Algebra and Functions Alıştırma Soruları — GMAT — Sayfa 9

Soru 161Soru →

A software provider offers two cloud subscription plans for enterprise clients: Tier A and Tier B. The monthly fee per user is $\$x$ for Tier A and $\$y$ for Tier B. In 2024, a corporate customer purchased 50 user licenses for Tier A and 30 user licenses for Tier B, resulting in a total monthly cost of $\$13,500$ .

In 2025, the provider adjusted its rates by increasing the Tier A monthly fee per user by $20\%$ and increasing the Tier B monthly fee per user by $10\%$ . Additionally, the customer altered its order by increasing its Tier A licenses by $10\%$ and decreasing its Tier B licenses by $20\%$ . If the customer's total monthly cost in 2025 was $\$15,180$ , what was the original monthly fee per user for Tier A in 2024?

\$125

\$150

\$175

\$200

\$225

Cevabı ve açıklamayı göster

Cevap: $\$150$

Cevap

The original monthly fee per user for Tier A in 2024 was

\$150

.

The correct option correctly models the two-variable linear system. By expressing the 2024 total cost as

5x + 3y = 1,350

and the 2025 cost as

66x + 26.4y = 15,180

, solving the simultaneous equations yields

x = 150

for Tier A and

y = 200

for Tier B.

Adım Adım Çözüm

1

Formulate the linear equation for the 2024 monthly cost.

50x + 30y = 13,500

, which simplifies to

5x + 3y = 1,350

.

The total cost is the sum of the products of license quantities and their respective per-user prices.

2

Determine the 2025 price per user and license quantities for both tiers.

Tier A: New price =

1.20x

, New quantity =

50 \times 1.10 = 55

.
Tier B: New price =

1.10y

, New quantity =

30 \times 0.80 = 24

.

Price and quantity changes must be applied independently as percentage increases or decreases.

3

Formulate the linear equation for the 2025 total monthly cost.

55(1.20x) + 24(1.10y) = 15,180 \implies 66x + 26.4y = 15,180

.

Multiply the updated unit price by the updated quantity for each tier.

4

Simplify the 2025 equation and solve the system of linear equations.

Multiply

66x + 26.4y = 15,180

by

5

to eliminate decimals:

330x + 132y = 75,900

, which simplifies by dividing by

6

to

55x + 22y = 12,650

.
Multiply

5x + 3y = 1,350

by

11

:

55x + 33y = 14,850

.
Subtract the two equations:

(55x + 33y) - (55x + 22y) = 14,850 - 12,650 \implies 11y = 2,200 \implies y = 200

.
Substitute

y = 200

into

5x + 3(200) = 1,350 \implies 5x = 750 \implies x = 150

.

Eliminate variable

x

to solve for

y

, then back-substitute to find

x

.

Anahtar Kavram

Multi-variable system modeling with multi-step percentage modifications

Tahmini Süre:2m 30s

Soru 162Soru →

If $x$ is a real number satisfying the inequality $|x - 3| + 2|x + 1| \leq 7$ , what is the minimum possible value of $x^2 - 2x$ ?

-1

0

-2

3

8

Cevabı ve açıklamayı göster

Cevap: $-1$

Cevap

The minimum possible value of

x^2 - 2x

is

-1

.

The absolute value inequality

|x - 3| + 2|x + 1| \leq 7

restricts

x

to the closed interval

[-2, 2]

. The quadratic function

x^2 - 2x

can be rewritten as

(x - 1)^2 - 1

, which has a minimum value of

-1

at

x = 1

. Because

x = 1

is inside the interval

[-2, 2]

, the minimum possible value is

-1

.

Adım Adım Çözüm

1

Identify the critical points of the absolute value expressions to set up case analysis.

The critical points are

x = -1

and

x = 3

, dividing the real line into three intervals:

x < -1

,

-1 \leq x \leq 3

, and

x > 3

.

Absolute value expressions

|x - 3|

and

|x + 1|

change sign at

x = 3

and

x = -1

, respectively.

2

Solve the inequality

|x - 3| + 2|x + 1| \leq 7

across all three cases.

Case 1 (

x < -1

):

3 - x - 2(x + 1) \leq 7 \implies 1 - 3x \leq 7 \implies x \geq -2

. Combining gives

-2 \leq x < -1

.
Case 2 (

-1 \leq x \leq 3

):

3 - x + 2(x + 1) \leq 7 \implies x + 5 \leq 7 \implies x \leq 2

. Combining gives

-1 \leq x \leq 2

.
Case 3 (

x > 3

):

x - 3 + 2(x + 1) \leq 7 \implies 3x - 1 \leq 7 \implies x \leq \frac{8}{3}

. No solutions since

\frac{8}{3} < 3

.
The combined solution set is

-2 \leq x \leq 2

.

Piecewise analysis determines the exact range of valid values for

x

.

3

Minimize the quadratic expression

f(x) = x^2 - 2x

over the interval

[-2, 2]

.

Completing the square gives

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

. The vertex occurs at

x = 1

. Since

x = 1 \in [-2, 2]

, the minimum value is

f(1) = -1

.

A parabola opening upward achieves its global minimum at its vertex if the vertex lies within the allowed domain.

Anahtar Kavram

Solving piecewise absolute value inequalities and finding the extreme values of a quadratic function over a bounded interval.

Tahmini Süre:2m 0s

Soru 163Soru →

If $x$ is a real number satisfying the equation $|x - 4| = 2x - 1$ , what is the value of $|x - 5|$ ?

8

\frac{7}{3}

\frac{10}{3}

4

\frac{34}{3}

Cevabı ve açıklamayı göster

Cevap: $\frac{10}{3}$

Cevap

\frac{10}{3}

Solving

|x - 4| = 2x - 1

requires

2x - 1 \geq 0

, or

x \geq \frac{1}{2}

. Splitting into cases gives

x - 4 = 2x - 1 \implies x = -3

(invalid because

-3 < \frac{1}{2}

) and

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

(valid because

\frac{5}{3} \geq \frac{1}{2}

). Substituting

x = \frac{5}{3}

into

|x - 5|

yields

|\frac{5}{3} - 5| = |-\frac{10}{3}| = \frac{10}{3}

.

Adım Adım Çözüm

1

Determine the non-negativity constraint for the absolute value equation

Since absolute values are non-negative,

2x - 1 \geq 0 \implies x \geq \frac{1}{2}

.

The right-hand side of

|x - 4| = 2x - 1

must be greater than or equal to 0.

2

Split into linear cases and solve for candidate values of x

Case 1:

x - 4 = 2x - 1 \implies x = -3

.
Case 2:

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

.

An absolute value equation

|A| = B

splits into

A = B

or

A = -B

.

3

Eliminate extraneous solutions

x = -3

fails the condition

x \geq \frac{1}{2}

(since

2(-3) - 1 = -7 < 0

).

x = \frac{5}{3}

satisfies

x \geq \frac{1}{2}

(since

2(\frac{5}{3}) - 1 = \frac{7}{3} > 0

). Thus,

x = \frac{5}{3}

is the sole valid solution.

Candidate solutions must satisfy the original non-negativity constraint.

4

Substitute the valid solution into

|x - 5|

|\frac{5}{3} - 5| = |-\frac{10}{3}| = \frac{10}{3}

.

Calculate the value of the requested expression using

x = \frac{5}{3}

.

Anahtar Kavram

Absolute Value Equations and Extraneous Solutions

Soru 164Soru →

A café sells small cold brew coffees for $\$4$ each and large cold brew coffees for $\$6$ each. On Monday morning, the café sold a total of $40$ cold brew coffees for $\$196$ . How many large cold brew coffees were sold?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The café sold 18 large cold brew coffees.

Setting up the system

s + l = 40

and

4s + 6l = 196

, substituting

s = 40 - l

yields

4(40 - l) + 6l = 196

. Simplifying gives

160 + 2l = 196

, so

2l = 36

and

l = 18

.

Adım Adım Çözüm

1

Set up equations based on total count and total cost.

Let

s

= number of small coffees and

l

= number of large coffees. The system is

s + l = 40

and

4s + 6l = 196

.

The problem gives two distinct relationships between the quantity sold and the total dollar value.

2

Substitute

s = 40 - l

into the second equation.

4(40 - l) + 6l = 196 \implies 160 + 2l = 196

.

Substituting reduces the system from two variables to a single equation in

l

.

3

Solve the linear equation for

l

.

2l = 36 \implies l = 18

.

Subtracting 160 from both sides and dividing by 2 yields the number of large cold brews.

Anahtar Kavram

Solving systems of linear equations in two variables using substitution.

Soru 165Soru →

A financial analyst models a company's total annual revenue, $R(t)$ , and total annual expenses, $E(t)$ , in thousands of dollars, as linear functions of time $t$ , where $t$ represents the number of years since 2020. In 2022 ( $t = 2$ ), the annual revenue was $\$850,000$ and annual expenses were $\$610,000$ . In 2025 ( $t = 5$ ), the annual revenue reached $\$1,120,000$ while annual expenses reached $\$700,000$ . Assuming these linear trends continue, in which calendar year will the company's annual profit (defined as total annual revenue minus total annual expenses) reach exactly $\$540,000$ ?

Cevabı ve açıklamayı göster

Cevap: 2027

Cevap

The company's annual profit will reach $540,000 in the year 2027.

The linear profit function increases at a constant rate of

60,000 per year from a base profit of

240,000 in 2022 (

t = 2

). Solving

240 + 60(t - 2) = 540

yields

t = 7

, which corresponds to the calendar year 2027.

Adım Adım Çözüm

1

Calculate the annual profit in thousands of dollars for the known years 2022 (

t = 2

) and 2025 (

t = 5

).

In 2022 (

t = 2

), profit

P(2) = 850 - 610 = 240

thousand dollars. In 2025 (

t = 5

), profit

P(5) = 1120 - 700 = 420

thousand dollars.

Profit is defined as revenue minus expenses.

2

Determine the linear profit function

P(t) = mt + b

.

The rate of change of profit per year is

m = \frac{420 - 240}{5 - 2} = \frac{180}{3} = 60

thousand dollars per year.

Since both revenue and expenses are linear functions of time

t

, their difference

P(t) = R(t) - E(t)

is also a linear function of

t

.

3

Set up the linear equation for profit using point-slope form.

P(t) - 240 = 60(t - 2) \implies P(t) = 60t + 120

.

Using the point

(2, 240)

and slope

m = 60

establishes the complete linear equation for annual profit.

4

Solve the linear equation for

t

when profit

P(t) = 540

thousand dollars.

60t + 120 = 540 \implies 60t = 420 \implies t = 7

.

Setting the profit equal to 540 gives the value of

t

years after 2020.

5

Convert the value of

t

into the target calendar year.

Calendar Year

= 2020 + 7 = 2027

.

Since

t

represents years elapsed since 2020,

t = 7

corresponds to calendar year 2027.

Anahtar Kavram

Linear Modeling and Linear Equations in Two Variables

Soru 166Soru →

How many integer values of $x$ satisfy both $|x^2 - 10| \le 6$ and $|x - 1| > 2$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4 integer values satisfy both inequalities:

-4

,

-3

,

-2

, and

4

.

Solving

|x^2 - 10| \le 6

gives

4 \le x^2 \le 16

, which yields 6 integer solutions:

\{-4, -3, -2, 2, 3, 4\}

. Applying the condition

|x - 1| > 2

requires

x < -1

or

x > 3

. Only

-4, -3, -2

, and

4

satisfy both conditions, giving exactly 4 valid integer values.

Adım Adım Çözüm

1

Solve the quadratic absolute value inequality

|x^2 - 10| \le 6

.

-6 \le x^2 - 10 \le 6 \implies 4 \le x^2 \le 16

.

Removing the absolute value bars gives the bounded inequality

-6 \le x^2 - 10 \le 6

. Adding 10 across all parts isolates

x^2

.

2

Identify all candidate integer values of

x

for which

4 \le x^2 \le 16

.

x \in \{-4, -3, -2, 2, 3, 4\}

.

Taking the square root yields

-4 \le x \le -2

or

2 \le x \le 4

. The integer solutions are

-4, -3, -2, 2, 3, 4

.

3

Solve the linear absolute value inequality

|x - 1| > 2

.

x > 3

or

x < -1

.

An absolute value distance greater than 2 expands into two disjunctive linear inequalities:

x - 1 > 2

or

x - 1 < -2

.

4

Intersect the candidate integer set

\{-4, -3, -2, 2, 3, 4\}

with the domain

x < -1

or

x > 3

.

The valid integers are

-4, -3, -2,

and

4

. The values

2

and

3

fail the inequality since they fall inside the excluded interval

[-1, 3]

.

Testing each candidate shows that

-4, -3, -2

are strictly less than

-1

, and

4

is strictly greater than

3

.

Anahtar Kavram

Solving compound absolute value inequalities and finding integer solution set intersections.

Soru 167Soru →

If $x$ is a real number satisfying the equation $|2x - 3| = 3x + 1$ , what is the value of $x^2 + 2x$ ?

-\frac{16}{25}

\frac{8}{25}

\frac{24}{25}

8

\frac{56}{25}

Cevabı ve açıklamayı göster

Cevap: $\frac{24}{25}$

Cevap

The value of

x^2 + 2x

is

\frac{24}{25}

.

The equation

|2x - 3| = 3x + 1

requires

3x + 1 \ge 0

, or

x \ge -\frac{1}{3}

. Setting

2x - 3 = 3x + 1

gives

x = -4

, which violates

x \ge -\frac{1}{3}

and is extraneous. Setting

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

yields

-2x + 3 = 3x + 1

, which simplifies to

5x = 2

, or

x = \frac{2}{5}

. Since

\frac{2}{5} \ge -\frac{1}{3}

, this is the unique valid solution. Substituting

x = \frac{2}{5}

into

x^2 + 2x

gives

\left(\frac{2}{5}\right)^2 + 2\left(\frac{2}{5}\right) = \frac{4}{25} + \frac{20}{25} = \frac{24}{25}

.

Adım Adım Çözüm

1

Set up the conditions for the absolute value equation

|2x - 3| = 3x + 1

.

Since the absolute value

|2x - 3|

must be non-negative, the right-hand side must satisfy

3x + 1 \ge 0

, which means

x \ge -\frac{1}{3}

.

An absolute value expression cannot equal a negative number.

2

Solve Case 1:

2x - 3 \ge 0

(so

x \ge \frac{3}{2}

).

2x - 3 = 3x + 1 \implies x = -4

.

Remove the absolute value bars directly when the inside expression is non-negative.

3

Check Case 1 candidate solution against constraints.

x = -4

is extraneous because

-4 < \frac{3}{2}

and

3(-4) + 1 = -11 < 0

. Reject

x = -4

.

Plugging

x = -4

into the original equation yields

|-11| = -11

, which is false.

4

Solve Case 2:

2x - 3 < 0

(so

x < \frac{3}{2}

).

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

.

Negate the expression inside the absolute value when it is negative.

5

Check Case 2 candidate solution and evaluate the target expression.

Since

x = \frac{2}{5} \ge -\frac{1}{3}

, it is valid. Then

x^2 + 2x = \left(\frac{2}{5}\right)^2 + 2\left(\frac{2}{5}\right) = \frac{4}{25} + \frac{4}{5} = \frac{24}{25}

.

Substitute the single valid real solution into

x^2 + 2x

.

Anahtar Kavram

Absolute Value Equations and Extraneous Solution Elimination

Soru 168Soru →

An investment firm allocated a total of $\$120,000$ between two venture capital funds, Fund X and Fund Y. Fund X yielded an annual simple interest rate of $8\%$ , while Fund Y yielded an annual simple interest rate of $12\%$ . If the total interest earned from both funds combined after one year was $\$11,600$ , how much money was invested in Fund X?

\$70,000

\$50,000

\$60,000

\$58,000

\$75,000

Cevabı ve açıklamayı göster

Cevap: $\$70,000$

Cevap

\$70,000

The correct answer is

\$70,000

. By modeling the investment in Fund X as

x

, the investment in Fund Y becomes

120,000 - x

. Setting up the total annual interest equation

0.08x + 0.12(120,000 - x) = 11,600

yields

-0.04x = -2,800

, giving

x = 70,000

.

Adım Adım Çözüm

1

Define variables for the invested amounts

Let

x

be the amount invested in Fund X. The remaining amount invested in Fund Y is

\$120,000 - x

.

The total capital is

\$120,000

, so the investments must sum to this total.

2

Set up the interest equation

0.08x + 0.12(120,000 - x) = 11,600

Total interest is the sum of interest from Fund X (

8\%

of

x

) and Fund Y (

12\%

of

120,000 - x

).

3

Expand and simplify the linear equation

0.08x + 14,400 - 0.12x = 11,600 \Rightarrow -0.04x + 14,400 = 11,600

Distribute

0.12

across

(120,000 - x)

and combine like terms.

4

Isolate the variable

x

-0.04x = 11,600 - 14,400 \Rightarrow -0.04x = -2,800 \Rightarrow x = \frac{-2,800}{-0.04} = 70,000

Subtract

14,400

from both sides and divide by

-0.04

to find the investment in Fund X.

Anahtar Kavram

Linear Equation Modeling for Investment Allocations

Tahmini Süre:1m 40s

Soru 169Soru →

A production facility uses two automated machines, Alpha and Beta, which operate at constant hourly production rates. Machine Alpha produces $x$ units per hour and Machine Beta produces $y$ units per hour. When both machines operate together for 4 hours, followed by Machine Alpha operating alone for 3 hours, the facility produces a total of 310 units. When Machine Beta operates alone for 2 hours, followed by both machines operating together for 5 hours, the facility produces a total of 325 units. If Machine Alpha's hourly rate is increased by $20\%$ and Machine Beta's hourly rate is decreased by $20\%$ , how many total units will both machines produce if they work together for 8 hours at their new rates?

Cevabı ve açıklamayı göster

Cevap: 448

Cevap

The total number of units produced by both machines working together for 8 hours under the modified rates is 448.

Setting up the system of equations based on individual hours worked yields

7x + 4y = 310

and

5x + 7y = 325

. Solving this system gives original rates

x = 30

units/hr and

y = 25

units/hr. Applying the

20\%

increase to Alpha (

36

units/hr) and

20\%

decrease to Beta (

20

units/hr) gives a combined rate of

56

units/hr. Multiplying by

8

hours gives the final answer of

448

units.

Adım Adım Çözüm

1

Translate the given operational scenarios into a system of two linear equations.

Equation 1:

7x + 4y = 310

; Equation 2:

5x + 7y = 325

.

Operating both machines for 4 hours and Alpha alone for 3 hours means Alpha works

4 + 3 = 7

hours while Beta works

4

hours, giving

7x + 4y = 310

. Operating Beta alone for 2 hours and both for 5 hours means Alpha works

5

hours while Beta works

2 + 5 = 7

hours, giving

5x + 7y = 325

.

2

Solve the system of linear equations for variables

x

and

y

.

x = 30

and

y = 25

.

Eliminating

y

by multiplying the first equation by 7 and the second by 4 yields

49x - 20x = 2170 - 1300 \implies 29x = 870 \implies x = 30

. Substituting

x = 30

back into

7x + 4y = 310

gives

210 + 4y = 310 \implies 4y = 100 \implies y = 25

.

3

Calculate the modified production rates after percentage adjustments.

New rate for Alpha is

36

units/hr; new rate for Beta is

20

units/hr.

A

20\%

increase on

x = 30

yields

30 \times 1.20 = 36

. A

20\%

decrease on

y = 25

yields

25 \times 0.80 = 20

.

4

Compute total combined output over 8 hours.

Total units produced =

448

.

Combined modified rate is

36 + 20 = 56

units per hour. Total production over 8 hours is

56 \times 8 = 448

.

Anahtar Kavram

Linear Equations in One and Two Variables

Tahmini Süre:2m 30s

Soru 170Soru →

A logistics company dispatches a fleet of delivery vehicles consisting entirely of Van A models and Van B models for a day's operations. Each Van A carries $40$ packages and consumes $5$ gallons of fuel per route. Each Van B carries $60$ packages and consumes $8$ gallons of fuel per route. On a given day, $18$ vans were dispatched in total, delivering a combined total of $920$ packages. Which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

The total number of Van B models dispatched was

10

.

The total fuel consumed by all Van A models combined was

50

gallons.

Van B models accounted for more than

60\%

of the total fuel consumed by the fleet.

The ratio of the number of Van A models to Van B models dispatched was

5

to

4

.

The total fuel consumed by the entire fleet was

120

gallons.

Cevabı ve açıklamayı göster

Cevap: The total number of Van B models dispatched was $10$ .; Van B models accounted for more than $60\%$ of the total fuel consumed by the fleet.; The total fuel consumed by the entire fleet was $120$ gallons.

Cevap

The true statements are those stating that 10 Van B models were dispatched, that Van B models accounted for more than 60% of total fleet fuel consumed, and that the total fuel consumed by the entire fleet was 120 gallons.

Solving the linear system

A + B = 18

and

40A + 60B = 920

yields

A = 8

Van A models and

B = 10

Van B models. Evaluating the claims: 10 Van B models were dispatched (true); Van B fuel (

80

gallons) represents

80/120 = 66.67\%

of total fuel, which is greater than

60\%

(true); and total fleet fuel consumption is

40 + 80 = 120

gallons (true).

Adım Adım Çözüm

1

Define variables and construct the system of linear equations.

Let

A

be the number of Van A models and

B

be the number of Van B models. Total vans equation:

A + B = 18

. Total packages equation:

40A + 60B = 920

.

Word problem modeling requires translating fleet count and package delivery totals into algebraic linear equations.

2

Solve the system of equations for

A

and

B

.

Divide the package equation by

20

:

2A + 3B = 46

. Substitute

A = 18 - B

:

2(18 - B) + 3B = 46 \implies 36 + B = 46 \implies B = 10

. Then

A = 18 - 10 = 8

.

Determining exact vehicle quantities is required to evaluate all downstream statements.

3

Calculate total fuel consumption for each van model and the entire fleet.

Van A fuel:

8 \text{ vans} \times 5 \text{ gal/van} = 40 \text{ gallons}

. Van B fuel:

10 \text{ vans} \times 8 \text{ gal/van} = 80 \text{ gallons}

. Total fleet fuel:

40 + 80 = 120 \text{ gallons}

.

Provides exact quantities to test statement validity.

4

Evaluate each statement against the calculated values.

1) Van B count is 10 (True). 2) Van A fuel is 40 gallons, not 50 (False). 3) Van B fuel percentage is

80/120 = 66.67\% > 60\%

(True). 4) Van A to Van B ratio is

8:10 = 4:5

, not

5:4

(False). 5) Total fuel is 120 gallons (True).

Identifies all correct options for multiple selection.

Anahtar Kavram

Linear Equation System Modeling in Applied Contexts

Soru 171Soru →

If $x$ is a real number satisfying the equation $|x + 3| + |x - 5| = 8$ , what is the maximum possible value of $|x^2 - 4x - 12|$ ?

7

9

12

16

25

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

The equation

|x + 3| + |x - 5| = 8

represents the sum of the distances from

x

to

-3

and from

x

to

5

. Since the distance between

-3

and

5

is exactly

8

, this equation holds for all

x

in the interval

[-3, 5]

. Rewriting

x^2 - 4x - 12

as

(x - 2)^2 - 16

, the vertex occurs at

x = 2

, which lies inside

[-3, 5]

. At

x = 2

,

x^2 - 4x - 12 = -16

, so

|x^2 - 4x - 12| = |-16| = 16

. At the endpoints

x = -3

and

x = 5

, the values are

|9| = 9

and

|-7| = 7

, respectively. Thus, the maximum possible value of

|x^2 - 4x - 12|

on the interval is 16.

Adım Adım Çözüm

1

Determine the solution set of the absolute value equation

|x + 3| + |x - 5| = 8

.

The domain of valid

x

values is the continuous closed interval

[-3, 5]

.

By the geometric distance interpretation, the sum of distances from

x

to

-3

and

x

to

5

equals the total distance between

-3

and

5

(which is

8

) if and only if

x

lies between

-3

and

5

inclusive.

2

Express the quadratic expression

g(x) = x^2 - 4x - 12

in vertex form.

g(x) = (x - 2)^2 - 16

.

Completing the square allows straightforward evaluation of the vertex and minimum/maximum values of the quadratic on

[-3, 5]

.

3

Find the range of

g(x) = (x - 2)^2 - 16

for

x \in [-3, 5]

.

The minimum value occurs at the vertex

x = 2

, where

g(2) = -16

. The maximum value occurs at the endpoint

x = -3

, where

g(-3) = 9

. Thus,

-16 \le g(x) \le 9

.

The parabola opens upwards with vertex at

x = 2 \in [-3, 5]

. The distance from

x = 2

to

x = -3

is

5

, while the distance to

x = 5

is

3

.

4

Evaluate the maximum value of

|g(x)| = |x^2 - 4x - 12|

on

[-3, 5]

.

The absolute value

|g(x)|

ranges from

0

to

\max(|-16|, |9|) = 16

.

The absolute value converts negative outputs to positive, so the extreme magnitude

|-16| = 16

at

x = 2

is the maximum value.

Anahtar Kavram

Absolute Value Distance Interpretation and Quadratic Range Optimization

Tahmini Süre:2m 0s

Soru 172Soru →

If $5^x = 100^y = 20^z$ for non-zero real numbers $x$ , $y$ , and $z$ , which of the following expressions is equal to $z$ ?

\frac{xy}{x - y}

\frac{xy}{x + y}

\frac{xy}{y - x}

x - y

y - x

Cevabı ve açıklamayı göster

Cevap: $\frac{xy}{x - y}$

Cevap

\frac{xy}{x - y}

By setting

5^x = 100^y = 20^z = K

, we rewrite each base as a power of

K

:

5 = K^{1/x}

,

100 = K^{1/y}

, and

20 = K^{1/z}

. Using the numerical relationship

20 = \frac{100}{5}

, we substitute to obtain

K^{1/z} = \frac{K^{1/y}}{K^{1/x}} = K^{\frac{1}{y} - \frac{1}{x}}

. Equating exponents gives

\frac{1}{z} = \frac{1}{y} - \frac{1}{x} = \frac{x - y}{xy}

. Taking the reciprocal yields the expression stating that

z

equals

\frac{xy}{x - y}

.

Adım Adım Çözüm

1

Set the common exponential expressions equal to a constant

K

.

Let

5^x = 100^y = 20^z = K

, where

K > 0

and

K \neq 1

.

Equating exponential terms to a shared parameter allows conversion from variable powers to fractional exponents.

2

Express the base numbers

5

,

100

, and

20

in terms of

K

.

5 = K^{1/x}

,

100 = K^{1/y}

, and

20 = K^{1/z}

.

Taking the

x

-th,

y

-th, and

z

-th roots of both sides isolates each base number.

3

Relate the base numbers using arithmetic division.

20 = \frac{100}{5} \implies K^{1/z} = \frac{K^{1/y}}{K^{1/x}} = K^{\frac{1}{y} - \frac{1}{x}}

.

Dividing powers with the same base subtracts their exponents.

4

Equate the exponents and solve for

z

.

\frac{1}{z} = \frac{1}{y} - \frac{1}{x} = \frac{x - y}{xy} \implies z = \frac{xy}{x - y}

.

Taking common denominators and inverting the fraction yields

z

in terms of

x

and

y

.

Anahtar Kavram

Relating different bases with variable exponents by setting them equal to a common constant and converting to reciprocal exponents.

Tahmini Süre:2m 0s

Soru 173Soru →

A corporate advisory firm executes two types of client projects: Type A and Type B. Each Type A project requires $x$ hours of financial modeling and $y$ hours of executive reporting. Each Type B project requires $x + 3$ hours of financial modeling and $2y - 2$ hours of executive reporting. During the first quarter, the firm completed 6 Type A projects and 4 Type B projects. If the total time spent on financial modeling across all 10 projects was 72 hours and the total time spent on executive reporting was 62 hours, what is the value of $3x + 4y$ ?

Cevabı ve açıklamayı göster

Cevap: 38

Cevap

The value of

3x + 4y

is 38.

Setting up total financial modeling hours gives

6x + 4(x + 3) = 72

, which yields

x = 6

. Setting up total executive reporting hours gives

6y + 4(2y - 2) = 62

, which yields

y = 5

. Substituting these values into

3x + 4y

yields

3(6) + 4(5) = 38

.

Adım Adım Çözüm

1

Formulate and solve the linear equation for financial modeling hours.

x = 6

6 Type A projects take

6x

hours and 4 Type B projects take

4(x + 3)

hours. Summing them yields

6x + 4x + 12 = 72

, which simplifies to

10x = 60

, giving

x = 6

.

2

Formulate and solve the linear equation for executive reporting hours.

y = 5

6 Type A projects take

6y

hours and 4 Type B projects take

4(2y - 2)

hours. Summing them yields

6y + 8y - 8 = 62

, which simplifies to

14y = 70

, giving

y = 5

.

3

Substitute

x = 6

and

y = 5

into the targeted expression

3x + 4y

.

38

3(6) + 4(5) = 18 + 20 = 38

.

Anahtar Kavram

Linear Equations in One and Two Variables

Soru 174Soru →

A courier service dispatches a delivery driver from Facility A to Facility B along a straight $120$ -mile route at a constant speed of $r$ miles per hour. On the return journey from Facility B to Facility A along the same route, traffic congestion reduces the driver's constant speed by $20$ miles per hour. If the total driving time for the entire round trip is $5$ hours, what was the driver's speed, in miles per hour, on the return trip?

38

40

48

60

8

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

The driver's speed on the return trip was 40 miles per hour.

Setting the total time equation to 120/r + 120/(r - 20) = 5 leads to the quadratic equation r^2 - 68r + 480 = 0. Factoring gives roots r = 60 and r = 8. Since r = 8 yields a negative return speed, r must be 60. Subtracting 20 gives the correct return speed of 40 mph.

Adım Adım Çözüm

1

Set up expressions for outbound and return travel times in terms of r.

Outbound time = \frac{120}{r} hours; Return time = \frac{120}{r - 20} hours.

Time equals distance divided by speed.

2

Formulate the equation for total round-trip time.

\frac{120}{r} + \frac{120}{r - 20} = 5

The sum of the travel times for both legs equals 5 hours.

3

Simplify the equation and convert it into standard quadratic form.

r^2 - 68r + 480 = 0

Dividing by 5 gives \frac{24}{r} + \frac{24}{r - 20} = 1, and multiplying by r(r - 20) yields 24(r - 20) + 24r = r(r - 20).

4

Solve the quadratic equation for r and eliminate extraneous solutions.

r = 60 mph (since r = 8 gives a negative return speed).

Factoring gives (r - 60)(r - 8) = 0. The root r = 8 is physically invalid because r - 20 must be positive.

5

Calculate the return speed r - 20.

60 - 20 = 40 mph

The return speed is 20 mph slower than the outbound speed.

Anahtar Kavram

Algebraic Modeling of Motion & Distance-Rate-Time Relationships

Tahmini Süre:2m 0s

Soru 175Soru →

If $n$ is an integer such that $|3 - 2n| \le 9$ and $|n - 1| \ge 2$ , what is the sum of all possible values of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The sum of all possible integer values of

n

is 12.

Solving

|3 - 2n| \le 9

gives

-3 \le n \le 6

. Solving

|n - 1| \ge 2

restricts

n

to

n \le -1

or

n \ge 3

. Taking the intersection yields the integer set

\{-3, -2, -1, 3, 4, 5, 6\}

, whose sum equals 12.

Adım Adım Çözüm

1

Unpack and solve

|3 - 2n| \le 9

-3 \le n \le 6

Removing absolute value yields

-9 \le 3 - 2n \le 9

. Dividing by

-2

requires reversing the inequality direction.

2

Unpack and solve

|n - 1| \ge 2

n \le -1

or

n \ge 3

An absolute value greater than or equal to 2 implies distance from 1 is at least 2 units in either direction.

3

Determine the intersection set of integers

\{-3, -2, -1, 3, 4, 5, 6\}

Filters out integers 0, 1, and 2 from the continuous range

[-3, 6]

.

4

Calculate the sum of the valid integers

12

Summing

(-3) + (-2) + (-1) + 3 + 4 + 5 + 6 = 12

.

Anahtar Kavram

Combining system of absolute value inequalities and handling inequality sign flips when dividing by negative quantities.

Soru 176Soru →

What is the sum of all valid real solutions to the equation $|2x + 15| = 1 - 5x$ ?

-2

\frac{10}{3}

\frac{16}{3}

-\frac{30}{7}

2

Cevabı ve açıklamayı göster

Cevap: $-2$

Cevap

The sum of all valid real solutions is

-2

.

Solving the absolute value equation

|2x + 15| = 1 - 5x

yields two candidate values:

x = -2

and

x = \frac{16}{3}

. Substituting

x = -2

into the right-hand side gives

1 - 5(-2) = 11

, which is non-negative and matches

|2(-2) + 15| = 11

. Substituting

x = \frac{16}{3}

gives

1 - 5\left(\frac{16}{3}\right) = -\frac{77}{3}

, which is negative and therefore invalid. Thus,

x = -2

is the unique valid solution, making the sum

-2

.

Adım Adım Çözüm

1

Set up the two cases for the absolute value equation

|2x + 15| = 1 - 5x

.

Case 1:

2x + 15 = 1 - 5x

; Case 2:

2x + 15 = -(1 - 5x)

.

By definition,

|u| = v

implies

u = v

or

u = -v

, provided

v \geq 0

.

2

Solve Case 1 for

x

.

7x = -14 \implies x = -2

.

Adding

5x

and subtracting

15

from both sides isolates

x

.

3

Solve Case 2 for

x

.

2x + 15 = -1 + 5x \implies 3x = 16 \implies x = \frac{16}{3}

.

Distributing the negative sign and combining like terms yields

x = \frac{16}{3}

.

4

Check candidate solutions against the non-negativity constraint

1 - 5x \geq 0

.

For

x = -2

:

1 - 5(-2) = 11 \geq 0

(Valid). For

x = \frac{16}{3}

:

1 - 5\left(\frac{16}{3}\right) = -\frac{77}{3} < 0

(Extraneous).

An absolute value cannot equal a negative number, so candidate solutions that make the right-hand side negative must be discarded.

5

Sum all valid real solutions.

The only valid solution is

x = -2

, so the sum is

-2

.

Extraneous solutions are excluded from the final sum.

Anahtar Kavram

Solving absolute value linear equations requires checking candidate solutions against domain constraints to filter out extraneous roots.

Tahmini Süre:2m 0s

Soru 177Soru →

For all real numbers $x > 0$ such that $x \neq 1$ , the algebraic expression $\frac{\sqrt{x} - 1}{\sqrt[4]{x} + 1} + \frac{\sqrt{x} + 1}{\sqrt[4]{x} - 1}$ is equivalent to $\frac{2(\sqrt[4]{x}^3 + 1)}{\sqrt{x} - 1}$ .

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

The statement is True.

The statement is true because substituting

u = \sqrt[4]{x}

allows both expressions to be simplified via algebraic factoring identities to the identical expression

\frac{2(u^2 - u + 1)}{u - 1}

.

Adım Adım Çözüm

1

Perform a substitution to simplify the radical exponents.

Let

u = \sqrt[4]{x}

, so that

\sqrt{x} = u^2

. The left-hand side becomes

\frac{u^2 - 1}{u + 1} + \frac{u^2 + 1}{u - 1}

.

Converting fourth roots and square roots into polynomial terms makes factoring easier.

2

Simplify the left-hand side expression.

\frac{(u-1)(u+1)}{u+1} + \frac{u^2+1}{u-1} = (u-1) + \frac{u^2+1}{u-1} = \frac{(u-1)^2 + u^2 + 1}{u-1} = \frac{2(u^2 - u + 1)}{u-1}.

Factoring the numerator of the first fraction cancels out the

(u+1)

term prior to combining terms.

3

Factor the right-hand side expression using polynomial identities.

\frac{2(u^3+1)}{u^2-1} = \frac{2(u+1)(u^2-u+1)}{(u+1)(u-1)} = \frac{2(u^2-u+1)}{u-1}.

Applying the sum of cubes identity

u^3 + 1 = (u + 1)(u^2 - u + 1)

allows cancellation of

(u+1)

from the denominator.

4

Compare the simplified left-hand side and right-hand side expressions.

Both expressions reduce to

\frac{2(\sqrt[4]{x}^2 - \sqrt[4]{x} + 1)}{\sqrt[4]{x} - 1}

.

Because both sides reduce to the identical simplified form for all

x > 0, x \neq 1

, the equivalence holds true.

Anahtar Kavram

Algebraic manipulation of radicals using fractional exponent substitution, difference of squares, and sum of cubes factoring.

Soru 178Soru →

A retail store sets the selling price, $P$ , of a custom item based on its wholesale cost, $C$ , according to the linear equation $P = 1.4C + 20$ . The store's profit on each item is defined as $P - C$ . If the profit on a certain item is also equal to $0.2P + 40$ , what is the wholesale cost, $C$ , of the item in dollars?

Cevabı ve açıklamayı göster

Cevap: 200

Cevap

The wholesale cost

C

of the item is $200.

By writing profit both as

P - C = 0.4C + 20

and as

0.2P + 40 = 0.28C + 44

, we obtain a single linear equation in terms of

C

:

0.4C + 20 = 0.28C + 44

. Subtracting

0.28C

and

20

from both sides gives

0.12C = 24

, which simplifies to

C = 200

.

Adım Adım Çözüm

1

Substitute

P = 1.4C + 20

into the standard profit expression

P - C

.

Profit =

0.4C + 20

This expresses the profit solely as a linear function of the wholesale cost

C

.

2

Substitute

P = 1.4C + 20

into the alternative profit expression

0.2P + 40

.

Profit =

0.28C + 44

This converts the given percentage-based profit condition into an expression dependent only on

C

.

3

Set the two profit expressions equal to each other and solve the resulting single-variable linear equation.

0.12C = 24 \implies C = 200

Equating two valid expressions for the same quantity allows solving for the unknown variable

C

.

Anahtar Kavram

Solving Systems of Linear Equations by Algebraic Substitution

Tahmini Süre:2m 0s

Soru 179Soru →

If $k$ is an integer such that $|2k - 7| \le 11$ and $|k + 2| > 4$ , how many possible values of $k$ exist?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The total number of possible integer values of

k

is 7.

Unfolding

|2k - 7| \le 11

gives

-2 \le k \le 9

, representing 12 possible integers. Unfolding

|k + 2| > 4

gives

k > 2

or

k < -6

. Intersecting these two regions for real integers eliminates

k < -6

(since

k \ge -2

) and restricts the set to

3 \le k \le 9

. The integers in this range are

3, 4, 5, 6, 7, 8,

and

9

, yielding a total of 7 valid values.

Adım Adım Çözüm

1

Solve the first absolute value inequality

|2k - 7| \le 11

.

-2 \le k \le 9

Expanding the absolute value gives

-11 \le 2k - 7 \le 11

. Adding 7 to all parts yields

-4 \le 2k \le 18

, and dividing by 2 results in

-2 \le k \le 9

.

2

Solve the second absolute value inequality

|k + 2| > 4

.

k > 2

or

k < -6

Expanding the strict absolute value inequality gives two cases:

k + 2 > 4 \implies k > 2

, or

k + 2 < -4 \implies k < -6

.

3

Find the intersection of the two solution sets for integer values of

k

.

3 \le k \le 9

Since the first inequality requires

k \ge -2

, no integer can satisfy both

k \ge -2

and

k < -6

. Thus,

k

must satisfy

-2 \le k \le 9

and

k > 2

, which reduces to

3 \le k \le 9

.

4

Count the number of integers in the range

3 \le k \le 9

.

7

The valid integers in this range are

3, 4, 5, 6, 7, 8,

and

9

. The total count is

9 - 3 + 1 = 7

.

Anahtar Kavram

Solving compound absolute value inequalities for integer solution counts

Soru 180Soru →

A chemical processing plant blends Solution X, which contains $25\%$ acid by volume, with Solution Y, which contains $65\%$ acid by volume, to create a $100$ -liter mixture that is $40\%$ acid by volume. If Solution X costs $\$3$ per liter and Solution Y costs $\$7$ per liter, what is the total cost of the solutions used to produce the mixture?

\$450

\$500

\$530

\$540

\$550

Cevabı ve açıklamayı göster

Cevap: $\$450$

Cevap

The total cost of the solutions used to produce the mixture is

\$450

.

To determine the total cost, first solve for the exact volume of each solution needed. Let

x

be the volume of Solution X in liters and

y

be the volume of Solution Y in liters. Since the total mixture volume is

100

liters,

x + y = 100

, so

y = 100 - x

. Setting up the acid balance equation gives

0.25x + 0.65y = 0.40(100)

. Substituting

y = 100 - x

yields

0.25x + 0.65(100 - x) = 40

, which simplifies to

-0.40x + 65 = 40

, leading to

0.40x = 25

and

x = 62.5

. Thus,

62.5

liters of Solution X and

37.5

liters of Solution Y are required. Multiplying each volume by its price per liter gives

3(62.5) + 7(37.5) = 187.50 + 262.50 = 450

. The total cost is

\$450

.

Adım Adım Çözüm

1

Define variables and write the equation for total volume.

Let

x

be the volume of Solution X in liters and

y

be the volume of Solution Y in liters. Then

x + y = 100

, which implies

y = 100 - x

.

The total required volume of the mixture is

100

liters.

2

Write the acid balance equation and solve for

x

and

y

.

Equation:

0.25x + 0.65(100 - x) = 0.40(100) \implies 0.25x + 65 - 0.65x = 40 \implies -0.40x = -25 \implies x = 62.5

liters. Thus,

y = 100 - 62.5 = 37.5

liters.

The total volume of pure acid in the final mixture must equal the sum of the pure acid contributed by Solution X and Solution Y.

3

Calculate the total cost of the mixture components.

Total Cost =

3(62.5) + 7(37.5) = 187.5 + 262.5 = 450

.

Multiply the volume of each solution by its respective price per liter and sum the results.

Anahtar Kavram

Algebraic Modeling of Mixture Problems and Systems of Linear Equations

Tahmini Süre:2m 0s

Algebra and Functions

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram