Algebra Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB)

Question 1Question →

Consider the algebraic expression: $3x^2 + 6x - 24$ . When this expression is factored completely, which of the following represents the correct product?

3(x + 4)(x - 2)

3(x - 4)(x + 2)

(3x + 12)(x - 2)

3(x + 4)(x + 2)

Show answer & explanation

Answer: $3(x + 4)(x - 2)$

Answer

The completely factored form is

3(x + 4)(x - 2)

.

To factor the expression completely, you must first factor out the greatest common factor (GCF) of 3 from all terms. This leaves

3(x^2 + 2x - 8)

. Next, you factor the quadratic trinomial inside the parentheses. You need two numbers that multiply to -8 and add to +2. Those numbers are +4 and -2. Therefore, combining the GCF and the binomials gives the completely factored form:

3(x + 4)(x - 2)

.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) from all three terms in the expression

3x^2 + 6x - 24

.

The GCF is 3. Factoring it out yields

3(x^2 + 2x - 8)

.

Factoring out the GCF simplifies the remaining trinomial, making it easier to factor and ensuring the final answer is factored completely.

2

Factor the remaining quadratic trinomial

x^2 + 2x - 8

by finding two integers that multiply to the constant term (-8) and add up to the middle coefficient (2).

The correct integers are 4 and -2 (since 4 * -2 = -8 and 4 + -2 = 2). This gives the binomial factors

(x + 4)(x - 2)

.

This process decomposes the monic quadratic trinomial into its constituent binomial factors.

3

Combine the GCF and the binomial factors to write the final factored expression.

The final product is

3(x + 4)(x - 2)

.

This represents the original expression as a product of its simplest prime factors.

Key Concept

Factoring Polynomials using GCF and Trinomial Decomposition

Question 2Question →

Find the greatest integer value of $x$ that makes the following inequality true:

\frac{2}{3}(6x - 9) - 4(2x + 1) > \frac{1}{4}(8x - 20) - 3x

Show answer & explanation

Answer: -2

Answer

The greatest integer value that satisfies the inequality is -2.

Expanding both sides yields

-4x - 10 > -x - 5

. Isolating

x

on one side gives

-3x > 5

. Dividing by -3 requires flipping the inequality symbol, resulting in

x < -\frac{5}{3}

(or

x < -1.67

). The greatest integer strictly less than -1.67 is -2.

Step-by-Step Solution

1

Distribute the constants and fractions into the parentheses on both sides.

4x - 6 - 8x - 4 > 2x - 5 - 3x

Applying the distributive property removes the parentheses so like terms can be combined.

2

Combine like terms on each side of the inequality.

-4x - 10 > -x - 5

Simplifying both expressions makes it easier to isolate the variable.

3

Add

x

to both sides and add

10

to both sides to separate the variable terms from the constant terms.

-3x > 5

This isolates the

x

term on one side of the inequality.

4

Divide both sides by -3 and flip the inequality sign.

x < -\frac{5}{3}

When multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed.

5

Identify the greatest integer less than

-\frac{5}{3}

.

Since

-\frac{5}{3} \approx -1.67

, the integers strictly less than this are

-2, -3, -4, \dots

The greatest of these is

-2

.

The question specifically asks for the greatest integer value that makes the inequality true.

Key Concept

Solving complex linear inequalities involves applying the distributive property, combining like terms, and reversing the inequality symbol when dividing or multiplying by a negative number.

Question 3Question →

Evaluate whether the following statement is true or false:

The only positive integer solutions to the linear inequality

-\frac{3}{5}\left(\frac{10}{3}x - 15\right) + 4(x - 2) \geq -\frac{5}{2}(-2x + 4) + 2

are 1, 2, and 3.

True

False

Show answer & explanation

Answer: True

Answer

The statement is True.

By successfully distributing the fractional negative coefficients and combining like terms, the inequality simplifies to

-3x \geq -9

. Dividing by

-3

requires reversing the inequality to

x \leq 3

. The positive integers in this solution set are exactly 1, 2, and 3, meaning the statement is factually correct.

Step-by-Step Solution

1

Distribute the negative fractions and constants on the left side of the inequality.

(-2x + 9) + (4x - 8)

Multiplying

-\frac{3}{5}

by

\frac{10}{3}x

gives

-2x

, and

-\frac{3}{5}

by

-15

gives

+9

. Multiplying

4

by

(x-2)

gives

4x - 8

.

2

Distribute the negative fraction on the right side of the inequality.

(5x - 10) + 2

Multiplying

-\frac{5}{2}

by

-2x

gives

5x

, and

-\frac{5}{2}

by

4

gives

-10

.

3

Combine the like terms on both sides.

2x + 1 \geq 5x - 8

On the left side,

-2x + 4x = 2x

and

9 - 8 = 1

. On the right side,

-10 + 2 = -8

.

4

Isolate the variable

x

by subtracting

5x

and

1

from both sides.

-3x \geq -9

Grouping all variables on one side and constants on the other prepares the equation for the final division.

5

Divide both sides by

-3

and reverse the inequality symbol.

x \leq 3

Dividing an inequality by a negative number requires flipping the inequality direction. Since

x \leq 3

, the only integers greater than zero that satisfy this are 1, 2, and 3.

Key Concept

Solving complex linear inequalities by distributing negative fractions, combining like terms, and properly applying the inequality sign flip rule.

Alternative Method

You can verify the statement by plugging the boundary values into the original inequality. Substitute

x=3

(which evaluates to

7 \geq 7

, a true statement),

x=4

(which evaluates to

9 \geq 12

, a false statement), and recognize that since it is a linear relation, solutions are valid downwards. Zero is not a positive integer, leaving 1, 2, and 3 as the only valid positive integers.

Estimated Time:2m 30s

Question 4Question →

Determine the missing constants in the completely factored form of the given polynomial.

Fill in the blanks below

The polynomial

5x^2 - 5x - 60

can be factored completely. When written in the form

a(x - b)(x + c)

, where

b

and

c

are positive integers, the value of

a

is , the value of

b

is , and the value of

c

is .

Show answer & explanation

Answer

The constants are a = 5, b = 4, and c = 3.

The correct values are found by first factoring out the greatest common factor of

5

, leaving the quadratic expression

x^2 - x - 12

. This quadratic expression factors into two binomials by finding two numbers that multiply to

-12

and add to

-1

, which are

-4

and

3

. This yields the fully factored form

5(x - 4)(x + 3)

. By matching this result to the required algebraic structure

a(x - b)(x + c)

, we can directly identify that

a = 5

,

b = 4

, and

c = 3

.

Step-by-Step Solution

1

Find the greatest common factor (GCF) of the terms in the polynomial

5x^2 - 5x - 60

.

The GCF is

5

. Factoring it out gives

5(x^2 - x - 12)

.

Extracting the GCF simplifies the trinomial and makes it easier to factor into binomials.

2

Factor the remaining quadratic trinomial

x^2 - x - 12

.

We look for two integers that multiply to

-12

and add to

-1

.

This is the standard procedure for factoring a monic quadratic trinomial of the form

x^2 + bx + c

.

3

Identify the proper integers and write the binomial factors.

The integers are

-4

and

3

. The factored trinomial is

(x - 4)(x + 3)

.

(-4) \times 3 = -12

and

-4 + 3 = -1

.

4

Combine the factors and match them to the target form

a(x - b)(x + c)

.

The completely factored expression is

5(x - 4)(x + 3)

. Matching this to

a(x - b)(x + c)

yields

a = 5

,

b = 4

, and

c = 3

.

This establishes the precise values corresponding to the blanks in the question.

Key Concept

Factoring out a greatest common factor followed by factoring a quadratic trinomial.

Question 5Question →

The area of a rectangular solar panel is modeled by the algebraic expression $w^2 - 5w + 6$ . Which of the following represents the completely factored form of this expression?

(w - 2)(w - 3)

(w + 2)(w + 3)

(w - 6)(w + 1)

(w + 6)(w - 1)

Show answer & explanation

Answer: $(w - 2)(w - 3)$

Answer

The completely factored form of the expression is

(w - 2)(w - 3)

.

To factor the quadratic expression

w^2 - 5w + 6

, you must find two integers that multiply to the constant term (6) and add to the middle term's coefficient (-5). The integers -2 and -3 satisfy both conditions because

(-2) \times (-3) = 6

and

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

. Placing these into binomials gives the correct factored form.

Step-by-Step Solution

1

Identify the type of expression

The expression is a quadratic trinomial in the form

w^2 + bw + c

, where

b = -5

and

c = 6

.

Recognizing the structure helps determine the factoring method.

2

Find the factors of the constant term

The factors of 6 are (1, 6), (-1, -6), (2, 3), and (-2, -3).

The binomial factors must contain two numbers that multiply to the constant term.

3

Select the factor pair that adds to the middle coefficient

The sum of -2 and -3 is -5.

The two numbers must add up to the coefficient of the middle term.

4

Write the final factored expression

(w - 2)(w - 3)

Substituting the correct numbers into the binomial format

(w + p)(w + q)

.

Key Concept

Factoring Monic Quadratic Trinomials

Question 6Question →

Assume

x

and

y

are non-zero real numbers. When the algebraic expression

\frac{(2x^{-3}y^4)^3}{4x^5y^{-2}}

is fully simplified to the form

cx^ay^b

, what is the integer value of

b

?

Show answer & explanation

Answer: 14

Answer

14

The expression simplifies to

2x^{-14}y^{14}

. Because the question asks for the exponent of

y

, which is denoted by

b

in the target expression

cx^ay^b

, the correct value is 14.

Step-by-Step Solution

1

Apply the power of a product rule to the numerator.

(2x^{-3}y^4)^3 = 2^3(x^{-3})^3(y^4)^3 = 8x^{-9}y^{12}

When an algebraic term with coefficients and variables is raised to a power, the outer exponent must be distributed to each individual factor inside the parentheses.

2

Set up the division by applying the quotient rule for each base.

\frac{8x^{-9}y^{12}}{4x^5y^{-2}} = \left(\frac{8}{4}\right) \cdot \left(\frac{x^{-9}}{x^5}\right) \cdot \left(\frac{y^{12}}{y^{-2}}\right)

Separating the expression by identical bases allows the independent application of the quotient rule for exponents.

3

Evaluate the exponent for the

y

term to find the value of

b

.

y^{12} \div y^{-2} = y^{12 - (-2)} = y^{12 + 2} = y^{14}

When dividing exponential terms with the same base, subtract the denominator's exponent from the numerator's exponent. Care must be taken with the double negative.

4

Identify the requested integer value.

b = 14

The question specifically asks for the integer value of

b

in the final simplified form

cx^ay^b

, which represents the exponent of the

y

term.

Key Concept

Applying the power of a power and quotient of powers rules for variable exponents.

Question 7Question →

Match each polynomial expression on the left with its completely factored equivalent on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches pair each polynomial with its completely factored form by applying techniques such as factoring out the greatest common factor (GCF), recognizing difference of squares, and factoring trinomials.

Each polynomial is completely factored by extracting any Greatest Common Factors and correctly applying fundamental algebraic patterns, including the difference of squares, perfect square trinomials, and standard quadratic trinomial factoring.

Step-by-Step Solution

1

Analyze the first polynomial:

x^2 - 3x - 10

.

It factors to

(x - 5)(x + 2)

.

This is a basic trinomial where we need factors of -10 that add up to -3.

2

Analyze the second polynomial:

2x^2 - 18

.

It factors to

2(x - 3)(x + 3)

.

Factoring out the GCF of 2 leaves

x^2 - 9

, which is a difference of squares and factors further into

(x - 3)(x + 3)

.

3

Analyze the third polynomial:

3x^2 - 6x

.

It factors to

3x(x - 2)

.

The two terms share a common factor of

3x

. Extracting this GCF directly yields the factored form.

4

Analyze the fourth polynomial:

x^2 - 10x + 25

.

It factors to

(x - 5)^2

.

This matches the perfect square trinomial pattern

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

with

a=x

and

b=5

.

Key Concept

Factoring Polynomials

Question 8Question →

Find the completely factored form of the given polynomial:

2x^2 + 2x - 40

Assume all missing values in the blanks are positive integers.

Fill in the blanks below

The completely factored expression is (x + )(x - ).

Show answer & explanation

Answer

The missing values are 2, 5, and 4. The completely factored expression is 2(x + 5)(x - 4).

Factoring out the greatest common factor of 2 leaves

x^2 + x - 20

. This trinomial factors into

(x + 5)(x - 4)

because 5 and -4 multiply to give -20 and add to give 1. Thus, the complete factored form is

2(x + 5)(x - 4)

, corresponding to the blank values 2, 5, and 4.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF) from all the terms in the polynomial.

2(x^2 + x - 20)

Identifying and extracting the GCF of 2 simplifies the remaining trinomial, making it easier to factor.

2

Factor the resulting quadratic trinomial

x^2 + x - 20

.

Find two numbers that multiply to -20 and add to 1 (the coefficient of x).

To factor a trinomial of the form

x^2 + bx + c

, you must find two factors of

c

that add up to

b

.

3

Identify the two numbers that satisfy the conditions.

The numbers are 5 and -4.

5 \times (-4) = -20

and

5 + (-4) = 1

.

4

Write the complete factorization by combining the GCF and the factored binomials.

2(x + 5)(x - 4)

This represents the original polynomial completely factored into its simplest terms.

Key Concept

Factoring Polynomials with a Greatest Common Factor

Question 9Question →

The total revenue from selling a batch of electronic components is modeled by the polynomial $15x^2 - 14x - 8$ . If the number of units sold is given by the expression $(5x + 2)$ , what binomial represents the price per unit?

Show answer & explanation

Answer: 3x-4; (3x-4); 3x - 4; (3x - 4)

Answer

3x - 4

The correct answer is derived by factoring the revenue polynomial

15x^2 - 14x - 8

. Because revenue equals the number of units multiplied by the price per unit, the missing price expression is the binomial that, when multiplied by

(5x + 2)

, yields the given polynomial. By factoring the expression, we find that

15x^2 - 14x - 8 = (5x + 2)(3x - 4)

, confirming

(3x - 4)

is the correct binomial.

Step-by-Step Solution

1

Set up the algebraic relationship based on the context.

Total Revenue = (Number of Units) × (Price per Unit), which translates to

15x^2 - 14x - 8 = (5x + 2) \times (\text{Price per Unit})

.

This establishes that the price per unit is the missing factor of the given polynomial.

2

Determine the factoring method for the trinomial

15x^2 - 14x - 8

.

Using the ac-method, multiply the leading coefficient (15) by the constant term (-8) to get

-120

.

This is the standard approach for factoring a trinomial where the leading coefficient is greater than 1.

3

Find two integers that multiply to

-120

and add to the middle coefficient,

-14

.

The required integers are

-20

and

6

, because

-20 \times 6 = -120

and

-20 + 6 = -14

.

These integers allow us to correctly split the middle term for factoring by grouping.

4

Rewrite the trinomial and factor by grouping.

15x^2 - 20x + 6x - 8 = 5x(3x - 4) + 2(3x - 4) = (5x + 2)(3x - 4)

.

Grouping extracts the greatest common factor from pairs of terms, revealing the two binomial factors.

5

Identify the missing factor.

Since one factor is given as

(5x + 2)

, the other factor representing the price is

(3x - 4)

.

This completes the solution, satisfying the initial equation.

Key Concept

Factoring Non-Monic Trinomials

Question 10Question →

Factor the trinomial $x^2 + 2x - 15$ . Which of the following choices shows the correct factorization?

(x + 5)(x - 3)

(x - 5)(x + 3)

(x + 5)(x + 3)

(x - 15)(x + 2)

Show answer & explanation

Answer: $(x + 5)(x - 3)$

Answer

The correct factorization is

(x + 5)(x - 3)

.

To factor the quadratic expression

x^2 + 2x - 15

, we need to find two numbers that multiply to

-15

and add up to

2

. The numbers

5

and

-3

meet these criteria (

5 \times -3 = -15

and

5 + -3 = 2

). Therefore, the completely factored form is

(x + 5)(x - 3)

.

Step-by-Step Solution

1

Identify the target sum and product for the factors.

The target sum is the coefficient of the middle term (

2

), and the target product is the constant term (

-15

).

To factor a trinomial

x^2 + bx + c

into

(x + p)(x + q)

, we need to find two integers

p

and

q

that multiply to

c

and add up to

b

.

2

Find the two integers that satisfy both conditions.

The factor pairs of

-15

are

(-1, 15)

,

(1, -15)

,

(-3, 5)

, and

(3, -5)

. The pair that adds to

2

is

-3

and

5

.

Checking the sums, only

5 + (-3) = 2

provides the correct middle coefficient.

3

Write the factored form using the selected integers.

The factored form is

(x + 5)(x - 3)

.

Substituting

5

and

-3

into the binomial framework

(x + p)(x + q)

completes the factorization.

Key Concept

Factoring Monic Quadratic Trinomials

Question 11Question →

Simplify the given algebraic expression by factoring it completely, then determine the missing positive integer values.

Fill in the blanks below

The completely factored form of the expression

2y^2 - 10y - 28

can be written in the format

2(y - a)(y + b)

, where

a

and

b

are positive integers.

The value of

a

is and the value of

b

is .

Show answer & explanation

Answer

The value of a is 7 and the value of b is 2.

Factoring out the GCF of 2 yields

2(y^2 - 5y - 14)

. The trinomial

y^2 - 5y - 14

factors into

(y - 7)(y + 2)

. Matching this with the given structure

2(y - a)(y + b)

shows that the positive integers are

a = 7

and

b = 2

.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF) from all terms in the polynomial.

2(y^2 - 5y - 14)

Identifying and factoring out the GCF of 2 simplifies the polynomial, revealing a trinomial with a leading coefficient of 1, which is easier to factor.

2

Factor the resulting quadratic trinomial

y^2 - 5y - 14

.

The binomial factors are

(y - 7)

and

(y + 2)

.

To factor

y^2 - 5y - 14

, we need to find two numbers that multiply to the constant term (-14) and add up to the middle term's coefficient (-5). The numbers -7 and 2 satisfy both conditions (-7 * 2 = -14 and -7 + 2 = -5).

3

Write the complete factorization and match it to the provided template

2(y - a)(y + b)

.

The expression is

2(y - 7)(y + 2)

. Matching this to the template gives

a = 7

and

b = 2

.

Comparing the forms directly, the term subtracted from y is 7 (so a = 7) and the term added to y is 2 (so b = 2).

Key Concept

Factoring polynomials using the Greatest Common Factor (GCF) and factoring quadratic trinomials.

Question 12Question →

Determine the exact ordered pair $(x, y)$ that satisfies the system of linear equations shown below.

4x - 3y = 25

x + 2y = 9

Enter your answer in the format $(x, y)$ .

Show answer & explanation

Answer: (7, 1); (7,1); 7, 1; 7,1

Answer

The correct ordered pair is (7, 1).

The correct solution is (7, 1). To verify, substitute x = 7 and y = 1 into both original equations. For the first equation: 4(7) - 3(1) = 28 - 3 = 25. For the second equation: 7 + 2(1) = 7 + 2 = 9. Since both equations hold true simultaneously, (7, 1) is the accurate solution.

Step-by-Step Solution

1

Isolate the variable

x

in the second equation.

x = 9 - 2y

The second equation has an

x

with a coefficient of 1, making it the most straightforward variable to isolate for substitution without creating fractions.

2

Substitute the expression for

x

into the first equation.

4(9 - 2y) - 3y = 25

This creates a single equation containing only the variable

y

, which can be solved algebraically.

3

Distribute the 4 and combine like terms.

36 - 8y - 3y = 25 \Rightarrow 36 - 11y = 25

Simplifying the equation prepares it for isolating the

y

term.

4

Solve for

y

by subtracting 36 from both sides and dividing by -11.

-11y = -11 \Rightarrow y = 1

This determines the numerical value for the y-coordinate of the solution.

5

Substitute

y = 1

back into the isolated

x

equation from Step 1.

x = 9 - 2(1) \Rightarrow x = 9 - 2 = 7

This provides the corresponding x-coordinate to complete the ordered pair.

Key Concept

Solving Systems of Linear Equations by Substitution

Question 13Question →

Match each non-monic quadratic polynomial to its completely factored form. Pay close attention to the signs in each binomial factor.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings are:

2m^2 - 11m + 15

with

(2m - 5)(m - 3)

;

2m^2 + m - 15

with

(2m - 5)(m + 3)

;

2m^2 + 11m + 15

with

(2m + 5)(m + 3)

; and

2m^2 - m - 15

with

(2m + 5)(m - 3)

.

Each polynomial requires finding two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient. The correct pairings map the expanded polynomials directly to their unique, fully factored equivalents.

Step-by-Step Solution

1

Analyze the signs of the constants in each polynomial.

Polynomials with a positive constant term (+15) must have binomial factors with identical signs (both positive or both negative). Polynomials with a negative constant term (-15) must have binomial factors with opposite signs.

The product of the constant terms in the binomials must equal the constant term of the polynomial.

2

Factor

2m^2 - 11m + 15

.

Since the constant is positive and the middle term is negative, both factors must be negative.

2m^2 - 6m - 5m + 15 = 2m(m - 3) - 5(m - 3) = (2m - 5)(m - 3)

.

Applying factoring by grouping.

3

Factor

2m^2 + m - 15

.

The constant is negative, so signs differ. The middle term is positive, so the larger product in FOIL must be positive.

2m^2 + 6m - 5m - 15 = 2m(m + 3) - 5(m + 3) = (2m - 5)(m + 3)

.

Applying factoring by grouping.

4

Factor

2m^2 + 11m + 15

.

Since both the constant and middle term are positive, both factors must be positive.

2m^2 + 6m + 5m + 15 = 2m(m + 3) + 5(m + 3) = (2m + 5)(m + 3)

.

Applying factoring by grouping.

5

Factor

2m^2 - m - 15

.

The constant is negative, so signs differ. The middle term is negative, so the larger product in FOIL must be negative.

2m^2 - 6m + 5m - 15 = 2m(m - 3) + 5(m - 3) = (2m + 5)(m - 3)

.

Applying factoring by grouping.

Key Concept

Factoring Non-Monic Trinomials and Sign Analysis

Estimated Time:2m 30s

Question 14Question →

Consider the following algebraic expression:

(2x - 5y)^2(x + 2y)

If you were to fully expand this expression and combine all like terms, what would be the numerical coefficient of the $x^2y$ term?

Fill in the blanks below

The numerical coefficient of the

x^2y

term is .

Show answer & explanation

Answer

-12

The correct answer is derived by first expanding the squared binomial:

(2x - 5y)^2 = 4x^2 - 20xy + 25y^2

. Next, multiply this resulting trinomial by the remaining binomial:

(4x^2 - 20xy + 25y^2)(x + 2y)

. To find the coefficient of the

x^2y

term without expanding the entire expression, identify the term pairs whose product yields

x^2y

. These pairs are

(4x^2)(2y) = 8x^2y

and

(-20xy)(x) = -20x^2y

. Combining these like terms gives

8x^2y - 20x^2y = -12x^2y

. Thus, the numerical coefficient of the

x^2y

term is

-12

.

Step-by-Step Solution

1

Expand the squared binomial

(2x - 5y)^2

.

(2x)^2 - 2(2x)(5y) + (5y)^2 = 4x^2 - 20xy + 25y^2

Order of operations requires evaluating exponents before multiplying by the binomial

(x + 2y)

.

2

Set up the multiplication of the resulting trinomial and the remaining binomial.

(4x^2 - 20xy + 25y^2)(x + 2y)

Substitute the expanded form back into the original expression to continue the multiplication process.

3

Identify only the specific multiplications that will produce an

x^2y

term.

The term

4x^2

multiplied by

2y

, and the term

-20xy

multiplied by

x

.

To find the coefficient of a specific term, it is most efficient to calculate only the products that result in that exact variable combination.

4

Calculate and combine the identified products to find the final coefficient.

(4x^2)(2y) + (-20xy)(x) = 8x^2y - 20x^2y = -12x^2y

. The numerical coefficient is

-12

.

Combining these like terms gives the total

x^2y

term in the fully expanded polynomial.

Key Concept

Multiplying Polynomials

Question 15Question →

For the inequality below, which of the following expresses all possible values of $y$ ?

2(3y - 5) - 3(y - 4) \geq 6y - 16

y \leq 6

y \geq 6

y \leq -2

y \geq -2

Show answer & explanation

Answer: $y \leq 6$

Answer

The correct answer is

y \leq 6

.

The correct answer is derived by distributing the terms to get

6y - 10 - 3y + 12 \geq 6y - 16

, which simplifies to

3y + 2 \geq 6y - 16

. Subtracting

6y

from both sides and subtracting 2 from both sides yields

-3y \geq -18

. Finally, dividing by -3 requires flipping the inequality symbol, resulting in

y \leq 6

.

Step-by-Step Solution

1

Distribute the constants on the left side of the inequality.

6y - 10 - 3y + 12 \geq 6y - 16

To clear the parentheses. Note that multiplying -3 by -4 results in a positive 12.

2

Combine the like terms on the left side.

3y + 2 \geq 6y - 16

To simplify the expression before isolating the variable.

3

Move all variable terms to the left side and constant terms to the right side.

3y - 6y \geq -16 - 2

, which simplifies to

-3y \geq -18

To group the unknown variable on one side of the inequality.

4

Divide both sides by -3.

y \leq 6

To solve for y. Whenever multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed.

Key Concept

Solving One-Variable Inequalities

Question 16Question →

For the linear inequality $-\frac{3}{4}(8x - 12) + 2x > -3(x - 4) - 2x$ , the value $x = 3$ is included in the solution set.

True

False

Show answer & explanation

Answer: False

Answer

False, because the solution set is

x > 3

, which strictly excludes the boundary value of 3.

Simplifying both sides of the inequality yields

-4x + 9 > -5x + 12

. Isolating

x

provides the solution set

x > 3

. Because the inequality is strictly 'greater than', it does not include the boundary value of 3, making the statement false.

Step-by-Step Solution

1

Distribute the negative fraction

-\frac{3}{4}

on the left side of the inequality.

-\frac{3}{4}(8x) - (-\frac{3}{4})(12) + 2x = -6x + 9 + 2x

To clear the parentheses and prepare the left side for combining like terms.

2

Distribute the negative integer

-3

on the right side of the inequality.

-3(x) - (-3)(4) - 2x = -3x + 12 - 2x

To clear the parentheses on the right side of the inequality.

3

Combine the like

x

terms on both sides of the inequality.

Left side:

-6x + 2x + 9 = -4x + 9

. Right side:

-3x - 2x + 12 = -5x + 12

. The simplified inequality is

-4x + 9 > -5x + 12

.

To simplify the expression before isolating the variable.

4

Isolate the variable

x

by adding

5x

and subtracting

9

on both sides.

-4x + 5x > 12 - 9

, which simplifies to

x > 3

.

To determine the final solution set for the inequality.

5

Evaluate whether

x = 3

satisfies the solution set

x > 3

.

Since

3

is not greater than

3

, the value is not included in the solution set.

To definitively determine if the original statement is true or false.

Key Concept

Solving multi-step linear inequalities and determining boundary value inclusion

Question 17Question →

Which of the following represents the correct solution set for the inequality below?

4(x - 3) - 3(2x + 5) \geq x - 12

x \leq -5

x \geq -5

x \leq 5

x \geq 5

Show answer & explanation

Answer: $x \leq -5$

Answer

The correct solution is

x \leq -5

.

The correct answer is derived by carefully distributing the values across the parentheses (yielding

4x - 12 - 6x - 15 \geq x - 12

), combining the like terms to form

-2x - 27 \geq x - 12

, and isolating

x

to get

-3x \geq 15

. Crucially, dividing both sides by

-3

requires reversing the direction of the inequality, which correctly results in

x \leq -5

.

Step-by-Step Solution

1

Distribute the coefficients outside the parentheses into the terms inside.

4x - 12 - 6x - 15 \geq x - 12

This removes the parentheses. Care must be taken to multiply -3 by +5 correctly to get -15.

2

Combine the like terms on the left side of the inequality.

-2x - 27 \geq x - 12

Simplifying both sides makes it easier to isolate the variable.

3

Move all terms containing

x

to one side and all constant terms to the other side.

-3x \geq 15

Subtracting

x

from both sides gives

-3x

, and adding

27

to both sides gives

15

.

4

Divide both sides by -3 and reverse the inequality symbol.

x \leq -5

Whenever you multiply or divide an inequality by a negative number, the direction of the inequality sign must be flipped to maintain a true mathematical statement.

Key Concept

Solving multi-step linear inequalities involving distribution and the inequality sign flip rule.

Question 18Question →

What is the complete solution set for the given inequality?

\frac{1}{2}(-8x - 10) - \frac{2}{3}(9x + 15) \leq -4(x - 2) + 2x + 1

x \geq -3

x \leq -3

x \geq -1

x \geq -2

Show answer & explanation

Answer: $x \geq -3$

Answer

The correct solution is

x \geq -3

.

By systematically distributing the fractions and negative integers on both sides, we simplify the expression to

-10x - 15 \leq -2x + 9

. Moving the variable terms to the left and constants to the right results in

-8x \leq 24

. Finally, dividing by

-8

isolates

x

and requires flipping the inequality symbol, providing the correct solution.

Step-by-Step Solution

1

Distribute the fractions into the terms inside the parentheses on the left side.

(-4x - 5) - (6x + 10) \leq -4(x - 2) + 2x + 1

To eliminate the parentheses and clear the fractions for easier manipulation.

2

Distribute the

-4

on the right side of the inequality.

-4x - 5 - 6x - 10 \leq -4x + 8 + 2x + 1

To expand all terms fully before attempting to combine them.

3

Combine the like terms on both the left and right sides independently.

-10x - 15 \leq -2x + 9

To simplify the inequality into a standard two-step format.

4

Add

2x

to both sides and add

15

to both sides to group variable terms and constant terms.

-8x \leq 24

To isolate the

x

term on one side of the inequality.

5

Divide both sides by

-8

and reverse the direction of the inequality symbol.

x \geq -3

Multiplying or dividing an inequality by a negative number requires flipping the inequality sign to maintain the mathematical truth.

Key Concept

Solving Complex Linear Inequalities

Question 19Question →

Consider the algebraic expression below, where $x \neq 0$ and $y \neq 0$ . Which of the following is its fully simplified form?

\left( \frac{3x^{-2}y^3}{2xy^{-2}} \right)^{-2} \times \left( \frac{9x^2}{4y^4} \right)^{-1}

\frac{16}{81x^{12}y^{40}}

\frac{27x^4}{8y^6}

\frac{16x^4}{81y^6}

\frac{16y^2}{81}

Show answer & explanation

Answer: $\frac{16x^4}{81y^6}$

Answer

The expression fully simplifies to

\frac{16x^4}{81y^6}

.

The correct expression is derived by sequentially simplifying the contents of the parentheses using the quotient rule, applying the outer negative exponents to both the numerical coefficients and variables, and then multiplying the resulting terms by adding the exponents of like bases.

Step-by-Step Solution

1

Simplify the fraction inside the first set of parentheses using the quotient rule for exponents (

a^m / a^n = a^{m-n}

).

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

Simplifying inside the parentheses first follows the order of operations and makes subsequent calculations more manageable.

2

Apply the outer negative exponent (

-2

) to all components of the first simplified term, including the numerical coefficient.

\left( \frac{3}{2} x^{-3} y^5 \right)^{-2} = \left( \frac{3}{2} \right)^{-2} (x^{-3})^{-2} (y^5)^{-2} = \frac{4}{9} x^6 y^{-10}

The power of a product rule states that

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

, and applying a negative power to a fraction flips it:

(a/b)^{-n} = (b/a)^n

.

3

Apply the outer negative exponent (

-1

) to the second term.

\left( \frac{9x^2}{4y^4} \right)^{-1} = \frac{4y^4}{9x^2} = \frac{4}{9} x^{-2} y^4

A negative exponent of

-1

simply takes the reciprocal (flips the fraction).

4

Multiply the two resulting expressions by grouping coefficients and using the product rule for exponents (

a^m \times a^n = a^{m+n}

).

\left( \frac{4}{9} x^6 y^{-10} \right) \times \left( \frac{4}{9} x^{-2} y^4 \right) = \left( \frac{4}{9} \times \frac{4}{9} \right) (x^6 \times x^{-2}) (y^{-10} \times y^4) = \frac{16}{81} x^4 y^{-6}

When multiplying terms with the same base, their exponents are added together.

5

Rewrite the final expression using only positive exponents.

\frac{16}{81} x^4 y^{-6} = \frac{16x^4}{81y^6}

Standard mathematical notation requires fully simplified algebraic expressions to be written without negative exponents.

Key Concept

Simplifying rational expressions using the laws of exponents, including the quotient rule, power rule, product rule, and handling negative exponents alongside numerical coefficients.

Question 20Question →

Which of the following represents the completely factored form of the algebraic expression $2x^2 - 4x - 30$ ?

2(x + 5)(x - 3)

2(x^2 - 2x - 15)

2(x - 5)(x + 3)

2(x - 5)(x - 3)

Show answer & explanation

Answer: $2(x - 5)(x + 3)$

Answer

The completely factored form is

2(x - 5)(x + 3)

.

To find the completely factored form, first factor out the greatest common factor (GCF), which is

2

. This gives

2(x^2 - 2x - 15)

. Next, factor the trinomial inside the parentheses. You need two numbers that multiply to

-15

and add up to

-2

. These numbers are

-5

and

3

. Substituting these into the binomial factors yields the final completely factored form:

2(x - 5)(x + 3)

.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF) from all terms in the expression

2x^2 - 4x - 30

.

The GCF is

2

. Factoring it out leaves

2(x^2 - 2x - 15)

.

Identifying and extracting the GCF is always the first step in simplifying and completely factoring a polynomial.

2

Factor the remaining quadratic trinomial

x^2 - 2x - 15

by finding two integers that multiply to

-15

and add to

-2

.

The two integers are

-5

and

3

.

For a trinomial in the form

x^2 + bx + c

, the constants in the binomial factors must multiply to

c

and add to

b

.

3

Write the expression as the product of the GCF and the two binomial factors.

The fully factored expression is

2(x - 5)(x + 3)

.

This combines all factored components into their most simplified, prime form.

Key Concept

Factoring out a greatest common factor (GCF) followed by factoring a quadratic trinomial.

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