Algebra Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 7

Match each unsimplified algebraic expression on the left with its correctly simplified form on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches pair each unsimplified expression with its corresponding simplified form: 4m + 7m - 2 pairs with 11m - 2; 3(2m + 5) pairs with 6m + 15; 9m - (4m - 3) pairs with 5m + 3; and m(2m - 1) pairs with 2m^2 - m.

Each unsimplified expression is paired with its mathematically equivalent simplified form by correctly applying standard algebraic rules: combining like terms and utilizing the distributive property.

Step-by-Step Solution

1

Simplify the first expression by identifying and combining like terms.

4m + 7m - 2 = 11m - 2

The terms 4m and 7m have the exact same variable part and can be added together.

2

Simplify the second expression by applying the distributive property.

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

The outside factor of 3 must be multiplied by every term contained inside the parentheses.

3

Simplify the third expression by distributing the negative sign, then combining the resulting like terms.

9m - 4m + 3 = 5m + 3

Subtracting a grouped polynomial requires reversing the sign of each term within the group.

4

Simplify the fourth expression by multiplying the variable through the parentheses.

m(2m - 1) = 2m^2 - m

Multiplying a variable by itself increases its exponent based on the laws of exponents.

Key Concept

Simplifying Algebraic Expressions

Question 122Question →

Pair each unsimplified algebraic expression with its correctly simplified polynomial equivalent in standard form.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The expressions simplify as follows: the first expression simplifies to 4p² - 11pq; the second to -2p² + 14pq; the third to 4p² + 5pq; and the fourth to -2p² + 3pq.

Each expression is correctly simplified by first distributing the outer multipliers to every term inside their respective parentheses, paying close attention to negative signs. Once expanded, like terms (terms containing p² and terms containing pq) are combined by adding or subtracting their coefficients.

Step-by-Step Solution

1

Simplify the first expression: 3p(2p - q) - 2(p² + 4pq)

6p² - 3pq - 2p² - 8pq = 4p² - 11pq

Distribution of the outside terms into the parentheses, followed by grouping like terms.

2

Simplify the second expression: -4(p² - 2pq) + 2p(p + 3q)

-4p² + 8pq + 2p² + 6pq = -2p² + 14pq

Carefully distribute the negative constant (-4), changing the inner sign to positive 8pq, then combine like terms.

3

Simplify the third expression: p(5p + 2q) - (p² - 3pq)

5p² + 2pq - p² + 3pq = 4p² + 5pq

Distribute the variable p and distribute the negative sign to flip the inner signs, then collect like terms.

4

Simplify the fourth expression: 2(3p² - pq) - p(8p - 5q)

6p² - 2pq - 8p² + 5pq = -2p² + 3pq

Distribute the 2 and the -p, ensuring the multiplication of -p and -5q results in +5pq, then combine.

Key Concept

Simplifying Algebraic Expressions

Question 123Question →

Evaluate the following multivariable algebraic expressions. Pair each original expression with its simplified form in standard notation.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Each unsimplified expression matches exactly one standard form polynomial upon successful expansion and collection of like terms.

Each pairing is correct because the unsimplified expressions rigidly evaluate to the exact standard forms provided when algebraic expansion and simplification rules are followed without computational errors.

Step-by-Step Solution

1

Simplify the first expression by expanding and subtracting the grouped terms.

-3a(2a - b) - (a - 2b)(2a + b)

simplifies to

-8a^2 + 6ab + 2b^2

.

Applying the distributive property and FOIL method accurately, followed by subtracting the terms (which reverses their signs).

2

Simplify the second expression by applying distribution to both halves.

2ab(a - 3b) - a(2ab - 5b^2)

simplifies to

-ab^2

.

Distributing the multipliers across the parentheses isolates terms that naturally cancel or combine.

3

Simplify the third expression by expanding the perfect square.

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

simplifies to

3b^2

.

The expansion of

(2a - b)^2

creates terms that precisely offset the

a^2

and

ab

terms in the distributed trinomial.

4

Simplify the fourth expression by expanding all grouped and squared segments.

a^2(2b - 1) - b(a^2 - 2a) + (a - b)^2 - b^2

simplifies to

a^2b

.

Combining the resulting elements reveals that nearly all terms possess additive inverses, leaving only the combination of

2a^2b - a^2b

.

Key Concept

Expanding polynomials, applying the distributive property with negative signs, and collecting like variables.

Question 124Question →

Match each unsimplified algebraic expression on the left with its correctly simplified polynomial equivalent on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings link each unsimplified expression with its resulting simplified polynomial after applying the distributive property and combining like terms.

Each unsimplified expression is accurately paired with its simplified equivalent by first applying the distributive property to eliminate all parentheses, rigorously managing negative multipliers, and finally grouping and combining all like terms according to their respective exponents.

Step-by-Step Solution

1

Expand and simplify the first expression.

6x^2 - 12x - 2x^2 + 10x

, which simplifies to

4x^2 - 2x

.

Applying the distributive property and combining the

x^2

terms and the

x

terms.

2

Expand and simplify the second expression.

5x^2 - 15x + 10 - 4x^2 + 8x

, which simplifies to

x^2 - 7x + 10

.

Distributing the constants and variables across parentheses, paying special attention to the negative signs.

3

Expand and simplify the third expression.

-6x^2 - 2x + 5x^2 - 4x

, which simplifies to

-x^2 - 6x

.

Multiplying each term inside the parentheses by the outside factor and combining the resulting like terms.

4

Expand and simplify the fourth expression.

8x^2 - 4x - 3x^2 - 6x + 3

, which simplifies to

5x^2 - 10x + 3

.

Distributing the

-3

correctly reverses the signs of all terms within the second polynomial.

Key Concept

Simplifying Algebraic Expressions

Question 125Question →

Evaluate the algebraic expression below given that $x = -4$ , $y = -2$ , and $z = -3$ :

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

Calculate the intermediate values and the final result to complete the statements.

Fill in the blanks below

The value of the numerator is .
The value of the denominator is .
The final evaluated result of the expression is .

Show answer & explanation

Answer

The numerator evaluates to -60, the denominator evaluates to -5, and the final evaluated result of the expression is 12.

By strictly adhering to the order of operations and properly handling negative substitutions, the numerator simplifies to -60 and the denominator simplifies to -5. Dividing these yields a final positive integer result of 12.

Step-by-Step Solution

1

Substitute the given values for

x

,

y

, and

z

into the expression.

\frac{(-4)^3 - (-2)^2(-4 - (-3))}{|-4| - (-3)^2}

Direct substitution is the required first step before performing any operations.

2

Evaluate the exponents in the numerator and denominator.

Numerator exponents:

(-4)^3 = -64

and

(-2)^2 = 4

. Denominator exponent:

(-3)^2 = 9

.

Following the order of operations, exponents must be evaluated before multiplication and subtraction.

3

Evaluate the expression inside the parentheses in the numerator and the absolute value in the denominator.

Numerator parentheses:

(-4 - (-3)) = (-4 + 3) = -1

. Denominator absolute value:

|-4| = 4

.

Operations inside grouping symbols and absolute value operations take precedence.

4

Perform the remaining multiplication in the numerator, then subtract to find the final numerator value.

-64 - (4)(-1) = -64 - (-4) = -64 + 4 = -60

. (Value for blank 1)

Multiplication is performed before addition or subtraction according to the order of operations.

5

Perform the subtraction in the denominator.

4 - 9 = -5

. (Value for blank 2)

This fully simplifies the expression in the denominator.

6

Divide the fully simplified numerator by the simplified denominator.

\frac{-60}{-5} = 12

. (Value for blank 3)

The fraction bar represents division, which is the final operation applied to the evaluated top and bottom terms.

Key Concept

Evaluating Complex Algebraic Expressions

Question 126Question →

7(y - 2) - 4y > 5y + 10

What are all the possible values of $y$ that satisfy the inequality shown above?

y < -12

y > -12

y < 12

y > 2

Show answer & explanation

Answer: $y < -12$

Answer

The correct solution set is

y < -12

.

The correct answer is found by properly applying the distributive property, grouping like terms, isolating the variable, and correctly reversing the inequality sign when dividing by -2.

Step-by-Step Solution

1

Distribute the 7 to both terms inside the parentheses.

7y - 14 - 4y > 5y + 10

This clears the parentheses, allowing the variable terms to be grouped.

2

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

3y - 14 > 5y + 10

Subtracting

4y

from

7y

simplifies the left side to

3y

.

3

Subtract

5y

from both sides to group the variable terms on the left.

-2y - 14 > 10

This moves all terms containing

y

to one side of the inequality.

4

Add 14 to both sides to isolate the variable term.

-2y > 24

This leaves the

-2y

term by itself on the left side.

5

Divide both sides by -2 and reverse the inequality sign.

y < -12

According to the properties of inequalities, multiplying or dividing by a negative number requires flipping the direction of the inequality symbol.

Key Concept

Solving One-Variable Inequalities

Question 127Question →

What is the value of $x$ in the equation below?

3 - \frac{2}{5}(10x - 15) = \frac{1}{2}(4x + 8) - 7x

-13

7

-5

3

Show answer & explanation

Answer: -5

Answer

-5

The correct answer is -5. By distributing the fractional coefficients, combining like terms, and isolating the variable, the equation simplifies to -4x + 9 = -5x + 4, which yields x = -5.

Step-by-Step Solution

1

Distribute the fractional coefficients on both sides of the equation.

3 - 4x + 6 = 2x + 4 - 7x

To eliminate the parentheses and prepare the equation for combining like terms. Carefully manage the negative sign when distributing -2/5.

2

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

-4x + 9 = -5x + 4

To simplify both sides of the equation before isolating the variable.

3

Add 5x to both sides of the equation.

x + 9 = 4

To keep the equation balanced while moving all x terms to one side.

4

Subtract 9 from both sides of the equation.

x = -5

To isolate the variable x and find its final value.

Key Concept

Solving multi-step linear equations involving fractions, distribution, and variables on both sides.

Question 128Question →

Which of the following expressions is equivalent to

15 - 3(2k + 4)

after applying the proper order of operations and combining like terms?

-6k + 3

24k + 48

-6k + 27

-3k

Show answer & explanation

Answer: $-6k + 3$

Answer

-6k + 3

The correct answer is found by first distributing the -3 to both the 2k and the 4, yielding -6k - 12. Then, combining the constant terms 15 and -12 results in a final, standard form simplified expression of -6k + 3.

Step-by-Step Solution

1

Distribute the -3 to both terms inside the parentheses.

15 - 6k - 12

According to the order of operations, multiplication (distribution) must be performed before subtraction.

2

Identify and group the like terms.

-6k + (15 - 12)

Constant numbers are like terms and must be combined to fully simplify the expression.

3

Perform the arithmetic to combine the constants.

-6k + 3

Subtracting 12 from 15 yields the final constant value in the expression.

Key Concept

Distributive property and combining like terms

Question 129Question →

Evaluate the following mathematical statement:

-4x + 10 \geq -14

Which of the following represents the correct solution for $x$ ?

x \leq 6

x \geq 6

x \leq -6

x \leq 1

Show answer & explanation

Answer: $x \leq 6$

Answer

The correct answer is

x \leq 6

.

To solve the inequality, first subtract

10

from both sides to get

-4x \geq -24

. Next, isolate

x

by dividing both sides by

-4

. Because you are dividing by a negative number, you must flip the inequality sign from

\geq

to

\leq

, resulting in

x \leq 6

.

Step-by-Step Solution

1

Subtract

10

from both sides of the inequality.

-4x \geq -24

This isolates the term containing the variable

x

on one side of the inequality.

2

Divide both sides by

-4

and reverse the inequality sign.

x \leq 6

When multiplying or dividing an inequality by a negative number, the direction of the inequality symbol must be flipped to maintain a true statement.

Key Concept

Solving one-variable inequalities requires the same algebraic steps as solving linear equations, with the critical exception that the inequality sign must be reversed when multiplying or dividing by a negative number.

Question 130Question →

Simplify the following algebraic expression:

12 - 3(2y - 4)

Which expression represents the fully simplified result?

-6y

18y - 36

-6y + 24

18y

Show answer & explanation

Answer: $-6y + 24$

Answer

The correct simplified expression is

-6y + 24

.

The expression is correctly simplified by first distributing the

-3

across the terms in the parentheses, which results in

-6y + 12

. Adding the initial

12

to this new constant gives a final expression of

-6y + 24

.

Step-by-Step Solution

1

Distribute the

-3

to both terms inside the parentheses.

12 - 6y + 12

The distributive property requires multiplying the term immediately outside the parentheses by each term inside.

2

Combine the constant terms.

-6y + 24

Like terms (the constants

12

and

12

) must be added together to fully simplify the expression.

Key Concept

Distributive Property and Combining Like Terms

Question 131Question →

An algebraic expression is given below:

-2p(3q - p) - [4pq - p(2p + 5q)]

When fully simplified, which of the following represents the equivalent polynomial?

-5pq

-p^2q

4p^2 - 5pq

-7pq

Show answer & explanation

Answer: $4p^2 - 5pq$

Answer

4p^2 - 5pq

The expression simplifies correctly by first distributing the multipliers:

-2p(3q - p) = -6pq + 2p^2

and

-p(2p + 5q) = -2p^2 - 5pq

. The bracket becomes

[4pq - 2p^2 - 5pq] = [-pq - 2p^2]

. Distributing the negative sign outside the bracket gives

+pq + 2p^2

. Finally, combining all terms:

(-6pq + 2p^2) + (pq + 2p^2) = 4p^2 - 5pq

.

Step-by-Step Solution

1

Distribute the outer multiplier

-2p

into the first binomial.

-6pq + 2p^2

Applying the distributive property:

-2p(3q) = -6pq

and

-2p(-p) = +2p^2

.

2

Distribute

-p

into the binomial within the square brackets.

4pq - 2p^2 - 5pq

Applying the distributive property inside the grouping:

-p(2p) = -2p^2

and

-p(5q) = -5pq

.

3

Combine like terms inside the square brackets.

-pq - 2p^2

Combining the like terms

4pq

and

-5pq

yields

-pq

.

4

Distribute the negative sign outside the square brackets to the simplified contents.

+pq + 2p^2

Subtracting a quantity is equivalent to adding its opposite:

-(-pq - 2p^2) = +pq + 2p^2

.

5

Combine all remaining terms from the entire expression.

2p^2 - 6pq + pq + 2p^2 = 4p^2 - 5pq

Grouping like terms:

(2p^2 + 2p^2) + (-6pq + pq) = 4p^2 - 5pq

.

Key Concept

Simplifying expressions involves correctly applying the distributive property, especially with negative multipliers, and combining like terms.

Question 132Question →

What is the greatest integer value of $x$ that satisfies the following inequality?

3(4 - 2x) - 2x > 5(x - 3) - 10x

Show answer & explanation

Answer: 8

Answer

8

Solving the inequality yields x < 9. Since the inequality is strictly less than 9, the greatest integer that satisfies the condition is 8.

Step-by-Step Solution

1

Distribute the 3 and 5

12 - 6x - 2x > 5x - 15 - 10x

To eliminate the parentheses on both sides of the inequality.

2

Combine like terms

12 - 8x > -5x - 15

To simplify both sides before isolating the variable.

3

Isolate the variable terms

-3x > -27

Subtract 12 from both sides and add 5x to both sides to get all x terms on the left and constants on the right.

4

Divide by -3

x < 9

Dividing an inequality by a negative number requires flipping the inequality sign.

5

Find the greatest integer

8

The inequality states x must be strictly less than 9, so the largest possible integer is 8.

Key Concept

Solving linear inequalities, applying the negative division flip rule, and interpreting integer boundaries.

Question 133Question →

What is the minimum integer value of

x

that satisfies the following inequality?

-3x + 2 \leq 14

Show answer & explanation

Answer: -4

Answer

-4

By isolating the variable and correctly reversing the inequality sign when dividing by a negative number, we find the solution set is

x \geq -4

. The smallest integer in this set is -4.

Step-by-Step Solution

1

Subtract 2 from both sides of the inequality.

-3x \leq 12

To isolate the variable term.

2

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

x \geq -4

Dividing an inequality by a negative number requires reversing the direction of the inequality symbol to maintain the true mathematical relationship.

3

Find the minimum integer value that satisfies the solution set.

-4

The solution set includes all numbers greater than or equal to -4. The smallest integer in this set is -4.

Key Concept

Solving two-step linear inequalities and applying the rule for dividing by a negative number.

Question 134Question →

A mathematical model for a fluid dynamics problem reduces to the following algebraic expression:

-2xy(3x - 4y) - [5x^2y - 2xy(x + 3y)]

When completely simplified, which of the following represents the equivalent standard form?

-13x^2y + 2xy^2

-9x^2y + 2xy^2

-9x^2y + 14xy^2

5x^3y^3

Show answer & explanation

Answer: $-9x^2y + 14xy^2$

Answer

The correct expression is

-9x^2y + 14xy^2

.

The correct expression is derived by applying the order of operations and the distributive property. First, evaluate the terms inside the brackets to get

3x^2y - 6xy^2

. Then expand the initial term to

-6x^2y + 8xy^2

. Subtracting the bracket flips the inner signs, yielding

-6x^2y + 8xy^2 - 3x^2y + 6xy^2

. Combining like terms results in

-9x^2y + 14xy^2

.

Step-by-Step Solution

1

Expand the inner terms inside the brackets by distributing

-2xy

to

(x + 3y)

.

5x^2y - 2x^2y - 6xy^2

Following the order of operations, simplify grouping symbols first.

2

Combine like terms within the brackets.

3x^2y - 6xy^2

To minimize the expression before subtracting it from the leading terms.

3

Distribute

-2xy

to the terms in the first parenthesis

(3x - 4y)

.

-6x^2y + 8xy^2

To expand the first part of the expression.

4

Subtract the simplified bracketed expression from the expanded first part.

-6x^2y + 8xy^2 - (3x^2y - 6xy^2) = -6x^2y + 8xy^2 - 3x^2y + 6xy^2

To remove the remaining grouping symbols and properly distribute the negative sign.

5

Combine the remaining like terms to reach the final form.

-9x^2y + 14xy^2

To express the polynomial in its simplest standard form.

Key Concept

Simplifying complex expressions using the distributive property, managing negative multipliers, and working with nested grouping symbols.

Question 135Question →

Match each complex verbal phrase with its correct algebraic expression representation.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings match each complex verbal description with its respective mathematical translation: 'Twice the difference...' pairs with

2(3x - 7)

; 'Three times a number...' pairs with

(x + 5)^2 - 3x

; 'The quotient of...' pairs with

\frac{2x - 5}{x + 3}

; and 'The sum of...' pairs with

\frac{x + 4}{3x} - 8

.

Each verbal phrase contains specific operational keywords ('quotient', 'product', 'sum', 'difference') and ordering phrases ('subtracted from', 'less than'). Correct translation requires accurately identifying the operations, maintaining the proper sequence, and utilizing grouping symbols where compound expressions are modified.

Step-by-Step Solution

1

Analyze the first phrase: 'Twice the difference when seven is subtracted from three times a number

x

'.

The phrase translates to

2(3x - 7)

.

The phrase 'subtracted from' reverses the order, making it

3x - 7

. 'Twice the difference' requires multiplying the entire grouping by 2.

2

Analyze the second phrase: 'Three times a number

x

subtracted from the square of five more than the number'.

The phrase translates to

(x + 5)^2 - 3x

.

The base quantity is 'five more than the number' (

x + 5

), which is then squared to become

(x + 5)^2

. Subtracting

3x

from this quantity gives the final expression.

3

Analyze the third phrase: 'The quotient of five less than twice a number

x

and three more than the number'.

The phrase translates to

\frac{2x - 5}{x + 3}

.

'Quotient' implies a division. The first part, 'five less than twice a number', forms the numerator (

2x - 5

). The second part, 'three more than the number', forms the denominator (

x + 3

).

4

Analyze the fourth phrase: 'The sum of a number

x

and four, divided by the product of three and the number, all decreased by eight'.

The phrase translates to

\frac{x + 4}{3x} - 8

.

The fraction is constructed with 'the sum' (

x + 4

) over 'the product' (

3x

). The phrase 'all decreased by eight' instructs to subtract 8 from the complete fraction.

Key Concept

Accurately identifying operational keywords and recognizing the correct order of translation for phrases like 'less than' and 'subtracted from'.

Estimated Time:2m 0s

Question 136Question →

Determine the completely simplified form of the following expression:

3(2a + 4) - 5a

Show answer & explanation

Answer: a + 12; a+12; 12 + a; 12+a; 1a + 12; 1a+12

Answer

a + 12

Applying the distributive property to

3(2a + 4)

yields

6a + 12

. The expression becomes

6a + 12 - 5a

. Subtracting

5a

from

6a

leaves

a

. The constant

12

cannot be combined with

a

, making the final simplified expression

a + 12

.

Step-by-Step Solution

1

Use the distributive property to multiply 3 by each term inside the parentheses.

6a + 12 - 5a

Expanding the parentheses is the necessary first step to access the terms for combination.

2

Group and combine the like terms (

6a

and

-5a

).

a + 12

Subtracting

5a

from

6a

leaves

1a

(or

a

), while the constant

12

remains unchanged.

Key Concept

Applying the distributive property and combining like terms.

Estimated Time:45s

Question 137Question →

Simplify the following algebraic expression completely:

3(4m - 2n) - 2(3m + 5n) - (m - 12n)

Show answer & explanation

Answer: 5m - 4n; 5m-4n; -4n + 5m; -4n+5m

Answer

5m - 4n

To simplify the expression, first apply the distributive property to eliminate the parentheses, taking care to distribute the negative signs properly:

12m - 6n - 6m - 10n - m + 12n

. Then, combine the like terms for

m

(

12m - 6m - m = 5m

) and the like terms for

n

(

-6n - 10n + 12n = -4n

) to reach the final answer of

5m - 4n

.

Step-by-Step Solution

1

Distribute the coefficients to the terms inside each set of parentheses.

12m - 6n - 6m - 10n - m + 12n

The distributive property must be applied first to remove parentheses before like terms can be combined.

2

Group the like terms for variables

m

and

n

.

(12m - 6m - m) + (-6n - 10n + 12n)

Grouping like terms organizes the expression to easily combine their coefficients.

3

Combine the coefficients of the grouped terms.

5m - 4n

Evaluating the addition and subtraction of the coefficients yields the fully simplified expression.

Key Concept

Simplifying Algebraic Expressions

Question 138Question →

Solve the following inequality for $x$ :

-2x + 6 \leq 14

Which of the following is the correct solution?

x \geq -4

x \leq -4

x \geq 4

x \leq -10

Show answer & explanation

Answer: $x \geq -4$

Answer

The correct solution is

x \geq -4

.

The correct answer is found by subtracting 6 from 14 to get 8, and then dividing 8 by -2. Because we divided by a negative number, the less than or equal to symbol (

\leq

) must flip to a greater than or equal to symbol (

\geq

), yielding

x \geq -4

.

Step-by-Step Solution

1

Subtract 6 from both sides of the inequality to isolate the variable term.

-2x \leq 14 - 6

-2x \leq 8

This is the first step in using inverse operations to isolate

x

.

2

Divide both sides by

-2

and reverse the inequality symbol.

x \geq 8 \div (-2)

x \geq -4

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

Key Concept

Solving One-Variable Inequalities

Question 139Question →

What is the correct solution set for the inequality shown below?

4(x - 1) - 2(x + 3) \geq 5x + 2

x \leq -4

x \geq -4

x \leq 0

x \leq 4

Show answer & explanation

Answer: $x \leq -4$

Answer

The correct solution set is

x \leq -4

.

The correct answer is derived by carefully distributing the coefficients, combining like terms, isolating the variable

x

, and applying the rule to reverse the inequality symbol when dividing by a negative number.

Step-by-Step Solution

1

Distribute the values

4

and

-2

into their respective parentheses on the left side.

4x - 4 - 2x - 6 \geq 5x + 2

This step removes the parentheses and prepares the terms for grouping.

2

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

2x - 10 \geq 5x + 2

Simplifying expressions on each side makes it easier to isolate the variable.

3

Subtract

5x

from both sides, and add

10

to both sides.

-3x \geq 12

This moves all variable terms to the left side and all constant terms to the right side.

4

Divide both sides by

-3

and reverse the direction of the inequality symbol.

x \leq -4

When dividing or multiplying an inequality by a negative number, the relationship between the two sides inverses, so the sign must be flipped.

Key Concept

Solving one-variable linear inequalities requires isolating the variable using inverse operations, and reversing the inequality symbol when multiplying or dividing by a negative number.

Question 140Question →

What is the result of fully expanding and simplifying the given algebraic expression?

-4k(2k - 3m) + 5k(k - 2m) - 3k^2

-6k^2 + 2km

-6k^2 - 22km

-4k^3m

-6k^2 - 5m

Show answer & explanation

Answer: $-6k^2 + 2km$

Answer

The correct simplified expression is

-6k^2 + 2km

.

The correct answer accurately expands the expression by distributing

-4k

to get

-8k^2 + 12km

, and

5k

to get

5k^2 - 10km

. Grouping the

k^2

terms (

-8 + 5 - 3 = -6

) and the

km

terms (

12 - 10 = 2

) yields

-6k^2 + 2km

.

Step-by-Step Solution

1

Distribute

-4k

across the terms inside the first set of parentheses:

(2k - 3m)

.

-8k^2 + 12km

To eliminate the first grouping symbol, being careful to multiply the negative signs properly (

-4k \times -3m = +12km

).

2

Distribute

5k

across the terms inside the second set of parentheses:

(k - 2m)

.

5k^2 - 10km

To eliminate the second grouping symbol.

3

Write out the complete expanded expression by combining the results and the remaining

-3k^2

term.

-8k^2 + 12km + 5k^2 - 10km - 3k^2

This shows all individual terms ready for grouping.

4

Identify and combine the like terms: the

k^2

terms (

-8k^2

,

5k^2

,

-3k^2

) and the

km

terms (

12km

,

-10km

).

-6k^2 + 2km

Combining like coefficients (

-8 + 5 - 3 = -6

and

12 - 10 = 2

) provides the final simplified expression in standard form.

Key Concept

Applying the distributive property with negative numbers and combining multivariable like terms.

Algebra

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept