution

1

Calculate the radius squared of the circle using the distance formula between the center

(3, -4)

and the point on the circle

(6, 0)

.

r^2 = (6 - 3)^2 + (0 - (-4))^2 = 3^2 + 4^2 = 9 + 16 = 25

The distance between the center and any point on the circle is equal to the radius of the circle.

2

Write the standard form of the circle's equation using the center

(h, k) = (3, -4)

and the radius squared

r^2 = 25

.

(x - 3)^2 + (y + 4)^2 = 25

The standard equation of a circle is

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

.

3

Expand the squared binomials in the standard equation to convert it into general form.

x^2 - 6x + 9 + y^2 + 8y + 16 = 25

Expanding allows us to combine like terms and match the general form expressions in the options.

4

Simplify the expanded equation by combining constant terms and setting the equation to zero.

x^2 + y^2 - 6x + 8y + 25 = 25 \Rightarrow x^2 + y^2 - 6x + 8y = 0

Subtracting 25 from both sides yields the final simplified general form equation of the circle.

Key Concept

Deriving and expanding the equation of a circle from its center and a point.

Question 2208Question →

Many educators argue that high school start times should be delayed to improve student academic performance. To support this claim, a school district in Minnesota shifted its start time from $7:30$ a.m. to $8:30$ a.m. Following this change, the district reported that average student grades rose by $10\%$ and overall attendance increased. This example demonstrates a direct link between later school hours and improved student outcomes.

Based on the passage, the credibility of the argument for delaying high school start times is most strengthened by which of the following?

The broad assertion that later school hours are universally beneficial to student outcomes

The citation of similar start-time experiments conducted in neighboring states

The specific data showing a rise in grades and attendance after the start-time change

The assertion that student grades rose by

10\%

because students slept exactly one hour longer

Show answer & explanation

Answer: The specific data showing a rise in grades and attendance after the start-time change

Answer

The specific data showing a rise in grades and attendance after the start-time change

The correct answer is correct because the passage supports its claim with specific empirical evidence: after delaying the start time, average grades rose by

10\%

and attendance increased. This data provides concrete support for the argument.

Step-by-Step Solution

1

Identify the main argument in the passage.

The main argument is that delaying high school start times improves student academic performance.

To evaluate what strengthens the argument, we must first clearly understand the claim being made.

2

Locate the evidence provided in the passage to support this argument.

The author cites a school district in Minnesota that delayed its start time and subsequently saw a

10\%

rise in average grades and increased attendance.

Empirical results from a real-world application directly support and strengthen the credibility of the claim.

3

Match the identified evidence with the correct answer option.

The option referring to the specific data on grades and attendance matches the empirical evidence in the text.

This option accurately describes the factual support provided by the author to validate the argument.

Key Concept

Evaluating Argument Strength and Credibility

Question 2209Question →

Which of the following is the set of all real values of $p$ for which the inequality $|7 - 3p| \ge 14$ is true?

-\frac{7}{3} \le p \le 7

p \le -\frac{7}{3} \text{ or } p \ge 7

p \le -\frac{7}{3}

p \le -\frac{7}{2} \text{ or } p \ge \frac{7}{2}

p \ge -\frac{7}{3} \text{ or } p \le 7

Show answer & explanation

Answer: $p \le -\frac{7}{3} \text{ or } p \ge 7$

Answer

p \le -\frac{7}{3} \text{ or } p \ge 7

To solve the inequality

|7 - 3p| \ge 14

, we split it into two separate inequalities:

7 - 3p \ge 14

or

7 - 3p \le -14

. Solving the first inequality gives

-3p \ge 7

, which simplifies to

p \le -\frac{7}{3}

after dividing by

-3

and reversing the inequality sign. Solving the second inequality gives

-3p \le -21

, which simplifies to

p \ge 7

after dividing by

-3

and reversing the inequality sign. Combining these two cases yields the set of values

p \le -\frac{7}{3}

or

p \ge 7

.

Step-by-Step Solution

1

Set up the two compound inequalities representing the absolute value inequality

|7 - 3p| \ge 14

.

7 - 3p \ge 14

or

7 - 3p \le -14

An absolute value inequality of the form

|u| \ge c

(where

c > 0

) is equivalent to the union of

u \ge c

or

u \le -c

.

2

Solve the first inequality

7 - 3p \ge 14

.

p \le -\frac{7}{3}

Subtract

7

from both sides to get

-3p \ge 7

. Then, divide by

-3

and reverse the inequality sign because of division by a negative number.

3

Solve the second inequality

7 - 3p \le -14

.

p \ge 7

Subtract

7

from both sides to get

-3p \le -21

. Then, divide by

-3

and reverse the inequality sign because of division by a negative number.

4

Combine the two case solutions to state the final solution set.

p \le -\frac{7}{3} \text{ or } p \ge 7

The complete solution set is the union of the solutions from both individual cases.

Key Concept

Solving absolute value inequalities of the form

|ax + b| \ge c

Question 2210Question →

In the standard $(x, y)$ coordinate plane, line $j$ has the equation $y = -\frac{1}{4}x + 7$ . If line $k$ is perpendicular to line $j$ , what is the slope of line $k$ ?

Show answer & explanation

Answer: 4

Answer

The slope of line

k

is 4.

The slope of the perpendicular line is

4

because the negative reciprocal of the given slope,

-\frac{1}{4}

, is

4

.

Step-by-Step Solution

1

Identify the slope of line

j

.

The slope of line

j

is

-\frac{1}{4}

.

The equation of line

j

is given in slope-intercept form,

y = mx + b

, where the coefficient of

x

represents the slope.

2

Calculate the negative reciprocal of the slope of line

j

to find the slope of perpendicular line

k

.

The slope of line

k

is

4

.

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

-\frac{1}{4}

is

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

.

Key Concept

The slopes of perpendicular lines are negative reciprocals of each other.

Estimated Time:45s

Question 2211Question →

In the standard $(x,y)$ coordinate plane, point $P$ has coordinates $(3, 2)$ . If point $P$ is reflected across the $y$ -axis and then translated $4$ units up to create point $P'$ , what are the coordinates of $P'$ ?

(-3, 6)

(-3, 2)

(-3, -2)

(1, 2)

(3, 2)

Show answer & explanation

Answer: $(-3, 6)$

Answer

The coordinates

(-3, 6)

Reflecting the point

(3, 2)

across the

y

-axis changes the sign of the

x

-coordinate, producing

(-3, 2)

. Translating this resulting point

4

units up adds

4

to the

y

-coordinate, giving the final coordinates

(-3, 6)

.

Step-by-Step Solution

1

Reflect the point

P(3, 2)

across the

y

-axis.

The

x

-coordinate changes sign, while the

y

-coordinate remains unchanged, yielding

(-3, 2)

.

A reflection across the

y

-axis maps any point

(x, y)

to

(-x, y)

.

2

Translate the point

(-3, 2)

up by

4

units.

Add

4

to the

y

-coordinate:

2 + 4 = 6

, yielding

(-3, 6)

.

A translation of

d

units upward maps any point

(x, y)

to

(x, y + d)

.

Key Concept

Applying a sequence of transformations to a point in the coordinate plane by first reflecting it across an axis and then translating it.

Question 2212Question →

In $\triangle ABC$ , point $D$ lies on side $AC$ . If $AB = BD$ , the measure of $\angle A$ is $70^\circ$ , and the measure of $\angle C$ is $25^\circ$ , what is the measure, in degrees, of $\angle DBC$ ?

Show answer & explanation

Answer: 45

Answer

The measure of

\angle DBC

is 45 degrees.

In

\triangle ABD

, since

AB = BD

, the triangle is isosceles and the base angles opposite these sides are equal. Therefore, the measure of

\angle ADB

is equal to the measure of

\angle A

, which is

70^\circ

. Because points

A

,

D

, and

C

form a straight line, the angles

\angle ADB

and

\angle BDC

are supplementary, meaning the measure of

\angle BDC = 180^\circ - 70^\circ = 110^\circ

. The sum of the interior angles in

\triangle BCD

must be

180^\circ

, so the measure of

\angle DBC = 180^\circ - 110^\circ - 25^\circ = 45^\circ

.

Step-by-Step Solution

1

Determine the measure of

\angle ADB

using the properties of isosceles triangle

\triangle ABD

.

The measure of

\angle ADB

is

70^\circ

.

Since

AB = BD

, the angles opposite those sides,

\angle A

and

\angle ADB

, are equal.

2

Calculate the measure of the supplementary angle

\angle BDC

.

The measure of

\angle BDC

is

110^\circ

.

Points

A

,

D

, and

C

are collinear, meaning

\angle ADB

and

\angle BDC

form a linear pair and sum to

180^\circ

.

3

Determine the measure of

\angle DBC

using the triangle angle sum theorem on

\triangle BCD

.

The measure of

\angle DBC

is

45^\circ

.

The sum of the interior angles of

\triangle BCD

is

180^\circ

, so the measure of

\angle DBC

is

180^\circ - (110^\circ + 25^\circ) = 45^\circ

.

Key Concept

Using properties of isosceles triangles, linear pairs, and the triangle angle sum theorem to trace unknown angles.

Question 2213Question →

An architect is designing a triangular window with side lengths, in feet, represented by $x$ , $3x - 2$ , and $18$ . If the value of $x$ must be an integer, what is the sum of all possible values of $x$ ?

Show answer & explanation

Answer: 30

Answer

The sum of all possible integer values of

x

is 30.

By applying the Triangle Inequality Theorem, the sum of any two sides of a triangle must be strictly greater than the third side. This yields the inequalities

x + (3x - 2) > 18

(which simplifies to

x > 5

),

x + 18 > 3x - 2

(which simplifies to

x < 10

), and

(3x - 2) + 18 > x

(which simplifies to

x > -8

). The intersection of these inequalities is

5 < x < 10

. Since

x

must be an integer, the possible values are

6

,

7

,

8

, and

9

. The sum of these values is

6 + 7 + 8 + 9 = 30

.

Step-by-Step Solution

1

Set up the first triangle inequality constraint where the sum of the two variable sides is greater than the constant side.

x + (3x - 2) > 18 \implies 4x > 20 \implies x > 5

The Triangle Inequality Theorem states that the sum of any two sides of a triangle must be strictly greater than the third side.

2

Set up the second triangle inequality constraint where the sum of

x

and the constant side is greater than the other variable side.

x + 18 > 3x - 2 \implies 20 > 2x \implies x < 10

To satisfy the Triangle Inequality Theorem for all combinations of sides.

3

Set up the third triangle inequality constraint where the sum of the second variable side and the constant side is greater than the first variable side.

(3x - 2) + 18 > x \implies 2x > -16 \implies x > -8

To ensure the third side combination is mathematically valid.

4

Find the intersection of all three inequalities to determine the valid range for

x

.

5 < x < 10

The value of

x

must satisfy all three inequalities simultaneously.

5

Identify the integer values of

x

within the open interval

(5, 10)

and calculate their sum.

The integers are

6, 7, 8,

and

9

. Sum =

6 + 7 + 8 + 9 = 30

.

The problem specifies that

x

must be an integer, so we sum only the integers strictly between 5 and 10.

Key Concept

Triangle Inequality Theorem

Estimated Time:2m 0s

Question 2214Question →

A linear model predicts the height of a plant, $H$ , in inches, based on the number of weeks, $w$ , since it was planted. According to the model, the height of the plant after $2$ weeks is $3\frac{1}{2}$ inches, and its height after $4$ weeks is $4\frac{1}{4}$ inches. What was the initial height of the plant, in inches, when it was planted?

1\frac{5}{6}

\frac{3}{8}

2

2\frac{3}{4}

\frac{9}{16}

Show answer & explanation

Answer: $2\frac{3}{4}$

Answer

The initial height of the plant was

2\frac{3}{4}

inches.

The correct answer is

2\frac{3}{4}

inches. First, the two given data points are

(2, \frac{7}{2})

and

(4, \frac{17}{4})

. The rate of growth (slope) is calculated as

m = \frac{\frac{17}{4} - \frac{7}{2}}{4 - 2} = \frac{3}{8}

inches per week. Substituting this into the slope-intercept form equation

H = mw + b

at

w = 2

gives

\frac{7}{2} = \frac{3}{8}(2) + b

, which simplifies to

b = \frac{7}{2} - \frac{3}{4} = 2\frac{3}{4}

inches.

Step-by-Step Solution

1

Convert the mixed numbers for the plant heights into improper fractions.

The height at

2

weeks is

3\frac{1}{2} = \frac{7}{2}

inches, and the height at

4

weeks is

4\frac{1}{4} = \frac{17}{4}

inches.

Improper fractions are easier to use in slope and linear equation calculations.

2

Calculate the slope (weekly growth rate) of the linear function using the formula

m = \frac{H_2 - H_1}{w_2 - w_1}

.

m = \frac{\frac{17}{4} - \frac{7}{2}}{4 - 2} = \frac{\frac{3}{4}}{2} = \frac{3}{8}

The slope represents the constant rate of height increase per week.

3

Use the slope-intercept form

H = mw + b

and substitute one of the points to solve for the y-intercept,

b

.

Using the point

(2, \frac{7}{2})

:

\frac{7}{2} = \frac{3}{8}(2) + b \Rightarrow \frac{7}{2} = \frac{3}{4} + b \Rightarrow b = \frac{7}{2} - \frac{3}{4} = \frac{14}{4} - \frac{3}{4} = \frac{11}{4} = 2\frac{3}{4}

The y-intercept,

b

, represents the initial height of the plant at

w = 0

weeks.

Key Concept

Determining the equation of a line from two points and finding the y-intercept in a real-world context.

Question 2215Question →

An online bookstore offers two subscription options for downloading e-books. Store A charges a monthly membership fee of $6.00$ plus $1.25$ per e-book download. Store B has no membership fee and charges $2.75$ per e-book download. A customer downloads the same number of e-books from each store in a single month and spends a total of $46.00$ across both stores. How many e-books did the customer download from each store?

4

7

10

11.5

13

Show answer & explanation

Answer: 10

Answer

10 e-books from each store

The correct answer represents the number of downloads from each store that satisfies the total cost equation. Let

x

represent the number of books downloaded from each store. The cost at Store A is

6.00 + 1.25x

and the cost at Store B is

2.75x

. Since the total spent is

46.00

, we write the equation

(6.00 + 1.25x) + 2.75x = 46.00

. Combining like terms gives

6.00 + 4.00x = 46.00

. Subtracting

6.00

from both sides gives

4.00x = 40.00

, and dividing by

4.00

yields

x = 10

.

Step-by-Step Solution

1

Define the variable and write the expressions for the cost of each store.

Let

x

be the number of e-books downloaded from each store. Store A cost is

6.00 + 1.25x

, and Store B cost is

2.75x

.

Establishing algebraic expressions represents the verbal statements mathematically.

2

Set up the total cost equation by summing the costs of both stores and setting it equal to the total spent.

(6.00 + 1.25x) + 2.75x = 46.00

The customer spends a combined total of

46.00

across both bookstores.

3

Combine like terms and solve for the variable

x

.

6.00 + 4.00x = 46.00 \implies 4.00x = 40.00 \implies x = 10

Isolating the variable determines the number of e-books downloaded from each store.

Key Concept

Solving linear equations in one variable derived from real-world scenarios.

Estimated Time:1m 30s

Question 2216Question →

In the standard $(x, y)$ coordinate plane, line $L_1$ is defined by the equation $3x - ky = 8$ , where $k$ is a non-zero constant. Line $L_2$ is perpendicular to $L_1$ and passes through the points $(k, 2)$ and $(-1, k+5)$ . What is the sum of all possible real values of $k$ ?

-9

-6

-2

0

2

Show answer & explanation

Answer: $2$

Answer

The sum of all possible real values of

k

is

2

.

The slope of line

L_1

is found by converting its equation to slope-intercept form, yielding

m_1 = \frac{3}{k}

. Since line

L_2

is perpendicular, its slope must be the negative reciprocal,

m_2 = -\frac{k}{3}

. Using the slope formula with the given points on line

L_2

, we also have

m_2 = \frac{(k+5) - 2}{-1 - k} = -\frac{k+3}{k+1}

. Setting these two expressions for

m_2

equal to each other and solving the resulting proportion gives the quadratic equation

k^2 - 2k - 9 = 0

. The discriminant of this equation is positive (

40 > 0

), confirming the existence of two real solutions. By Vieta's formulas, the sum of the roots is

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

.

Step-by-Step Solution

1

Determine the slope of line

L_1

in terms of

k

.

The slope of line

L_1

is

m_1 = \frac{3}{k}

.

Rewriting the equation

3x - ky = 8

in slope-intercept form (

y = mx + b

) gives

ky = 3x - 8

, which simplifies to

y = \frac{3}{k}x - \frac{8}{k}

. The coefficient of

x

represents the slope of the line.

2

Find the perpendicular slope

m_2

of line

L_2

in terms of

k

.

The slope of line

L_2

is

m_2 = -\frac{k}{3}

.

Since line

L_2

is perpendicular to line

L_1

, its slope must be the negative reciprocal of

m_1

(i.e.,

m_2 = -\frac{1}{m_1}

).

3

Express the slope of line

L_2

using the two given points

(k, 2)

and

(-1, k+5)

.

The slope of line

L_2

is

m_2 = -\frac{k+3}{k+1}

.

Using the slope formula

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

with the points

(k, 2)

and

(-1, k+5)

, we get

m_2 = \frac{(k+5) - 2}{-1 - k} = \frac{k+3}{-(k+1)} = -\frac{k+3}{k+1}

.

4

Set the two expressions for the slope of line

L_2

equal to each other and solve for

k

.

The quadratic equation is

k^2 - 2k - 9 = 0

.

Equating the slopes gives

-\frac{k}{3} = -\frac{k+3}{k+1}

. Multiplying both sides by

-1

gives

\frac{k}{3} = \frac{k+3}{k+1}

. Cross-multiplying yields

k(k+1) = 3(k+3) \implies k^2 + k = 3k + 9 \implies k^2 - 2k - 9 = 0

.

5

Calculate the sum of all possible real values of

k

.

The sum of all possible real values of

k

is

2

.

The discriminant of the quadratic equation

k^2 - 2k - 9 = 0

is

(-2)^2 - 4(1)(-9) = 40 > 0

, confirming that two distinct real solutions for

k

exist. According to Vieta's formulas, the sum of the roots of a quadratic equation

ax^2 + bx + c = 0

is

-\frac{b}{a}

. Here,

a=1

and

b=-2

, so the sum is

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

.

Key Concept

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

m_1 \cdot m_2 = -1

).

Question 2217Question →

A scientist maps the movement of a cell on a coordinate grid. The cell is initially located at the point $(2, -5)$ . The cell then moves to a new position after being translated 6 units to the left and 4 units up. What are the coordinates of the cell's new position?

(-4, -9)

(8, -1)

(-4, -1)

(8, -9)

(-1, -4)

Show answer & explanation

Answer: $(-4, -1)$

Answer

The correct position of the cell is

(-4, -1)

.

To find the coordinates of the new position, apply the translation to the initial coordinates

(2, -5)

. A translation of 6 units to the left subtracts 6 from the x-coordinate:

2 - 6 = -4

. A translation of 4 units up adds 4 to the y-coordinate:

-5 + 4 = -1

. This results in the coordinates

(-4, -1)

.

Step-by-Step Solution

1

Calculate the new x-coordinate by applying the horizontal translation.

The initial x-coordinate is

2

. Translating 6 units to the left means subtracting 6:

2 - 6 = -4

.

A horizontal shift to the left decreases the x-coordinate value.

2

Calculate the new y-coordinate by applying the vertical translation.

The initial y-coordinate is

-5

. Translating 4 units up means adding 4:

-5 + 4 = -1

.

A vertical shift upward increases the y-coordinate value.

3

Combine the new coordinates into an ordered pair

(x, y)

.

The new coordinates are

(-4, -1)

.

The coordinates are represented by the new horizontal position followed by the new vertical position.

Key Concept

Translating a point in the coordinate plane by modifying its coordinates:

(x, y) \rightarrow (x + h, y + k)

, where

h

is the horizontal shift and

k

is the vertical shift.

Question 2218Question →

A technician uses a linear model to estimate the time, $t$ hours, required to complete a project. The estimate satisfies the equation:

\frac{2}{3}(t - 12) + \frac{5}{6}(t + 6) = 18

What is the value of $t$ ?

Show answer & explanation

Answer: 14

Answer

The value of

t

is

14

.

To solve the linear equation, we first multiply all terms by the least common multiple of the denominators, which is

6

. This simplifies the equation to

4(t - 12) + 5(t + 6) = 108

. Distributing the terms gives

4t - 48 + 5t + 30 = 108

. Combining like terms yields

9t - 18 = 108

. Adding

18

to both sides gives

9t = 126

. Finally, dividing by

9

gives

t = 14

.

Step-by-Step Solution

1

Multiply both sides of the equation by

6

to eliminate the denominators.

4(t - 12) + 5(t + 6) = 108

Multiplying by the least common multiple of the denominators clears the fractions, making the linear equation easier to solve.

2

Apply the distributive property to expand the terms.

4t - 48 + 5t + 30 = 108

Distributing the constants allows us to group like terms.

3

Combine the variable terms and the constant terms on the left side.

9t - 18 = 108

Simplifying the expression is a necessary step before isolating the variable.

4

Add

18

to both sides of the equation.

9t = 126

Adding the constant to both sides isolates the variable term on the left side.

5

Divide both sides of the equation by

9

to solve for

t

.

t = 14

Dividing by the coefficient of the variable yields the final solution.

Key Concept

Solving linear equations with fractional coefficients by clearing denominators and isolating the variable.

Question 2219Question →

In the standard $(x,y)$ coordinate plane, triangle $PQR$ has vertices $P(2, 3)$ , $Q(6, 3)$ , and $R(2, 6)$ . The triangle undergoes a sequence of three transformations:

1. A dilation centered at the point $(4, 2)$ with a scale factor of $-2$ .
2. A reflection across the line $y = -x$ .
3. A rotation of $90^\circ$ counterclockwise about the origin.

What are the coordinates of the image of vertex $R$ after this sequence of transformations?

(6, -8)

(-4, 12)

(8, 6)

(-8, -6)

(0, -10)

Show answer & explanation

Answer: $(8, 6)$

Answer

The coordinate pair

(8, 6)

Applying the dilation formula centered at

(4, 2)

with scale factor

-2

to

R(2, 6)

gives the point

R'(8, -6)

. Reflecting this point across the line

y = -x

negates and swaps the coordinates, yielding

R''(6, -8)

. Finally, a

90^\circ

counterclockwise rotation about the origin swaps the coordinates and negates the new

x

-coordinate, producing the final coordinates

(8, 6)

.

Step-by-Step Solution

1

Perform the dilation of vertex

R(2, 6)

centered at

C(4, 2)

with a scale factor of

k = -2

.

The intermediate image is

R'(8, -6)

.

For a dilation centered at

(x_c, y_c)

with scale factor

k

, the coordinates of the image are

(x_c + k(x - x_c), y_c + k(y - y_c))

. Substituting

R(2, 6)

,

C(4, 2)

, and

k = -2

gives:

x' = 4 - 2(2 - 4) = 8

and

y' = 2 - 2(6 - 2) = -6

.

2

Reflect the point

R'(8, -6)

across the line

y = -x

.

The intermediate image is

R''(6, -8)

.

A reflection across the line

y = -x

maps any point

(x, y)

to

(-y, -x)

. Applying this rule to

R'(8, -6)

yields

R''(-(-6), -8) = (6, -8)

.

3

Rotate the point

R''(6, -8)

by

90^\circ

counterclockwise about the origin.

The final image is

R'''(8, 6)

.

A rotation of

90^\circ

counterclockwise about the origin maps any point

(x, y)

to

(-y, x)

. Applying this rule to

R''(6, -8)

yields

R'''(-(-8), 6) = (8, 6)

.

Key Concept

Composite transformations in the coordinate plane combining dilation from a non-origin center, reflection across diagonal lines, and rotation about the origin.

Alternative Method

Instead of applying the transformations step-by-step to the point, we can track the transformations vectorially. For the dilation, the vector

\vec{CR} = R - C = (-2, 4)

is scaled by

-2

to get

-2\vec{CR} = (4, -8)

, which added back to

C(4, 2)

yields

R'(8, -6)

. Reflecting across

y = -x

exchanges the coordinates and negates them, giving

R''(6, -8)

. Rotating

90^\circ

counterclockwise about the origin maps

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

, yielding

R'''(8, 6)

.

Estimated Time:3m 0s

Question 2220Question →

In $\triangle XYZ$ , the measure of $\angle X$ is $40^\circ$ . If the measure of $\angle Y$ is three times the measure of $\angle X$ , what is the measure, in degrees, of $\angle Z$ ?

Show answer & explanation

Answer: 20

Answer

The measure of

\angle Z

is

20

degrees.

To find the measure of

\angle Z

, we first determine the measure of

\angle Y

. Since

\angle Y

is three times the measure of

\angle X

(

40^\circ

), we calculate

3 \times 40^\circ = 120^\circ

. Because the interior angles of a triangle must sum to

180^\circ

, the remaining angle

\angle Z

is found by subtracting the measures of

\angle X

and

\angle Y

from

180^\circ

:

180^\circ - 40^\circ - 120^\circ = 20^\circ

.

Step-by-Step Solution

1

Calculate the measure of

\angle Y

.

The measure of

\angle Y

is

120^\circ

.

The measure of

\angle Y

is specified to be three times the measure of

\angle X

, which is given as

40^\circ

. Multiplying

40^\circ

by

3

gives

120^\circ

.

2

Calculate the measure of

\angle Z

.

The measure of

\angle Z

is

20^\circ

.

The interior angles of any triangle sum to

180^\circ

. Subtracting the sum of the measures of

\angle X

(

40^\circ

) and

\angle Y

(

120^\circ

) from

180^\circ

yields the measure of

\angle Z

.

Key Concept

The sum of the interior angles of a triangle is always

180^\circ

.

Estimated Time:45s

Answer

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