Algebra and Functions Practice Questions — GMAT — Page 6

Question 101Question →

For all real numbers $a$ and $b$ such that $a + b \neq 0$ , the custom operator $\star$ is defined by $a \star b = \frac{a - b}{a + b}$ . What is the value of $(3 \star 1) \star 2$ ?

-\frac{5}{3}

-\frac{3}{5}

0

\frac{3}{5}

\frac{5}{3}

Show answer & explanation

Answer: $-\frac{3}{5}$

Answer

-\frac{3}{5}

Evaluating the expression step-by-step according to parentheses yields

3 \star 1 = \frac{3-1}{3+1} = \frac{1}{2}

. Substituting

\frac{1}{2}

as the first input and

2

as the second input gives

\frac{1/2 - 2}{1/2 + 2} = \frac{-3/2}{5/2} = -\frac{3}{5}

.

Step-by-Step Solution

1

Evaluate the inner custom operation inside parentheses:

3 \star 1

.

3 \star 1 = \frac{3 - 1}{3 + 1} = \frac{2}{4} = \frac{1}{2}

.

Follow the order of operations by resolving the grouped expression first using the definition

a \star b = \frac{a - b}{a + b}

with

a = 3

and

b = 1

.

2

Substitute the result

\frac{1}{2}

back into the main expression to compute

(\frac{1}{2}) \star 2

.

(\frac{1}{2}) \star 2 = \frac{\frac{1}{2} - 2}{\frac{1}{2} + 2}

.

Apply the definition of the custom operator again, where the left operand is

\frac{1}{2}

and the right operand is

2

.

3

Simplify the complex fraction.

\frac{\frac{1}{2} - \frac{4}{2}}{\frac{1}{2} + \frac{4}{2}} = \frac{-\frac{3}{2}}{\frac{5}{2}} = -\frac{3}{5}.

Combine the fractions in the numerator and denominator, then divide.

Key Concept

Custom Operators and Order of Operations

Estimated Time:1m 30s

Question 102Question →

The quadratic equation $x^2 - 6x + a = 0$ has two distinct real roots $\alpha$ and $\gamma$ , and the quadratic equation $y^2 - by + 21 = 0$ has two distinct real roots $\beta$ and $\delta$ . If $\alpha < \beta < \gamma < \delta$ and the four roots form an arithmetic progression in that order, what is the value of $a + b$ ?

Show answer & explanation

Answer: 15

Answer

The value of

a + b

is 15.

By representing the four ordered roots as

r, r+d, r+2d, r+3d

, Vieta's formula for the sum of roots of the first equation yields

r + (r+2d) = 2(r+d) = 6

, which implies

r+d = 3

. Thus, the second root is

\beta = 3

. Using the product of roots for the second equation,

3\delta = 21

gives

\delta = 7

. The common difference is

d = (7-3)/2 = 2

, which gives the roots

1, 3, 5, 7

. Finally,

a = 1 \times 5 = 5

and

b = 3 + 7 = 10

, so

a + b = 15

.

Step-by-Step Solution

1

Apply Vieta's formulas to both quadratic equations.

\alpha + \gamma = 6

,

\alpha\gamma = a

,

\beta + \delta = b

, and

\beta\delta = 21

.

Vieta's relations connect the coefficients of a quadratic polynomial to the sum and product of its roots.

2

Set up the arithmetic progression representation for the roots.

\alpha = r

,

\beta = r + d

,

\gamma = r + 2d

, and

\delta = r + 3d

, where

d > 0

.

The roots form an increasing arithmetic progression in the order

\alpha, \beta, \gamma, \delta

.

3

Substitute the expressions for

\alpha

and

\gamma

into

\alpha + \gamma = 6

.

r + (r + 2d) = 2r + 2d = 2(r + d) = 6 \implies r + d = 3

.

Combining terms simplifies the sum of the first and third terms of the arithmetic progression.

4

Identify the value of

\beta

and solve for

\delta

.

\beta = r + d = 3

, so

\beta\delta = 21 \implies 3\delta = 21 \implies \delta = 7

.

Since

\beta = r + d

, its value is directly determined as 3, allowing

\delta

to be solved from the product relation.

5

Calculate the common difference

d

and the first term

r

.

\delta - \beta = 2d = 7 - 3 = 4 \implies d = 2

, and

r = 3 - 2 = 1

.

The difference between the fourth and second terms of an AP is equal to

2d

.

6

Find

a

,

b

, and their sum

a + b

.

a = \alpha\gamma = 1 \times 5 = 5

,

b = \beta + \delta = 3 + 7 = 10

, so

a + b = 5 + 10 = 15

.

With all four roots determined (

1, 3, 5, 7

), the missing coefficients are calculated using Vieta's formulas.

Key Concept

Combining Vieta's Formulas with Arithmetic Progressions to Solve Quadratic Systems

Question 103Question →

For any real numbers $u$ and $v$ , the binary operator $\star$ is defined by $u \star v = 2u^2 - 3v$ . A function $f$ is defined by $f(x) = x \star 4$ , and a function $g$ is defined by $g(x) = 3x + 1$ . If $m$ is a real number such that $f(g(m)) = 38$ , what is the product of all possible values of $m$ ?

-\frac{8}{3}

-1

-\frac{16}{9}

\frac{4}{3}

\frac{8}{3}

Show answer & explanation

Answer: $-\frac{8}{3}$

Answer

The product of all possible values of

m

is

-\frac{8}{3}

.

First, evaluate

f(x) = x \star 4 = 2x^2 - 3(4) = 2x^2 - 12

. Next, substitute

g(m) = 3m + 1

into

f(x)

to get

f(g(m)) = 2(3m + 1)^2 - 12

. Setting this expression equal to

38

gives

2(3m + 1)^2 - 12 = 38

, which simplifies to

(3m + 1)^2 = 25

. Taking the square root gives two possible linear equations:

3m + 1 = 5

(which gives

m = \frac{4}{3}

) and

3m + 1 = -5

(which gives

m = -2

). Multiplying these two solutions yields

\left(\frac{4}{3}\right) \times (-2) = -\frac{8}{3}

. Thus, the option equal to

-\frac{8}{3}

is correct.

Step-by-Step Solution

1

Evaluate the function

f(x)

using the definition of the custom operator

\star

.

f(x) = x \star 4 = 2x^2 - 3(4) = 2x^2 - 12

Substitute

u = x

and

v = 4

into the formula

u \star v = 2u^2 - 3v

.

2

Express the nested function

f(g(m))

in terms of

m

.

f(g(m)) = 2(g(m))^2 - 12 = 2(3m + 1)^2 - 12

Substitute

g(m) = 3m + 1

into

f(x)

.

3

Set

f(g(m))

equal to

38

and solve for

(3m + 1)^2

.

2(3m + 1)^2 - 12 = 38 \implies 2(3m + 1)^2 = 50 \implies (3m + 1)^2 = 25

Isolate the squared binomial term using basic algebraic manipulation.

4

Take the square root of both sides to find all possible values of

m

.

3m + 1 = 5 \implies m = \frac{4}{3}

or

3m + 1 = -5 \implies m = -2

A positive real number has both positive and negative square roots.

5

Calculate the product of the two solutions for

m

.

\left(\frac{4}{3}\right) \times (-2) = -\frac{8}{3}

Multiply the two roots together as requested by the question stem.

Key Concept

Evaluating custom binary operators and nested composite functions, solving quadratic equations, and finding products of roots.

Question 104Question →

If $(x + 3)^2 = 25$ and $x > 0$ , what is the value of $x^2 - 1$ ?

3

4

15

24

63

Show answer & explanation

Answer: 3

Answer

3

Taking the square root of both sides of

(x + 3)^2 = 25

gives

x + 3 = \pm 5

. Since

x > 0

, we choose

x + 3 = 5

, which gives

x = 2

. Substituting

x = 2

into

x^2 - 1

produces

2^2 - 1 = 3

.

Step-by-Step Solution

1

Take the square root of both sides of the given quadratic equation

x + 3 = 5

or

x + 3 = -5

The equation

(x + 3)^2 = 25

implies

x + 3

can be either positive or negative 5.

2

Solve for

x

under the constraint

x > 0

x = 2

From

x + 3 = 5

, we obtain

x = 2

. The second root

x = -8

is discarded because

x > 0

.

3

Substitute

x = 2

into the target expression

x^2 - 1

2^2 - 1 = 3

Evaluating

x^2 - 1

at

x = 2

yields

4 - 1 = 3

.

Key Concept

Solving Quadratic Equations by Taking Square Roots

Estimated Time:1m 0s

Question 105Question →

For all non-zero real numbers $x$ and $y$ , the custom operation $\star$ is defined by $x \star y = \frac{x}{y} - \frac{y}{x}$ . If $p$ and $q$ are positive numbers such that $p + q = 5$ and $p \star q = \frac{5}{6}$ , what is the value of $(p + 1) \star q$ ?

\frac{3}{2}

-\frac{3}{2}

\frac{24}{5}

0

\frac{21}{10}

Show answer & explanation

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

Answer

\frac{3}{2}

By simplifying

p \star q = \frac{p^2 - q^2}{pq} = \frac{(p - q)(p + q)}{pq}

, we substitute

p + q = 5

to get

\frac{5(p - q)}{pq} = \frac{5}{6}

, which reduces to

6(p - q) = pq

. Substituting

q = 5 - p

yields the quadratic equation

p^2 + 7p - 30 = 0

. Since

p > 0

, we find

p = 3

and

q = 2

. Evaluating

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

.

Step-by-Step Solution

1

Express the custom operation using algebraic fractions and factor the numerator

p \star q = \frac{p}{q} - \frac{q}{p} = \frac{p^2 - q^2}{pq} = \frac{(p - q)(p + q)}{pq}

Finding a common denominator converts the expression into a form where known identities can be substituted.

2

Substitute

p + q = 5

and

p \star q = \frac{5}{6}

into the simplified expression

\frac{5(p - q)}{pq} = \frac{5}{6} \implies \frac{p - q}{pq} = \frac{1}{6} \implies 6(p - q) = pq

Simplifying by dividing both sides by 5 establishes a direct relation between the difference and product of p and q.

3

Substitute

q = 5 - p

to form and solve a quadratic equation for

p

6(2p - 5) = p(5 - p) \implies 12p - 30 = 5p - p^2 \implies p^2 + 7p - 30 = 0 \implies (p + 10)(p - 3) = 0

Since p and q are positive numbers, p must be positive, giving p = 3 and q = 2.

4

Evaluate

(p + 1) \star q

with

p = 3

and

q = 2

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

Applying the custom operator definition to inputs 4 and 2 yields the final value.

Key Concept

Custom Operator Evaluation and Quadratic System Solving

Question 106Question →

What is the sum of all distinct real solutions to the equation

\frac{x^2 - 8x + 15}{x - 3} = (x - 5)^2

?

6

9

11

14

1

Show answer & explanation

Answer: 11

Answer

11

Factoring the numerator

x^2 - 8x + 15

gives

(x - 3)(x - 5)

. For all

x \neq 3

, the original equation simplifies to

x - 5 = (x - 5)^2

. Rearranging into standard factored form

(x - 5)(x - 6) = 0

yields two distinct real solutions:

x = 5

and

x = 6

. Neither value violates the restriction

x \neq 3

. Adding these values together yields

5 + 6 = 11

.

Step-by-Step Solution

1

Determine the domain restriction for the rational expression

The expression

\frac{x^2 - 8x + 15}{x - 3}

requires that the denominator

x - 3 \neq 0

, so

x \neq 3

.

Division by zero is undefined in real numbers.

2

Factor the numerator and simplify the left side of the equation

Since

x^2 - 8x + 15 = (x - 3)(x - 5)

, for

x \neq 3

,

\frac{(x - 3)(x - 5)}{x - 3} = x - 5

.

Canceling the common non-zero factor

(x - 3)

simplifies the rational equation.

3

Solve the simplified quadratic equation for

x

x - 5 = (x - 5)^2 \implies (x - 5)^2 - (x - 5) = 0 \implies (x - 5)(x - 5 - 1) = 0 \implies (x - 5)(x - 6) = 0

. Thus,

x = 5

or

x = 6

.

Factoring out the common factor

(x - 5)

preserves all valid solutions.

4

Verify solutions against the domain restriction and calculate the sum

Both

x = 5

and

x = 6

satisfy

x \neq 3

. The sum of all distinct real solutions is

5 + 6 = 11

.

Combining the valid distinct roots gives the required total.

Key Concept

Polynomial factoring, domain restrictions on rational expressions, and avoiding root loss when solving quadratic equations.

Estimated Time:2m 0s

Question 107Question →

A rectangular garden has an area of $48$ square meters. If the length of the garden is $2$ meters greater than its width, what is the perimeter of the garden, in meters?

Show answer & explanation

Answer: 28

Answer

The perimeter of the garden is 28 meters.

By setting up the area equation

w(w + 2) = 48

, we obtain the quadratic equation

w^2 + 2w - 48 = 0

. Factoring gives

(w + 8)(w - 6) = 0

. Since width must be positive,

w = 6

meters. The length is

6 + 2 = 8

meters. Thus, the perimeter is

2(6 + 8) = 28

meters.

Step-by-Step Solution

1

Define variables for width and length.

Width =

w

, Length =

w + 2

.

The length is given as 2 meters greater than the width.

2

Formulate and rearrange the quadratic equation for area.

w^2 + 2w - 48 = 0

Area is length multiplied by width, set equal to 48.

3

Factor the quadratic polynomial.

(w + 8)(w - 6) = 0

Find two numbers that multiply to -48 and add up to +2.

4

Determine the valid physical dimension.

w = 6

meters and length

l = 8

meters.

A physical dimension cannot be negative, so

w = -8

is discarded.

5

Compute the perimeter.

Perimeter =

2(6 + 8) = 28

meters.

Perimeter of a rectangle is twice the sum of its length and width.

Key Concept

Solving quadratic equations by factoring in word problem contexts.

Question 108Question →

For any real number $x$ , the function $h$ is defined by $h(x) = 3x^2 - 5x + 4$ . What is the value of $h(3)$ ?

Show answer & explanation

Answer: 16

Answer

16

To find

h(3)

, substitute

x = 3

into the function

h(x) = 3x^2 - 5x + 4

. First, compute

3^2 = 9

. Multiplying by 3 yields 27. Subtracting

5(3) = 15

gives 12, and adding 4 results in the final value of 16.

Step-by-Step Solution

1

Substitute the value

x = 3

into the definition of

h(x)

.

h(3) = 3(3)^2 - 5(3) + 4

To evaluate a function at a specific value, replace all instances of the input variable with that value.

2

Evaluate the exponent and carry out the multiplications.

h(3) = 3(9) - 15 + 4 = 27 - 15 + 4

Follow standard order of operations: compute powers before performing multiplication.

3

Perform the remaining addition and subtraction.

h(3) = 16

Complete the calculation from left to right.

Key Concept

Function Evaluation

Question 109Question →

If $x^2 - 9x + 18 = 0$ , and $x_1$ and $x_2$ are the real roots of the equation such that $x_1 > x_2$ , what is the value of $2x_1 - x_2$ ?

Show answer & explanation

Answer: 9

Answer

The value of

2x_1 - x_2

is 9.

Factoring the quadratic equation

x^2 - 9x + 18 = 0

yields

(x - 6)(x - 3) = 0

. The roots are

x = 6

and

x = 3

. Given that

x_1 > x_2

, we assign

x_1 = 6

and

x_2 = 3

. Evaluating

2x_1 - x_2

gives

2(6) - 3 = 9

.

Step-by-Step Solution

1

Factor the quadratic expression

(x - 6)(x - 3) = 0

Find two numbers that multiply to

18

and add to

-9

, which are

-6

and

-3

.

2

Solve for the roots of the equation

x = 6

or

x = 3

Set each factor equal to zero:

x - 6 = 0 \implies x = 6

and

x - 3 = 0 \implies x = 3

.

3

Assign root values based on the condition

x_1 > x_2

x_1 = 6

and

x_2 = 3

Since

6 > 3

, the larger root is assigned to

x_1

and the smaller root to

x_2

.

4

Evaluate the target expression

9

Substitute

x_1 = 6

and

x_2 = 3

into

2x_1 - x_2 = 2(6) - 3 = 12 - 3 = 9

.

Key Concept

Solving Quadratic Equations by Factoring

Question 110Question →

For all real numbers $x$ and $y$ , the custom operation $\Delta$ is defined by $x \Delta y = (x + 1)(y - 2)$ . If the function $g$ is defined by $g(n) = n^2 - 3$ for all real numbers $n$ , what is the value of $g(3) \Delta g(2)$ ?

Show answer & explanation

Answer: -7

Answer

The value of

g(3) \Delta g(2)

is

-7

.

Evaluating

g(3)

gives

3^2 - 3 = 6

, and evaluating

g(2)

gives

2^2 - 3 = 1

. Substituting these values as

x = 6

and

y = 1

into the custom operator definition

x \Delta y = (x + 1)(y - 2)

yields

(6 + 1)(1 - 2) = 7 \times (-1) = -7

.

Step-by-Step Solution

1

Evaluate the inner function

g(n)

at

n = 3

g(3) = 3^2 - 3 = 9 - 3 = 6

Before evaluating the custom operator, the inputs

g(3)

and

g(2)

must be explicitly calculated.

2

Evaluate the inner function

g(n)

at

n = 2

g(2) = 2^2 - 3 = 4 - 3 = 1

Determine the second operand for the custom operation.

3

Apply the custom operation

\Delta

to the results from Step 1 and Step 2

6 \Delta 1 = (6 + 1)(1 - 2) = 7 \times (-1) = -7

Substitute

x = 6

and

y = 1

into the formula

x \Delta y = (x + 1)(y - 2)

.

Key Concept

Evaluating functions and applying non-commutative custom operators

Question 111Question →

For how many integer values of $k$ does the quadratic equation $(k - 2)x^2 - 2(k + 1)x + (k + 3) = 0$ have two distinct real roots $\alpha$ and \beta $such that$ \frac{1}{\alpha} + \frac{1}{\beta} < 1$?

2

3

5

6

7

Show answer & explanation

Answer: 3

Answer

There are exactly 3 integer values of

k

that satisfy all given conditions.

To find the number of valid integer values for

k

, we apply the condition for distinct real roots (

D > 0

), which yields

k > -7

with

k \neq 2

. Expressing the sum of reciprocals as

\frac{\alpha + \beta}{\alpha\beta} = \frac{2(k+1)}{k+3} < 1

, solving

\frac{k-1}{k+3} < 0

yields

-3 < k < 1

. The integer values within this range are

-2, -1,

and

0

, giving a total of 3 valid integer values.

Step-by-Step Solution

1

Ensure the given equation is a valid quadratic equation and determine the condition for two distinct real roots.

For the equation to be quadratic, the coefficient of

x^2

must be non-zero, so

k - 2 \neq 0 \implies k \neq 2

. For distinct real roots, the discriminant

D > 0

.

D = [-2(k+1)]^2 - 4(k-2)(k+3) = 4(k^2 + 2k + 1) - 4(k^2 + k - 6) = 4(k + 7)

. Thus,

4(k+7) > 0 \implies k > -7

.

A quadratic equation has two distinct real roots if and only if its discriminant is strictly positive and the leading coefficient is non-zero.

2

Apply Vieta's formulas to express the sum and product of the roots in terms of

k

.

\alpha + \beta = \frac{2(k+1)}{k-2}

and

\alpha\beta = \frac{k+3}{k-2}

. Note that for

\frac{1}{\alpha} + \frac{1}{\beta}

to exist, neither root can be 0, which requires

\alpha\beta
eq 0 \implies k
eq -3$.

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.

3

Simplify the condition

\frac{1}{\alpha} + \frac{1}{\beta} < 1

using Vieta's relations.

\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{\frac{2(k+1)}{k-2}}{\frac{k+3}{k-2}} = \frac{2(k+1)}{k+3}

. We need

\frac{2k+2}{k+3} < 1$.

Combining the fractions over a common denominator allows direct substitution of the root sum and product.

4

Solve the algebraic inequality

\frac{2k+2}{k+3} < 1

.

\frac{2k+2}{k+3} - 1 < 0 \implies \frac{(2k+2) - (k+3)}{k+3} < 0 \implies \frac{k-1}{k+3} < 0

. The fraction is negative when the numerator and denominator have opposite signs, which gives the interval

-3 < k < 1$.

Subtracting 1 to get 0 on one side prevents sign errors that occur when cross-multiplying variable expressions.

5

Combine all constraints on

k

and count the valid integer values.

Constraints on

k

: (1)

k > -7

, (2)

k \neq 2

, (3)

k \neq -3

, and (4)

-3 < k < 1

. Intersecting these gives

-3 < k < 1

. The integer values in this open interval are

k = -2, -1, 0

. Total count = 3.

All structural and domain restrictions must be satisfied simultaneously.

Key Concept

Quadratic Equations and Polynomial Factoring

Estimated Time:3m 0s

Question 112Question →

If $x = 1$ is a root of the cubic polynomial $P(x) = x^3 + ax^2 - 11x + 12$ , where $a$ is a constant, what is the value of the largest real root of $P(x)$ ?

Show answer & explanation

Answer: 4

Answer

The largest real root of the polynomial

P(x)

is

4

.

Evaluating

P(1) = 0

yields

1 + a - 11 + 12 = 0

, giving

a = -2

. Factoring the polynomial

x^3 - 2x^2 - 11x + 12

by

(x - 1)

produces the quadratic expression

x^2 - x - 12

. Factoring this quadratic gives

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

. The roots of the polynomial are

-3

,

1

, and

4

. Comparing these values, the maximum root is

4

.

Step-by-Step Solution

1

Find the constant

a

using the given root

x = 1

.

a = -2

Since

x = 1

is a root,

P(1) = 1^3 + a(1)^2 - 11(1) + 12 = 0

, which simplifies to

a + 2 = 0

.

2

Factor

(x - 1)

out of

P(x) = x^3 - 2x^2 - 11x + 12

.

P(x) = (x - 1)(x^2 - x - 12)

Dividing

x^3 - 2x^2 - 11x + 12

by

(x - 1)

yields the quotient

x^2 - x - 12

.

3

Factor the quadratic factor

x^2 - x - 12

.

(x - 4)(x + 3)

The quadratic factors into two linear terms whose roots are

4

and

-3

.

4

Determine the largest of all real roots of

P(x)

.

4

The complete set of real roots is

\{-3, 1, 4\}

, of which

4

is the greatest.

Key Concept

Polynomial Factoring and Remainder/Factor Theorem

Question 113Question →

A specialty coffee shop sells two types of coffee beans: Type A for $\$8$ per pound and Type B for $\$12$ per pound. A customer buys a total of $x$ pounds of coffee, consisting of $a$ pounds of Type A and $b$ pounds of Type B, for a total cost of $C$ dollars. Which of the following equations correctly model the relationship between these quantities? Select all that apply.

Select all that apply

a = x - b

C = 12x - 4a

b = \frac{C - 8a}{12}

C = 20x

a = \frac{C - 12}{8x}

Show answer & explanation

Answer: $a = x - b$ ; $C = 12x - 4a$ ; $b = \frac{C - 8a}{12}$

Answer

The correct formulations are

a = x - b

,

C = 12x - 4a

, and

b = \frac{C - 8a}{12}

.

The system is defined by two fundamental linear relationships: weight total

a + b = x

and cost total

8a + 12b = C

. Isolating

a

from the weight total gives

a = x - b

. Substituting

b = x - a

into the cost equation yields

C = 8a + 12(x - a) = 12x - 4a

. Isolating

b

from the cost equation gives

12b = C - 8a \implies b = \frac{C - 8a}{12}

. All three expressions are mathematically equivalent and correct.

Step-by-Step Solution

1

Set up the basic linear system from the word problem.

Total weight equation:

a + b = x

; Total cost equation:

8a + 12b = C

.

Word problems with total quantity and total monetary value translate into two distinct linear equations.

2

Rearrange the total weight equation to express

a

in terms of

x

and

b

.

a = x - b

.

Subtracting

b

from both sides isolates

a

.

3

Substitute

b = x - a

into the total cost equation.

C = 8a + 12(x - a) = 8a + 12x - 12a = 12x - 4a

.

Expressing cost in terms of a single variable

a

and total weight

x

simplifies multi-variable models.

4

Isolate

b

in the total cost equation.

12b = C - 8a \implies b = \frac{C - 8a}{12}

.

Subtracting the cost contribution of Type A (

8a

) and dividing by the unit cost of Type B (

12

) isolates

b

.

Key Concept

Linear Equation Modeling in Word Problems

Question 114Question →

If $x^2(2x - 3) = 9(2x - 3)$ , what is the sum of all distinct real solutions for $x$ ?

-1.5

0

1.5

3

4.5

Show answer & explanation

Answer: $1.5$

Answer

The sum of all distinct real solutions for

x

is

1.5

.

To find all solutions without losing roots, set the equation to zero:

x^2(2x - 3) - 9(2x - 3) = 0

. Factoring out

(2x - 3)

gives

(x^2 - 9)(2x - 3) = 0

, which fully factors into

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

. The distinct real roots are

3

,

-3

, and

1.5

. Adding these values together yields

3 + (-3) + 1.5 = 1.5

.

Step-by-Step Solution

1

Move all terms to one side of the equation to avoid losing solutions by dividing by a variable expression.

x^2(2x - 3) - 9(2x - 3) = 0

Dividing by an expression containing variables can eliminate valid roots.

2

Factor out the common term

(2x - 3)

.

(x^2 - 9)(2x - 3) = 0

Factoring allows application of the zero-product property.

3

Factor the quadratic term

x^2 - 9

as a difference of squares.

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

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

.

4

Set each factor to zero to find all real solutions.

x - 3 = 0 \implies x = 3

,

x + 3 = 0 \implies x = -3

, and

2x - 3 = 0 \implies x = 1.5

.

If the product of factors is zero, at least one factor must equal zero.

5

Sum all distinct real solutions.

3 + (-3) + 1.5 = 1.5

The question asks for the sum of all distinct real solutions.

Key Concept

Factoring Polynomial Equations and Avoiding Variable Cancellation

Estimated Time:1m 30s

Question 115Question →

For all non-zero real numbers $a$ and $b$ , the custom binary operator $\star$ is defined by $a \star b = \frac{a^2 - b^2}{ab}$ . A function $f$ is defined for all positive numbers $x$ by $f(x) = (x + 1) \star x$ . What is the value of $f(3) \star f(2)$ ?

-\frac{3}{70}

\frac{51}{70}

-\frac{51}{70}

-\frac{56}{45}

-6

Show answer & explanation

Answer: $-\frac{51}{70}$

Answer

-\frac{51}{70}

First, evaluate the inner function values:

f(3) = 4 \star 3 = \frac{4^2 - 3^2}{4 \cdot 3} = \frac{7}{12}

and

f(2) = 3 \star 2 = \frac{3^2 - 2^2}{3 \cdot 2} = \frac{5}{6} = \frac{10}{12}

. Next, evaluate the outer operation

\left(\frac{7}{12}\right) \star \left(\frac{10}{12}\right) = \frac{(7/12)^2 - (10/12)^2}{(7/12)(10/12)} = \frac{49 - 100}{70} = -\frac{51}{70}

. Hence, the correct response is

-\frac{51}{70}

.

Step-by-Step Solution

1

Evaluate

f(3)

using the definition of

f(x)

and the custom operator

\star

.

f(3) = (3 + 1) \star 3 = 4 \star 3 = \frac{4^2 - 3^2}{4 \times 3} = \frac{16 - 9}{12} = \frac{7}{12}

By definition,

f(x) = (x+1) \star x

. Setting

x=3

yields

4 \star 3

.

2

Evaluate

f(2)

using the definition of

f(x)

and the custom operator

\star

.

f(2) = (2 + 1) \star 2 = 3 \star 2 = \frac{3^2 - 2^2}{3 \times 2} = \frac{9 - 4}{6} = \frac{5}{6}

Setting

x=2

in

f(x)

yields

3 \star 2

.

3

Evaluate

f(3) \star f(2) = \left(\frac{7}{12}\right) \star \left(\frac{5}{6}\right)

.

\left(\frac{7}{12}\right) \star \left(\frac{10}{12}\right) = \frac{\left(\frac{7}{12}\right)^2 - \left(\frac{10}{12}\right)^2}{\left(\frac{7}{12}\right)\left(\frac{10}{12}\right)} = \frac{\frac{49 - 100}{144}}{\frac{70}{144}} = -\frac{51}{70}

Apply the rule

u \star v = \frac{u^2 - v^2}{uv}

with

u = \frac{7}{12}

and

v = \frac{10}{12}

.

Key Concept

Custom non-commutative binary operators and function evaluation

Estimated Time:2m 0s

Question 116Question →

Working together at their respective constant rates, Machine X and Machine Y can complete a specialized production order in $12$ hours. If Machine X works alone for $4$ hours and Machine Y works alone for $9$ hours, they complete $\frac{7}{12}$ of the total order. Machine Z operates at a constant rate that is $50\%$ greater than the rate of Machine X. How many hours would it take Machine Y and Machine Z working together to complete the entire order?

10

12

15

20

40

Show answer & explanation

Answer: $10$

Answer

10

hours

The correct answer is

10

hours. Solving the system of equations yields individual rates of

\frac{1}{30}

order/hour for Machine X and

\frac{1}{20}

order/hour for Machine Y. A

50\%

rate increase gives Machine Z a rate of

\frac{1}{20}

order/hour. Adding the rates of Machine Y and Machine Z yields a combined rate of

\frac{1}{10}

order/hour, which requires

10

hours to complete one job.

Step-by-Step Solution

1

Define variables and model the joint rate equation.

Let

x

be the rate of Machine X (order/hour) and

y

be the rate of Machine Y (order/hour). Their combined rate equation is

x + y = \frac{1}{12}

.

Working together for

12

hours completes

1

full order.

2

Set up the second equation based on the partial work completed.

4x + 9y = \frac{7}{12}

.

Machine X works for

4

hours and Machine Y works for

9

hours to complete

\frac{7}{12}

of the order.

3

Solve the system of linear equations for

x

and

y

.

Express

4x + 9y

as

4(x + y) + 5y = \frac{7}{12}

. Substituting

x + y = \frac{1}{12}

gives

4\left(\frac{1}{12}\right) + 5y = \frac{7}{12} \implies \frac{4}{12} + 5y = \frac{7}{12} \implies 5y = \frac{3}{12} = \frac{1}{4} \implies y = \frac{1}{20}

. Subsequently,

x = \frac{1}{12} - \frac{1}{20} = \frac{5 - 3}{60} = \frac{2}{60} = \frac{1}{30}

.

Determines the individual work rates of Machine X and Machine Y.

4

Calculate the work rate of Machine Z.

Rate of Machine Z =

1.5 \times x = 1.5 \times \frac{1}{30} = \frac{3}{2} \times \frac{1}{30} = \frac{1}{20}

.

Machine Z is

50\%

faster than Machine X.

5

Calculate the combined rate and total time for Machine Y and Machine Z.

Combined rate =

y + z = \frac{1}{20} + \frac{1}{20} = \frac{2}{20} = \frac{1}{10}

. Total time =

\frac{1}{\frac{1}{10}} = 10

hours.

The inverse of the combined rate gives the total time required to finish one job.

Key Concept

Work Rates and Simultaneous Linear Modeling

Estimated Time:2m 30s

Question 117Question →

For all real numbers $a$ and $b$ , the custom binary operation $\star$ is defined by $a \star b = a^2 - 2b^2 + ab$ . The function $g$ is defined by $g(x) = x \star 3$ . What is the positive value of $x$ for which $g(x) = 10$ ?

4

1

7

\sqrt{28}

3

Show answer & explanation

Answer: 4

Answer

The positive value of

x

for which

g(x) = 10

is

4

.

Evaluating

g(x) = x \star 3

gives

x^2 - 2(3)^2 + x(3) = x^2 + 3x - 18

. Setting this expression equal to

10

produces the quadratic equation

x^2 + 3x - 28 = 0

. Factoring yields

(x + 7)(x - 4) = 0

, giving solutions

x = -7

and

x = 4

. Since

x

must be positive, the correct value is

4

.

Step-by-Step Solution

1

Apply the definition of the custom binary operator

\star

to find

g(x)

.

g(x) = x \star 3 = x^2 - 2(3)^2 + x(3) = x^2 - 18 + 3x = x^2 + 3x - 18

Substitute

a = x

and

b = 3

into the given formula

a \star b = a^2 - 2b^2 + ab

.

2

Set the function expression equal to

10

and rearrange into standard quadratic form.

x^2 + 3x - 18 = 10 \implies x^2 + 3x - 28 = 0

Subtract

10

from both sides to form a quadratic equation equal to zero.

3

Factor the quadratic equation to find the values of

x

.

(x + 7)(x - 4) = 0 \implies x = -7 \text{ or } x = 4

Find two numbers that multiply to

-28

and add to

3

, which are

7

and

-4

.

4

Select the positive value of

x

as specified in the stem.

x = 4

The question asks specifically for the positive value of

x

.

Key Concept

Evaluating non-commutative custom operators and solving polynomial equations resulting from function evaluations.

Estimated Time:2m 0s

Question 118Question →

A local theater sold a total of $120$ tickets for a weekend performance, consisting only of adult tickets and student tickets. Adult tickets were sold for $\$15$ each, and student tickets were sold for $\$10$ each. If the total revenue from the ticket sales was $\$1{}500$ , how many adult tickets were sold?

Show answer & explanation

Answer: 60

Answer

The correct answer is 60 adult tickets.

By setting up the total ticket equation

a + s = 120

and total revenue equation

15a + 10s = 1500

, substituting

s = 120 - a

gives

15a + 10(120 - a) = 1500

. Simplifying yields

5a = 300

, so

a = 60

. Thus, 60 adult tickets were sold.

Step-by-Step Solution

1

Set up the linear equation in one variable.

15a + 10(120 - a) = 1500

Express the number of student tickets as

(120 - a)

so the revenue equation relies on a single variable.

2

Expand and combine like terms.

5a + 1200 = 1500

Distribute

10

across

(120 - a)

to simplify the equation.

3

Isolate the variable

a

.

a = 60

Subtract

1200

from both sides and divide by

5

to find the number of adult tickets.

Key Concept

Algebraic Modeling of Linear Systems in Two Variables

Estimated Time:1m 0s

Question 119Question →

How many integer values of $x$ satisfy the inequality $|x - 3| \leq 4$ ?

Show answer & explanation

Answer: 9

Answer

There are 9 integer values of

x

that satisfy the inequality.

Unfolding

|x - 3| \leq 4

yields

-4 \leq x - 3 \leq 4

. Adding 3 across all terms gives

-1 \leq x \leq 7

. The integer values satisfying this range are

-1, 0, 1, 2, 3, 4, 5, 6, 7

, which totals 9 values.

Step-by-Step Solution

1

Convert the absolute value inequality into a compound linear inequality.

-4 \leq x - 3 \leq 4

The absolute value inequality

|k| \leq c

(for

c \geq 0

) is equivalent to

-c \leq k \leq c

.

2

Isolate

x

by adding 3 across all parts of the inequality.

-1 \leq x \leq 7

Adding the same constant to all parts of an inequality maintains the inequality relationships.

3

Count the total number of integer values within the inclusive range

[-1, 7]

.

9

The number of integers in an inclusive range

[a, b]

is calculated as

b - a + 1

, which gives

7 - (-1) + 1 = 9

.

Key Concept

Solving absolute value inequalities and counting integer solutions within a bounded interval

Question 120Question →

For all real numbers $a$ and $b$ , the custom binary operator $\star$ is defined by $a \star b = 2a - b$ . The function $f$ is defined for all real numbers by $f(x) = x^2 - 1$ . If $(x \star 3) \star f(3) = 6$ , what is the value of $x$ ?

-2

4

5

6

3

Show answer & explanation

Answer: 5

Answer

5

First, evaluate the function

f(3) = 3^2 - 1 = 8

. Next, evaluate the first custom operation

x \star 3 = 2x - 3

. Substituting these into the overall expression yields

(2x - 3) \star 8 = 2(2x - 3) - 8 = 4x - 14

. Setting this equal to 6 gives

4x - 14 = 6

, which simplifies to

4x = 20

, so

x = 5

.

Step-by-Step Solution

1

Evaluate the inner function

f(3)

f(3) = 3^2 - 1 = 9 - 1 = 8

Substitute

x = 3

into the given definition

f(x) = x^2 - 1

.

2

Evaluate the inner custom operation

x \star 3

x \star 3 = 2x - 3

Apply the custom operator rule

a \star b = 2a - b

with

a = x

and

b = 3

.

3

Evaluate the outer custom operation

(x \star 3) \star f(3)

(2x - 3) \star 8 = 2(2x - 3) - 8 = 4x - 6 - 8 = 4x - 14

Substitute the results from Step 1 and Step 2 into the custom operator definition.

4

Solve the algebraic equation for

x

4x - 14 = 6 \implies 4x = 20 \implies x = 5

Set the simplified expression equal to 6 as given in the problem stem and isolate

x

.

Key Concept

Custom Operator Evaluation and Nested Functions

Estimated Time:1m 30s

Algebra and Functions

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