Algebra Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 6

Match each unsimplified algebraic expression with its correctly simplified polynomial equivalent.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches pair each unsimplified expression with its corresponding standard form by applying the distributive property and carefully combining like terms.

Each correct pairing is derived by rigorously applying the distributive property, adhering to the order of operations for nested groupings, and grouping matching polynomial degrees correctly.

Step-by-Step Solution

1

Simplify the first expression:

3x - 2(x - 5) + 4

.

x + 14

Apply the distributive property to

-2(x - 5)

to get

-2x + 10

. Then combine like terms:

(3x - 2x) + (10 + 4)

.

2

Simplify the second expression:

5(2x^2 - x) - 3x(x + 2)

.

7x^2 - 11x

Distribute

5

to get

10x^2 - 5x

, and distribute

-3x

to get

-3x^2 - 6x

. Group like terms:

(10x^2 - 3x^2) + (-5x - 6x)

.

3

Simplify the third expression:

4x - [2x + 3(x - 1)]

.

-x + 3

Follow the order of operations by simplifying the innermost grouping first:

3(x - 1) = 3x - 3

. The brackets become

[5x - 3]

. Distributing the subtraction gives

4x - 5x + 3

.

4

Simplify the fourth expression:

\frac{1}{2}(4x - 6) + 3(x + 2)

.

5x + 3

Multiply coefficients:

\frac{1}{2}(4x) - \frac{1}{2}(6) = 2x - 3

, and

3(x) + 3(2) = 3x + 6

. Combine

x

terms and constants:

(2x + 3x) + (-3 + 6)

.

Key Concept

Simplifying Algebraic Expressions

Question 102Question →

Which of the following is the simplified form of the expression $4(2x - 3) + 5x$ ?

13x - 12

13x - 3

x

13x + 12

Show answer & explanation

Answer: $13x - 12$

Answer

13x - 12

The correct answer is obtained by first applying the distributive property to multiply

4

by both

2x

and

-3

, yielding

8x - 12

. Then, the like terms

8x

and

5x

are combined to get

13x

. The constant

-12

cannot be combined with the variable terms, so the fully simplified expression is

13x - 12

.

Step-by-Step Solution

1

Distribute the 4 into the parentheses.

8x - 12 + 5x

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

2

Group and combine the like terms.

13x - 12

The terms

8x

and

5x

are like terms and can be added together. The constant

-12

cannot be combined with variable terms.

Key Concept

Simplifying Algebraic Expressions via Distribution and Combining Like Terms

Question 103Question →

Which of the following is equivalent to the expression $8 - 2(3x - 4) + 5x$ ?

-x + 16

23x - 24

-x

7x + 8

Show answer & explanation

Answer: $-x + 16$

Answer

The correct simplified expression is

-x + 16

.

The expression correctly applies the distributive property and the rules for combining like terms. Distributing

-2

into

(3x - 4)

yields

-6x + 8

. Combining the constants (

8 + 8

) gives

16

, and combining the variable terms (

-6x + 5x

) gives

-x

.

Step-by-Step Solution

1

Distribute the

-2

to the terms inside the parentheses.

8 - 6x + 8 + 5x

According to the order of operations (PEMDAS), the multiplication involved in the distributive property must be performed before addition and subtraction.

2

Group the like terms together.

(-6x + 5x) + (8 + 8)

Only terms with the same variable configuration (or constants) can be added or subtracted together.

3

Combine the like terms.

-x + 16

Adding the coefficients of the variable terms (

-6 + 5 = -1

) and adding the constants (

8 + 8 = 16

) yields the fully simplified expression.

Key Concept

Simplifying Algebraic Expressions

Question 104Question →

Consider the algebraic expression:

$-2a(3a - 4b) - 3b(a - 2b) + 5a^2$

Which of the following represents the correctly simplified form of this expression?

-a^2 + 5ab + 6b^2

-a^2 + 5ab - 6b^2

-a^2 - 11ab + 6b^2

4ab + 6b^2

Show answer & explanation

Answer: $-a^2 + 5ab + 6b^2$

Answer

The correctly simplified form is

-a^2 + 5ab + 6b^2

.

The expression correctly applies the distributive property while carefully maintaining negative signs, and properly groups terms with identical variable combinations (

a^2

,

ab

, and

b^2

) to reach the final simplified form.

Step-by-Step Solution

1

Distribute

-2a

into the first set of parentheses.

-6a^2 + 8ab

Applying the distributive property:

(-2a \times 3a) + (-2a \times -4b)

. A negative term multiplied by a negative term results in a positive term.

2

Distribute

-3b

into the second set of parentheses.

-3ab + 6b^2

Applying the distributive property:

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

. Again, the two negative factors produce a positive result.

3

Write out the full expanded expression by combining the distributed parts with the remaining term.

-6a^2 + 8ab - 3ab + 6b^2 + 5a^2

Setting up the expression to easily identify and group like terms together.

4

Group and combine the like terms.

-a^2 + 5ab + 6b^2

Combining the

a^2

terms (

-6a^2 + 5a^2 = -a^2

) and the

ab

terms (

8ab - 3ab = 5ab

). The

b^2

term remains unchanged.

Key Concept

Simplifying Expressions using the Distributive Property

Estimated Time:1m 0s

Question 105Question →

Which of the following is equivalent to the expression below?

2x(3x - 4) - 3[x(x - 2) - 4(2x + 1)] + x^2

4x^2 + 22x + 12

4x^2 - 38x - 12

4x^2 + 34x

4x^2 - 18x - 4

Show answer & explanation

Answer: $4x^2 + 22x + 12$

Answer

The correct expression is

4x^2 + 22x + 12

.

The correct expression is found by following the order of operations: first expanding the terms inside the innermost parentheses, then combining like terms within the brackets, next distributing the negative multiplier to all terms in the brackets, and finally combining all remaining like terms.

Step-by-Step Solution

1

Expand the first part of the expression by distributing

2x

.

2x(3x - 4) = 6x^2 - 8x

Distribution is required to remove the initial parentheses.

2

Simplify the expression inside the square brackets by distributing and combining like terms.

x(x - 2) - 4(2x + 1) = x^2 - 2x - 8x - 4 = x^2 - 10x - 4

According to the order of operations, simplify the innermost grouping symbols first.

3

Distribute the

-3

across the simplified expression from the brackets.

-3(x^2 - 10x - 4) = -3x^2 + 30x + 12

The negative sign and the constant must be multiplied by every term inside the brackets.

4

Substitute all parts back into the main expression and combine like terms.

(6x^2 - 8x) + (-3x^2 + 30x + 12) + x^2 = (6x^2 - 3x^2 + x^2) + (-8x + 30x) + 12 = 4x^2 + 22x + 12

Combining coefficients of terms with the same variables and exponents yields the final simplified polynomial.

Key Concept

Simplifying complex algebraic expressions by applying the distributive property, handling negative multipliers, and combining like terms.

Question 106Question →

Match each complex algebraic expression with its completely simplified polynomial form.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The first expression maps to

-2x^2 - 2xy

, the second expression maps to

y^2

, the third expression maps to

-2xy + 9y^2

, and the fourth expression maps to

2x^2 - 4y^2

.

Each expression represents a distinct structural archetype for polynomial simplification. By systematically applying the order of operations—expanding binomials, distributing negative coefficients, and resolving nested grouping symbols from the inside out—all like terms can be properly combined to reach the correct canonical form.

Step-by-Step Solution

1

Simplify the first expression by resolving innermost parentheses, then brackets, then combining terms.

3x^2 - 6xy - [2x^2 - 4xy + 3x^2]

simplifies to

3x^2 - 6xy - 5x^2 + 4xy

, which yields

-2x^2 - 2xy

.

Following the order of operations from the innermost grouping symbols prevents sign errors when distributing negative coefficients.

2

Simplify the second expression by expanding the binomial square and distributing.

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

simplifies directly to

y^2

.

Proper binomial expansion

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

avoids the common error of dropping the middle term, allowing opposite terms to cancel.

3

Simplify the third expression using the difference of squares formula, then distribute the negative.

-(x^2 - 9y^2) + x^2 - 2xy

simplifies to

-x^2 + 9y^2 + x^2 - 2xy

, which reduces to

-2xy + 9y^2

.

Recognizing the difference of squares simplifies multiplication, and properly distributing the outer negative sign ensures the squared variable term has the correct positive sign.

4

Simplify the fourth expression by distributing within the brackets before multiplying by the outer coefficient.

4xy - 2[-x^2 + 2xy + 2y^2]

simplifies to

4xy + 2x^2 - 4xy - 4y^2 = 2x^2 - 4y^2

.

Combining like terms inside the bracket before multiplying by the outer constant significantly reduces the chance of sign and variable combination errors.

Key Concept

Simplifying Complex Multivariable Algebraic Expressions

Estimated Time:3m 0s

Question 107Question →

A student needs to simplify the expression $7k - 2(k + 4) + 1$ . What is the fully simplified form of this expression?

5k - 7

5k + 9

5k + 5

-2k

Show answer & explanation

Answer: $5k - 7$

Answer

The correct simplified expression is

5k - 7

.

The expression is simplified by first distributing the

-2

into

(k + 4)

to get

-2k - 8

. Next, the like terms are combined:

7k - 2k

becomes

5k

, and the constants

-8

and

1

combine to

-7

. This results in the final simplified expression of

5k - 7

.

Step-by-Step Solution

1

Apply the distributive property to the term

-2(k + 4)

.

7k - 2k - 8 + 1

The multiplier

-2

must be distributed to both

k

and

4

inside the parentheses.

2

Group the like terms together.

(7k - 2k) + (-8 + 1)

Like terms (variables with the same exponent, and constants) are grouped so they can be combined.

3

Combine the grouped like terms to simplify the expression.

5k - 7

Subtracting

2k

from

7k

gives

5k

, and adding

1

to

-8

gives

-7

.

Key Concept

Simplifying algebraic expressions by using the distributive property with negative multipliers and combining like terms.

Estimated Time:45s

Question 108Question →

Consider the following one-variable inequality:

12 - 3x \leq 27

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

x \geq -5

x \leq -5

x \geq 5

x \geq -13

Show answer & explanation

Answer: $x \geq -5$

Answer

x \geq -5

To solve the inequality, first subtract 12 from both sides to get

-3x \leq 15

. Next, divide both sides by -3. Because the division involves a negative number, the direction of the inequality symbol must be reversed, resulting in the correct solution of

x \geq -5

.

Step-by-Step Solution

1

Subtract

12

from both sides of the inequality.

-3x \leq 15

This step isolates the term containing the variable on one side of the inequality.

2

Divide both sides by

-3

and reverse the inequality symbol.

x \geq -5

According to the rules of algebra, whenever you multiply or divide an inequality by a negative number, the direction of the inequality sign must be flipped.

Key Concept

Solving One-Variable Inequalities

Estimated Time:45s

Question 109Question →

The solution to the one-variable inequality $-2x > 10$ is $x > -5$ .

True

False

Show answer & explanation

Answer: False

Answer

False

The correct answer is False because isolating

x

requires dividing by

-2

, which means the greater-than sign (

>

) must flip to a less-than sign (

<

). The actual mathematically correct solution is

x < -5

.

Step-by-Step Solution

1

Divide both sides of the inequality by

-2

.

\frac{-2x}{-2}

and

\frac{10}{-2}

To isolate the variable

x

on one side of the inequality.

2

Reverse the inequality symbol.

x < -5

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

Key Concept

Solving One-Variable Inequalities

Question 110Question →

An engineer is calculating the total weight supported by two structural pillars. The weight on the first pillar is given by the algebraic expression $5y - 2$ , and the weight on the second pillar is given by $3y + 6$ . What is the fully simplified algebraic expression for the total weight supported by both pillars combined?

Show answer & explanation

Answer: 8y + 4; 8y+4; 4 + 8y; 4+8y; 8y +4; 8y+ 4; 8y+ 4

Answer

8y + 4

The total weight is found by adding the two individual weight expressions. By identifying and grouping like terms, the variable terms are combined (

5y + 3y = 8y

) and the constant terms are combined (

-2 + 6 = 4

) to get the final simplified expression of

8y + 4

.

Step-by-Step Solution

1

Set up the addition of the two individual expressions

(5y - 2) + (3y + 6)

To find the total weight, both individual weight expressions must be added together.

2

Group the like terms

(5y + 3y) + (-2 + 6)

Like terms (terms with the same variable, and constant terms) must be grouped to simplify the expression.

3

Combine the grouped terms

8y + 4

Adding the coefficients of

y

(

5 + 3 = 8

) and adding the constants (

-2 + 6 = 4

) yields the fully simplified form.

Key Concept

Combining Like Terms

Question 111Question →

Match each complex unsimplified algebraic expression with its correctly simplified equivalent.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings match each multi-step expression to its exact simplified polynomial counterpart: L1 matches R1, L2 matches R2, L3 matches R3, and L4 matches R4.

Each correct match is formed by strictly following the order of operations: clearing innermost groupings first, distributing outer multipliers across all contained terms, and combining structurally identical variable terms into standard canonical form.

Step-by-Step Solution

1

Simplify the first expression

-3x[2y - (x - 4y)] + 2x(x - y)

.

The expression simplifies to

5x^2 - 20xy

.

By following the order of operations, innermost parentheses are cleared first, followed by distribution and combining like terms.

2

Simplify the second expression

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

.

The expression simplifies to

2x^2 + 4y^2

.

Expanding the terms inside the bracket yields

-x^2 + 2xy - 2y^2

. Distributing the

-2

flips the signs, and the

xy

terms successfully cancel.

3

Simplify the third expression

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

.

The expression simplifies to

3xy

.

Careful distribution of the negative sign across the simplified bracket

(x^2y - 2x^2 - 4xy)

allows multiple high-degree terms to cancel out perfectly.

4

Simplify the fourth expression

-(2x - 3y)(x + y) + x(2x + y) - y(x - 3y)

.

The expression simplifies to

xy + 6y^2

.

FOILing the first binomial and keeping the negative sign outside ensures signs are correctly flipped before combining with the other distributed terms.

Key Concept

Simplifying complex algebraic expressions through precise application of the distributive property, order of operations (PEMDAS), and combination of like terms.

Estimated Time:4m 0s

Question 112Question →

What is the simplified form of the algebraic expression below?

2(4x + 3) - x

13x

7x + 6

7x + 3

9x + 6

Show answer & explanation

Answer: $7x + 6$

Answer

The correctly simplified expression is

7x + 6

.

To simplify the expression, first apply the distributive property by multiplying 2 by both terms inside the parentheses:

2(4x) + 2(3) = 8x + 6

. Then, bring down the rest of the expression:

8x + 6 - x

. Finally, combine the like terms involving

x

:

8x - x = 7x

. The simplified result is

7x + 6

.

Step-by-Step Solution

1

Distribute the 2 to both terms inside the parentheses.

8x + 6 - x

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

2

Combine the like terms (

8x

and

-x

).

7x + 6

Only terms with the same variable and exponent can be combined.

Key Concept

Simplifying Algebraic Expressions (Distributive Property and Combining Like Terms)

Question 113Question →

Multiplying both sides of the inequality $-\frac{x}{3} < 5$ by $-3$ results in the equivalent inequality $x > -15$ .

True

False

Show answer & explanation

Answer: True

Answer

True

The statement correctly applies the algebraic rule that multiplying an inequality by a negative number reverses the direction of the inequality sign, properly yielding

x > -15

.

Step-by-Step Solution

1

Identify the operation needed to isolate the variable

x

.

Multiply both sides of the inequality

-\frac{x}{3} < 5

by

-3

.

The variable

x

is being divided by

-3

, so multiplying by

-3

is the inverse operation.

2

Perform the multiplication and apply the inequality division/multiplication rule.

-3 \times (-\frac{x}{3}) > -3 \times 5

, which simplifies to

x > -15

.

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

Key Concept

When solving inequalities, multiplying or dividing both sides by a negative number requires flipping the inequality sign.

Question 114Question →

Solve the following linear equation for $x$ :

\frac{2}{3}(6x - 9) - \frac{1}{2}(4x + 10) = -\frac{1}{4}(8 - 12x) + 3

Fill in the blanks below

The value of

x

that satisfies the equation is .

Show answer & explanation

Answer

The correct value for

x

is -12.

The correct answer is derived by methodically applying the distributive property to all grouped terms on both sides of the equation. By taking special care with negative signs during the multiplication of

-\frac{1}{2}

and

-\frac{1}{4}

, the equation simplifies cleanly to

2x - 11 = 3x + 1

. From there, standard algebraic isolation steps lead directly to

x = -12

.

Step-by-Step Solution

1

Distribute the fractional coefficients on the left side of the equation.

4x - 6 - 2x - 5

Applying the distributive property multiplies

\frac{2}{3}

by both

6x

and

-9

, and

-\frac{1}{2}

by both

4x

and

10

. Careful attention must be given to the negative sign.

2

Combine like terms on the left side.

2x - 11

Grouping the variable terms (

4x - 2x

) and constant terms (

-6 - 5

) simplifies the expression.

3

Distribute the fractional coefficient on the right side of the equation.

-2 + 3x + 3

Multiplying

-\frac{1}{4}

by

8

gives

-2

, and

-\frac{1}{4}

by

-12x

gives

+3x

. The negative times negative results in a positive term.

4

Combine like terms on the right side.

3x + 1

Adding the constant terms (

-2 + 3

) simplifies the right side.

5

Set the simplified left side equal to the simplified right side and isolate the variable

x

.

x = -12

Starting with

2x - 11 = 3x + 1

, subtract

2x

from both sides to get

-11 = x + 1

. Then, subtract

1

from both sides to solve for

x

.

Key Concept

Solving complex one-variable linear equations requiring careful fractional distribution and negative sign rules.

Question 115Question →

Which of the following describes the correct solution set for the inequality $-2x - 3 \leq 9$ ?

x \geq -6

x \leq -6

x \geq 6

x \leq 6

Show answer & explanation

Answer: $x \geq -6$

Answer

The correct solution is

x \geq -6

.

To solve the inequality, first isolate the variable term by adding 3 to both sides, which simplifies to

-2x \leq 12

. Next, divide both sides by -2. A critical rule of inequalities is that dividing or multiplying by a negative number requires reversing the inequality symbol. Therefore,

\leq

becomes

\geq

, and

12 \div -2 = -6

, resulting in the final answer of

x \geq -6

.

Step-by-Step Solution

1

Isolate the variable term by adding 3 to both sides of the inequality.

-2x \leq 12

This removes the constant term from the side with the variable, leaving only the term with x.

2

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

x \geq -6

When multiplying or dividing an inequality by a negative number, the direction of the inequality must flip to keep the mathematical statement true.

Key Concept

Solving one-variable inequalities and applying the inequality flip rule.

Estimated Time:45s

Question 116Question →

A student needs to solve the following algebraic inequality:

7 - 2(x - 3) \geq 4x - 5

What is the correct solution for $x$ ?

x \geq 3

x \geq 10

x \leq 3

x \leq -3

Show answer & explanation

Answer: $x \leq 3$

Answer

The correct solution is

x \leq 3

.

The correct answer is derived by carefully distributing the

-2

, combining the constants, grouping the variables on one side, and dividing by

-6

. Crucially, dividing by a negative coefficient triggers the rule to flip the

\geq

sign to

\leq

, resulting in

x \leq 3

.

Step-by-Step Solution

1

Distribute the

-2

into the parentheses on the left side of the inequality.

7 - 2x + 6 \geq 4x - 5

To eliminate the parentheses and allow like terms to be combined.

2

Combine the constant terms on the left side.

13 - 2x \geq 4x - 5

Adding

7

and

6

simplifies the left side of the algebraic expression.

3

Subtract

4x

from both sides of the inequality.

13 - 6x \geq -5

To group all terms containing the variable

x

onto the left side.

4

Subtract

13

from both sides.

-6x \geq -18

To isolate the variable term on one side and the constant values on the other.

5

Divide both sides by

-6

and reverse the inequality symbol.

x \leq 3

Whenever an inequality is multiplied or divided by a negative number, the direction of the inequality sign must be flipped.

Key Concept

Solving multi-step inequalities, including the distributive property and the inequality flip rule for negative coefficients.

Question 117Question →

Consider the algebraic expression:

$3a(2a - 5b) - 2b(a + 4b) + 7ab$

When completely simplified, what is the resulting polynomial?

6a^2 - 10ab - 8b^2

6a^2 - 10ab + 8b^2

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

-12ab

Show answer & explanation

Answer: $6a^2 - 10ab - 8b^2$

Answer

The simplified polynomial is

6a^2 - 10ab - 8b^2

.

The correct answer is found by first distributing the outer terms to eliminate parentheses, carefully observing negative signs. Distributing

3a

gives

6a^2 - 15ab

. Distributing

-2b

gives

-2ab - 8b^2

. Adding the

+7ab

at the end produces the expanded expression

6a^2 - 15ab - 2ab - 8b^2 + 7ab

. Combining the like terms (

-15ab - 2ab + 7ab

) yields

-10ab

, resulting in the final expression

6a^2 - 10ab - 8b^2

.

Step-by-Step Solution

1

Distribute

3a

to the terms inside the first set of parentheses.

3a(2a) - 3a(5b) = 6a^2 - 15ab

Applying the distributive property to eliminate the first set of parentheses.

2

Distribute

-2b

to the terms inside the second set of parentheses.

-2b(a) - 2b(4b) = -2ab - 8b^2

Applying the distributive property, ensuring the negative sign is multiplied into both terms.

3

Rewrite the full expression with the expanded terms.

6a^2 - 15ab - 2ab - 8b^2 + 7ab

Putting all terms together to prepare for combining like terms.

4

Identify and combine the like terms (the

ab

terms).

-15ab - 2ab + 7ab = -10ab

Only terms with the exact same variable combinations can be added or subtracted.

5

Construct the final polynomial.

6a^2 - 10ab - 8b^2

The

a^2

and

b^2

terms cannot be combined with anything else.

Key Concept

Simplifying algebraic expressions using the distributive property and combining like terms.

Question 118Question →

Which of the following algebraic expressions correctly represents the statement: 'Five less than the quotient of twice a number $p$ and three is subtracted from the product of four and the sum of the number and six'?

4(p + 6) - \left(\frac{2p}{3} - 5\right)

\left(\frac{2p}{3} - 5\right) - 4(p + 6)

4(p + 6) - \left(5 - \frac{2p}{3}\right)

4p + 6 - \left(\frac{2p}{3} - 5\right)

Show answer & explanation

Answer: $4(p + 6) - \left(\frac{2p}{3} - 5\right)$

Answer

The expression

4(p + 6) - \left(\frac{2p}{3} - 5\right)

correctly translates the word phrase into algebra.

The phrase requires breaking down into two parts. 'Five less than the quotient of twice a number

p

and three' translates to

(\frac{2p}{3} - 5)

because 'less than' reverses the order. 'The product of four and the sum of the number and six' translates to

4(p + 6)

because parentheses are needed to multiply the entire sum. Finally, 'is subtracted from' is a turnaround phrase meaning the first part is taken away from the second part, resulting in

4(p + 6) - \left(\frac{2p}{3} - 5\right)

.

Step-by-Step Solution

1

Translate the first phrase: 'Five less than the quotient of twice a number

p

and three'.

\frac{2p}{3} - 5

'Twice a number' is

2p

. The 'quotient... and three' means divide by

3

, giving

\frac{2p}{3}

. 'Five less than' indicates subtracting

5

from that quantity.

2

Translate the second phrase: 'the product of four and the sum of the number and six'.

4(p + 6)

'The sum of the number and six' requires grouping symbols:

(p + 6)

. 'The product of four and...' means multiplying

4

by the entire grouped sum.

3

Combine the two expressions using the operation 'is subtracted from'.

4(p + 6) - \left(\frac{2p}{3} - 5\right)

The phrase 'A is subtracted from B' translates to

B - A

. Therefore, the first expression must be subtracted from the second expression, requiring parentheses around the first expression.

Key Concept

Translating complex word phrases into algebraic expressions using correct order and grouping symbols.

Question 119Question →

Which of the following is the completely simplified form of the algebraic expression below?

4a(2a - 3b) - [3b(a - 4b) - 2a(a + b)]

10a^2 - 13ab + 12b^2

6a^2 - 17ab + 12b^2

9ab

6a^2 - 13ab - 12b^2

Show answer & explanation

Answer: $10a^2 - 13ab + 12b^2$

Answer

The simplified expression is

10a^2 - 13ab + 12b^2

.

The correct answer is derived by first expanding the inner products (

3ab - 12b^2

and

-2a^2 - 2ab

), combining them to get

ab - 12b^2 - 2a^2

, and then subtracting this entire grouped result from the expanded outer product (

8a^2 - 12ab

). Properly distributing the negative sign to all terms from the bracket yields

10a^2 - 13ab + 12b^2

.

Step-by-Step Solution

1

Distribute the terms inside the square brackets.

3b(a - 4b) = 3ab - 12b^2

and

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

According to the order of operations, simplify the innermost expressions first.

2

Combine like terms inside the brackets.

[3ab - 12b^2 - 2a^2 - 2ab] = [ab - 12b^2 - 2a^2]

Simplifying the bracket's contents reduces complexity before distributing the negative sign.

3

Distribute the terms outside the brackets.

4a(2a - 3b) = 8a^2 - 12ab

Expand the remaining multiplication.

4

Distribute the negative sign to the terms from the bracket.

-(ab - 12b^2 - 2a^2) = -ab + 12b^2 + 2a^2

Subtracting a polynomial requires reversing the sign of every term inside.

5

Combine all remaining like terms.

8a^2 - 12ab - ab + 12b^2 + 2a^2 = 10a^2 - 13ab + 12b^2

Group terms with identical variable parts to find the final simplified form.

Key Concept

Simplifying complex expressions by adhering strictly to the order of operations, accurately distributing terms and negative signs, and properly combining like terms.

Question 120Question →

The value $x = -2$ is a valid solution to the inequality $-4(x - 3) \leq 2x + 18$ .

True

False

Show answer & explanation

Answer: False

Answer

False

The statement is false because substituting

-2

into the inequality results in

20 \leq 14

, which is not mathematically accurate. Alternatively, algebraic solving yields

x \geq -1

, and

-2

does not satisfy this condition.

Step-by-Step Solution

1

Substitute

x = -2

into the left side of the inequality.

-4(-2 - 3) = -4(-5) = 20

To determine the numerical value of the left expression when

x = -2

.

2

Substitute

x = -2

into the right side of the inequality.

2(-2) + 18 = -4 + 18 = 14

To determine the numerical value of the right expression when

x = -2

.

3

Compare the two results using the given inequality symbol.

The statement

20 \leq 14

is false.

To verify if the substituted value satisfies the mathematical condition.

Key Concept

Solving One-Variable Inequalities

Algebra

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