Examkin

Algebra

282 questions

All types282Multiple choice147True / False7Numeric33Matching33Fill in the blank39Short answer23
Question 261Question →

What is the minimum integer value of xx that satisfies the inequality shown below?

3(2x−5)−8x≤4(x−1)−23 3(2x - 5) - 8x \leq 4(x - 1) - 23
Show answer & explanation

Answer: 2

Answer

2
Solving the inequality yields x≥2x \geq 2. Since xx must be greater than or equal to 2, the minimum integer value that makes this statement true is exactly 2.

Step-by-Step Solution

1
Distribute the constants into the parentheses on both sides of the inequality.
6x−15−8x≤4x−4−236x - 15 - 8x \leq 4x - 4 - 23
This removes the parentheses, allowing us to combine like terms.
2
Combine the like terms on each side of the inequality.
−2x−15≤4x−27-2x - 15 \leq 4x - 27
Simplifying each side makes it easier to isolate the variable.
3
Subtract 4x4x from both sides to group the variable terms on the left side.
−6x−15≤−27-6x - 15 \leq -27
We need all terms with xx on one side of the inequality to solve for it.
4
Add 1515 to both sides to isolate the term with the variable.
−6x≤−12-6x \leq -12
This leaves only the −6x-6x term on the left side.
5
Divide both sides by −6-6, remembering to flip the inequality symbol.
x≥2x \geq 2
Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.
6
Determine the minimum integer value that satisfies the simplified inequality.
2
The inequality states that xx must be greater than or equal to 22. The smallest integer that meets this condition is 22.

Key Concept

Solving linear inequalities requiring the distributive property, combining like terms, and reversing the inequality symbol when dividing by a negative coefficient.
Question 262Question →

Given the variable assignments x=−4x = -4, y=2y = 2, and z=−6z = -6, which of the following represents the evaluated result of the mathematical expression below?

x2y−∣x−z∣y+x x^2 y - \frac{|x - z|}{y + x}
Show answer & explanation

Answer: 33

Answer

33
Substituting the given values into the expression yields (−4)2(2)−∣−4−(−6)∣2+(−4)(-4)^2(2) - \frac{|-4 - (-6)|}{2 + (-4)}. Squaring −4-4 gives 1616, and multiplying by 22 results in 3232. Inside the absolute value, −4−(−6)-4 - (-6) becomes −4+6=2-4 + 6 = 2, so the numerator is 22. The denominator is 2+(−4)=−22 + (-4) = -2. The fraction simplifies to 2/−2=−12 / -2 = -1. Subtracting this from the first term gives 32−(−1)=3332 - (-1) = 33.

Step-by-Step Solution

1
Substitute the given values for x, y, and z into the expression.
(−4)2(2)−∣−4−(−6)∣2+(−4)(-4)^2 (2) - \frac{|-4 - (-6)|}{2 + (-4)}
Direct substitution is the first requirement to evaluate an algebraic expression.
2
Evaluate the first term with exponents and multiplication.
(−4)2=16(-4)^2 = 16, then 16×2=3216 \times 2 = 32.
Following the order of operations (PEMDAS), exponents are calculated before multiplication.
3
Simplify the expression inside the absolute value in the numerator.
∣−4−(−6)∣=∣−4+6∣=∣2∣=2|-4 - (-6)| = |-4 + 6| = |2| = 2.
Subtracting a negative is equivalent to adding a positive. Absolute value makes the result positive.
4
Simplify the denominator and evaluate the fraction.
Denominator is 2+(−4)=−22 + (-4) = -2. The fraction evaluates to 2−2=−1\frac{2}{-2} = -1.
The operations within the implied grouping of the fraction must be fully resolved before dividing.
5
Subtract the evaluated fraction from the first term.
32−(−1)=32+1=3332 - (-1) = 32 + 1 = 33.
Final subtraction, ensuring that subtracting a negative value is treated as addition.

Key Concept

Evaluating Complex Algebraic Expressions with Integers
Question 263Question →

Match each algebraic expression containing numerical coefficients on the left with its fully simplified equivalent form on the right.

Click a left item, then click its matching right item

Items

3y2⋅2y43y^2 \cdot 2y^4
12y74y3\frac{12y^7}{4y^3}
(2y3)3(2y^3)^3
5y−25y^{-2}

Matches

Show answer & explanation

Answer

The simplified pairs are: 3y2⋅2y43y^2 \cdot 2y^4 matches with 6y66y^6, 12y74y3\frac{12y^7}{4y^3} matches with 3y43y^4, (2y3)3(2y^3)^3 matches with 8y98y^9, and 5y−25y^{-2} matches with 5y2\frac{5}{y^2}.
Each left-side expression simplifies to exactly one unique form on the right by appropriately applying exponent rules to the variable bases and standard arithmetic operations to the numerical coefficients.

Step-by-Step Solution

1
Simplify the product expression 3y2⋅2y43y^2 \cdot 2y^4.
6y66y^6
When multiplying terms, handle coefficients and variables separately. Multiply coefficients (3×2=63 \times 2 = 6) and add exponents for identical bases (y2+4=y6y^{2+4} = y^6).
2
Simplify the quotient expression 12y74y3\frac{12y^7}{4y^3}.
3y43y^4
Divide the coefficients (12÷4=312 \div 4 = 3) and subtract the bottom exponent from the top exponent (y7−3=y4y^{7-3} = y^4).
3
Simplify the power expression (2y3)3(2y^3)^3.
8y98y^9
Raise both the coefficient and the variable to the power of 3. Calculate 23=82^3 = 8, and multiply the exponents for the variable 3×3=93 \times 3 = 9.
4
Simplify the negative exponent expression 5y−25y^{-2}.
5y2\frac{5}{y^2}
A negative exponent creates a reciprocal. Since the exponent only applies to yy, it moves to the denominator while the coefficient 5 stays in the numerator.

Key Concept

Handling numerical coefficients alongside basic operations with variable exponents.
Question 264Question →

Compute the value of the algebraic expression 3p2−2pq3p^2 - 2pq utilizing the parameters p=−3p = -3 and q=5q = 5.

Show answer & explanation

Answer: 57

Answer

57
The correct answer accurately substitutes the negative and positive integers into the formula and follows PEMDAS. Squaring −3-3 yields 99, which is multiplied by 33 to give 2727. The second term involves multiplying 22, −3-3, and 55 to yield −30-30. Subtracting −30-30 from 2727 is equivalent to 27+3027 + 30, which totals 5757.

Step-by-Step Solution

1
Substitute the assigned values for p and q into the expression.
3(−3)2−2(−3)(5)3(-3)^2 - 2(-3)(5)
Variable substitution requires replacing every instance of a letter with its numerical value.
2
Evaluate the exponent according to the order of operations.
3(9)−2(−3)(5)3(9) - 2(-3)(5)
Exponents must be calculated before multiplication. A negative number squared becomes positive.
3
Perform the multiplications for both terms.
27−(−30)27 - (-30)
Multiply 3×93 \times 9 to get 27, and multiply 2×(−3)×52 \times (-3) \times 5 to get −30-30.
4
Subtract the second term from the first term.
57
Subtracting a negative number is the same as adding its positive equivalent (27+30=5727 + 30 = 57).

Key Concept

Evaluating Algebraic Expressions
Estimated Time:45s
Question 265Question →

In a manufacturing process, the required cooling time in minutes is defined as eight subtracted from twice the sum of the baseline time, tt, and four. Which of the following algebraic expressions represents the required cooling time?

Show answer & explanation

Answer: 2(t+4)−82(t + 4) - 8

Answer

The correct algebraic expression is 2(t+4)−82(t + 4) - 8.
The correct expression carefully follows the algebraic translation rules: 'sum' dictates the grouping (t+4)(t + 4), 'twice' multiplies that entire group by 2 to get 2(t+4)2(t + 4), and 'eight subtracted from' places the −8- 8 at the end of the expression.

Step-by-Step Solution

1
Identify and group the addition operation described in the problem.
(t+4)(t + 4)
The phrase 'the sum of the baseline time, tt, and four' indicates that these two quantities must be added and kept together as a single unit.
2
Apply the multiplication to the grouped sum.
2(t+4)2(t + 4)
The word 'twice' means multiplying the entire group by 2.
3
Translate the subtraction phrase in the correct order.
2(t+4)−82(t + 4) - 8
The phrase 'eight subtracted from' is a turnaround phrase, meaning 8 must be taken away from the grouped expression.

Key Concept

Translating complex word phrases requires identifying grouping words (like 'sum') to use parentheses, and recognizing turnaround phrases (like 'subtracted from') that reverse the written order in the mathematical expression.
Question 266Question →

When p=−2p = -2 and q=−5q = -5, which of the following represents the value of 4p−3q4p - 3q?

Show answer & explanation

Answer: 7

Answer

7
The correct value is found by substituting p=−2p = -2 and q=−5q = -5 into the expression 4p−3q4p - 3q. Applying the order of operations, first multiply to get −8-8 and −15-15. Finally, evaluate −8−(−15)-8 - (-15), which correctly simplifies to −8+15=7-8 + 15 = 7.

Step-by-Step Solution

1
Substitute −2-2 for pp and −5-5 for qq into the algebraic expression.
4(−2)−3(−5)4(-2) - 3(-5)
Variables must be replaced with their given numerical values to evaluate the expression.
2
Multiply the terms following the order of operations.
−8−(−15)-8 - (-15)
Multiplication must be performed before subtraction.
3
Simplify the resulting subtraction.
−8+15=7-8 + 15 = 7
Subtracting a negative number is mathematically identical to adding a positive number.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:45s
Question 267Question →

Simplify the following algebraic expression completely:

(5x−2y4)(−2x5y−1)2 (5x^{-2}y^4)(-2x^5y^{-1})^2

Which of the following represents the correct equivalent form?

Show answer & explanation

Answer: 20x8y220x^8y^2

Answer

The correctly simplified expression is 20x8y220x^8y^2.
The correct result is achieved by following the order of operations and applying the laws of exponents. First, the exponent of 2 must be distributed to all parts of the second term: (−2)2(x5)2(y−1)2(-2)^2(x^5)^2(y^{-1})^2, which simplifies to 4x10y−24x^{10}y^{-2}. Next, this result is multiplied by the first term: (5x−2y4)(4x10y−2)(5x^{-2}y^4)(4x^{10}y^{-2}). Multiplying the coefficients gives 5×4=205 \times 4 = 20. Using the product of powers rule, adding the exponents for xx yields −2+10=8-2 + 10 = 8, and for yy yields 4+(−2)=24 + (-2) = 2. This results in the final expression 20x8y220x^8y^2.

Step-by-Step Solution

1
Apply the power of a product rule to the second term by squaring the coefficient and each variable.
(−2)2(x5)2(y−1)2=4x10y−2(-2)^2(x^5)^2(y^{-1})^2 = 4x^{10}y^{-2}
The order of operations (PEMDAS) requires evaluating exponents before performing multiplication.
2
Rewrite the full expression with the expanded term.
(5x−2y4)(4x10y−2)(5x^{-2}y^4)(4x^{10}y^{-2})
To clearly identify the coefficients and bases that need to be combined.
3
Multiply the numerical coefficients and apply the product of powers rule to the variables by adding their exponents.
(5×4)x−2+10y4+(−2)=20x8y2(5 \times 4)x^{-2+10}y^{4+(-2)} = 20x^8y^2
When multiplying exponential terms with the same base, you add the exponents.

Key Concept

Operations with Variable Exponents
Question 268Question →

Pair each algebraic operation on the left with its correctly simplified expression on the right. Assume all variables represent non-zero real numbers.

Click a left item, then click its matching right item

Items

(3m4n2)(4m2n5)(3m^4 n^2)(4m^2 n^5)
12m8n34m2n−4\frac{12m^8 n^3}{4m^2 n^{-4}}
(2m3n2)3(2m^3 n^2)^3
24m4n23m5n−1\frac{24m^4 n^2}{3m^5 n^{-1}}

Matches

Show answer & explanation

Answer

The correct matches require applying the product, quotient, and power rules of exponents systematically to each expression.
Each expression is correctly simplified by identifying the primary exponent rule required (product, quotient, or power) and applying it to both the numerical coefficients and the variables' exponents.

Step-by-Step Solution

1
Evaluate (3m4n2)(4m2n5)(3m^4 n^2)(4m^2 n^5) using the Product Rule.
12m6n712m^6 n^7
Multiply coefficients (3 * 4 = 12) and add exponents of corresponding variables (m4+2m^{4+2}, n2+5n^{2+5}).
2
Evaluate 12m8n34m2n−4\frac{12m^8 n^3}{4m^2 n^{-4}} using the Quotient Rule.
3m6n73m^6 n^7
Divide coefficients (12 / 4 = 3) and subtract denominator exponents from numerator exponents (m8−2m^{8-2}, n3−(−4)n^{3-(-4)}).
3
Evaluate (2m3n2)3(2m^3 n^2)^3 using the Power of a Product Rule.
8m9n68m^9 n^6
Cube the coefficient (23=82^3 = 8) and multiply inner exponents by the outer exponent (m3×3m^{3 \times 3}, n2×3n^{2 \times 3}).
4
Evaluate 24m4n23m5n−1\frac{24m^4 n^2}{3m^5 n^{-1}} using the Quotient Rule and Negative Exponent Rule.
8n3m\frac{8n^3}{m}
Divide coefficients (24 / 3 = 8), subtract exponents (m4−5=m−1m^{4-5} = m^{-1}, n2−(−1)=n3n^{2-(-1)} = n^3), and rewrite the negative exponent in the denominator.

Key Concept

Applying fundamental exponent laws (product rule, quotient rule, power of a power) and properly handling numerical coefficients separately from variable exponents.
Estimated Time:1m 30s
Question 269Question →

Which of the following is equivalent to the expression (3m−2n4)2(2m5n0)(3m^{-2}n^4)^2(2m^5n^0)?

Show answer & explanation

Answer: 18mn818mn^8

Answer

The simplified expression is 18mn818mn^8.
The correct answer is derived by first squaring each part of the initial term, making it 9m−4n89m^{-4}n^8. The second term simplifies to 2m52m^5 because n0=1n^0 = 1. Multiplying these together requires multiplying the coefficients (9×2=189 \times 2 = 18) and adding the exponents for mm (−4+5=1-4 + 5 = 1), resulting in 18mn818mn^8.

Step-by-Step Solution

1
Simplify the first factor by applying the power of 2 to the coefficient and the variable exponents.
(3)2(m−2)2(n4)2=9m−4n8(3)^2(m^{-2})^2(n^4)^2 = 9m^{-4}n^8
When a product is raised to a power, the exponent must be applied to every factor inside the parentheses, multiplying the variable exponents.
2
Simplify the second factor using the zero exponent rule.
2m5(1)=2m52m^5(1) = 2m^5
Any non-zero term raised to the power of 0 equals 1, so n0=1n^0 = 1.
3
Multiply the two simplified factors together.
(9m−4n8)(2m5)=(9⋅2)(m−4⋅m5)(n8)(9m^{-4}n^8)(2m^5) = (9 \cdot 2)(m^{-4} \cdot m^5)(n^8)
To multiply terms, multiply the numerical coefficients and add the exponents of like bases.
4
Perform the final arithmetic for the coefficients and exponents.
18m−4+5n8=18m1n8=18mn818m^{-4+5}n^8 = 18m^1n^8 = 18mn^8
Evaluating 9⋅2=189 \cdot 2 = 18 and combining the exponents of mm gives the final simplified expression.

Key Concept

Operations with Variable Exponents
Question 270Question →
Find the numerical value of the rational expression 3a2−4bc+2\frac{3a^2 - 4b}{c + 2} using the parameter values a=−2a = -2, b=−5b = -5, and c=−6c = -6.
Show answer & explanation

Answer: -8

Answer

The final numerical value is -8.
The correct value is found by carefully substituting the negative integer values into the rational expression, strictly applying the order of operations (exponents, then multiplication, then addition/subtraction), and accurately tracking the signs at each step to get 32 in the numerator and -4 in the denominator, which yields -8.

Step-by-Step Solution

1
Substitute the given values into the expression.
3(−2)2−4(−5)−6+2\frac{3(-2)^2 - 4(-5)}{-6 + 2}
Replace variables aa, bb, and cc with their assigned numerical equivalents to begin evaluation.
2
Evaluate the exponent in the numerator.
3(4)−4(−5)−6+2\frac{3(4) - 4(-5)}{-6 + 2}
According to the order of operations, exponents must be evaluated before multiplication. The square of a negative number is positive: (−2)2=4(-2)^2 = 4.
3
Perform the multiplications in the numerator.
12−(−20)−6+2\frac{12 - (-20)}{-6 + 2}
Calculate the products: 3×4=123 \times 4 = 12 and 4×(−5)=−204 \times (-5) = -20.
4
Simplify the subtraction in the numerator.
12+20−6+2=32−6+2\frac{12 + 20}{-6 + 2} = \frac{32}{-6 + 2}
Subtracting a negative number is equivalent to adding a positive number: 12−(−20)=12+20=3212 - (-20) = 12 + 20 = 32.
5
Evaluate the addition in the denominator.
32−4\frac{32}{-4}
Combine the constant terms in the denominator: −6+2=−4-6 + 2 = -4.
6
Divide the numerator by the denominator.
-8
A positive number divided by a negative number results in a negative quotient: 32÷(−4)=−832 \div (-4) = -8.

Key Concept

Evaluating Algebraic Expressions
Question 271Question →

Consider the following mathematical relationship:

34(m−8)+2=12m−1 \frac{3}{4}(m - 8) + 2 = \frac{1}{2}m - 1

Which of the following represents the correct solution for mm?

Show answer & explanation

Answer: 12

Answer

The correct solution is 12.
The correct solution is found by systematically applying the distributive property to eliminate parentheses, combining the constant terms, grouping the variable terms on one side, and fully isolating the variable. Following these steps accurately yields a value of 12.

Step-by-Step Solution

1
Apply the distributive property to the left side of the equation.
34m−6+2=12m−1 \frac{3}{4}m - 6 + 2 = \frac{1}{2}m - 1
This removes the parentheses by multiplying the fraction through the terms inside.
2
Combine the constant terms on the left side.
34m−4=12m−1 \frac{3}{4}m - 4 = \frac{1}{2}m - 1
Simplifying the expressions on both sides makes it easier to isolate the variable.
3
Subtract 12m \frac{1}{2}m from both sides to group the variable terms.
14m−4=−1 \frac{1}{4}m - 4 = -1
This places all terms containing the variable on one side of the equation. (Note: 34m−24m=14m \frac{3}{4}m - \frac{2}{4}m = \frac{1}{4}m )
4
Add 4 to both sides to isolate the variable term.
14m=3 \frac{1}{4}m = 3
This cancels out the constant on the left side, leaving only the variable term.
5
Multiply both sides by 4.
m=12 m = 12
This clears the fractional coefficient and solves for the final value of the variable.

Key Concept

Solving one-variable linear equations with fractions and distribution
Question 272Question →

A fitness center offers two membership plans. Plan A requires a one-time enrollment fee of 4545 dollars and costs 3030 dollars per month. Plan B requires a one-time enrollment fee of 105105 dollars and costs 2020 dollars per month. For how many months of membership will the total cost of both plans be exactly the same?

Show answer & explanation

Answer: 6

Answer

The plans will cost the exact same amount at 6 months.
Setting the cost of Plan A (45+30m45 + 30m) equal to the cost of Plan B (105+20m105 + 20m) creates the equation 45+30m=105+20m45 + 30m = 105 + 20m. By isolating the variable, we find that 10m=6010m = 60, which simplifies to m=6m = 6. Therefore, it takes exactly 66 months for both memberships to cost the same.

Step-by-Step Solution

1
Define the variable and formulate a one-variable linear equation.
45+30m=105+20m45 + 30m = 105 + 20m
To determine when the total costs are equal, the cost expression for Plan A (4545 fee plus 3030 per month) must be set equal to the cost expression for Plan B (105105 fee plus 2020 per month).
2
Subtract 20m20m from both sides of the equation.
45+10m=10545 + 10m = 105
This groups all terms containing the variable mm on one side of the equation.
3
Subtract 4545 from both sides of the equation.
10m=6010m = 60
This isolates the variable term by moving all constant values to the opposite side of the equation.
4
Divide both sides by 1010.
m=6m = 6
Dividing by the coefficient of the variable provides the final numerical solution for the unknown.

Key Concept

Translating a real-world scenario into a one-variable linear equation and isolating the variable to solve for an unknown value.
Estimated Time:1m 30s
Question 273Question →

Which of the following values for mm makes the equation true?

35(10m−15)−4m=12(8m+14)−34 \frac{3}{5}(10m - 15) - 4m = \frac{1}{2}(8m + 14) - 34
Show answer & explanation

Answer: 9

Answer

The correct answer is 9.
The correct answer is found by properly executing the distributive property on both sides to clear the parentheses. On the left side, 3/5 is distributed to yield 6m - 9. On the right side, 1/2 is distributed to yield 4m + 7. After combining the remaining like terms, the equation simplifies to 2m - 9 = 4m - 27. Adding 27 and subtracting 2m from both sides gives 18 = 2m, which simplifies to m = 9.

Step-by-Step Solution

1
Distribute the fractions into the parentheses on both sides of the equation.
6m−9−4m=4m+7−346m - 9 - 4m = 4m + 7 - 34
The distributive property must be applied first to expand the expressions and eliminate the parentheses.
2
Combine like terms on each side of the equal sign.
2m−9=4m−272m - 9 = 4m - 27
Simplifying each side individually makes it easier to isolate the variable in subsequent steps.
3
Subtract 2m from both sides.
−9=2m−27-9 = 2m - 27
This gathers all variable terms on one side of the equation.
4
Add 27 to both sides and divide the result by 2.
18=2m18 = 2m, which means m=9m = 9
This isolates the variable completely and provides the final solution.

Key Concept

Solving One-Variable Linear Equations
Estimated Time:1m 30s
Question 274Question →

A technician calculates a load factor using the expression x2−2yx+2\frac{x^2 - 2y}{x + 2}. If the inputs are x=−4x = -4 and y=3y = 3, what is the resulting load factor?

Show answer & explanation

Answer: -5

Answer

The resulting load factor is -5.
Substituting x=−4x = -4 and y=3y = 3 into the numerator gives (−4)2−2(3)=16−6=10(-4)^2 - 2(3) = 16 - 6 = 10. Substituting x=−4x = -4 into the denominator gives −4+2=−2-4 + 2 = -2. Finally, dividing the numerator by the denominator yields 10/−2=−510 / -2 = -5.

Step-by-Step Solution

1
Substitute the values x=−4x = -4 and y=3y = 3 into the numerator of the expression.
(−4)2−2(3)(-4)^2 - 2(3)
Variable substitution requires replacing the letters with their corresponding numerical values, using parentheses for negative numbers.
2
Evaluate the numerator following the standard order of operations.
16−6=1016 - 6 = 10
Squaring the negative integer −4-4 results in positive 1616. Multiplication (2×3=62 \times 3 = 6) must be performed before subtraction.
3
Substitute the value of xx into the denominator and evaluate.
−4+2=−2-4 + 2 = -2
Adding a positive integer to a negative integer reduces its absolute value while maintaining the negative sign.
4
Divide the evaluated numerator by the evaluated denominator.
10/−2=−510 / -2 = -5
A positive number divided by a negative number results in a negative quotient.

Key Concept

Evaluating rational algebraic expressions with negative integers
Question 275Question →

Which of the following expressions is equivalent to the product of (5x−2)(5x - 2) and (x2−3x−4)(x^2 - 3x - 4)?

Show answer & explanation

Answer: 5x3−17x2−14x+85x^3 - 17x^2 - 14x + 8

Answer

The correct answer is 5x3−17x2−14x+85x^3 - 17x^2 - 14x + 8.
The correct expression is found by systematically distributing each term in the first polynomial to each term in the second. First, multiplying 5x5x by (x2−3x−4)(x^2 - 3x - 4) yields 5x3−15x2−20x5x^3 - 15x^2 - 20x. Next, multiplying −2-2 by (x2−3x−4)(x^2 - 3x - 4) yields −2x2+6x+8-2x^2 + 6x + 8. Finally, combining the like terms (−15x2-15x^2 and −2x2-2x^2 becomes −17x2-17x^2; −20x-20x and 6x6x becomes −14x-14x) results in the simplified polynomial 5x3−17x2−14x+85x^3 - 17x^2 - 14x + 8.

Step-by-Step Solution

1
Distribute the first term of the binomial, 5x5x, to each term in the trinomial.
5x(x2−3x−4)=5x3−15x2−20x5x(x^2 - 3x - 4) = 5x^3 - 15x^2 - 20x
To expand the polynomials, every term in the first polynomial must be multiplied by every term in the second.
2
Distribute the second term of the binomial, −2-2, to each term in the trinomial.
−2(x2−3x−4)=−2x2+6x+8-2(x^2 - 3x - 4) = -2x^2 + 6x + 8
The second term, along with its negative sign, must also be multiplied by every term in the trinomial.
3
Add the results from Step 1 and Step 2 and combine like terms.
(5x3−15x2−20x)+(−2x2+6x+8)=5x3−17x2−14x+8(5x^3 - 15x^2 - 20x) + (-2x^2 + 6x + 8) = 5x^3 - 17x^2 - 14x + 8
Combining terms with the same variable to the same power fully simplifies the resulting polynomial.

Key Concept

Multiplying Polynomials using Distribution
Question 276Question →

Solve the following equation for the variable nn:

2n−43=n+52\frac{2n - 4}{3} = \frac{n + 5}{2}

What is the correct value of nn?

Show answer & explanation

Answer: 23

Answer

The correct value for the variable is 23.
The correct value is 23, which is found by properly cross-multiplying the fractions, distributing the values to all terms in the numerators, and maintaining equality while isolating the variable.

Step-by-Step Solution

1
Cross-multiply to eliminate the fractions.
2(2n−4)=3(n+5)2(2n - 4) = 3(n + 5)
Multiplying each side's numerator by the other side's denominator clears the fractions from the equation.
2
Distribute the constants on both sides of the equation.
4n−8=3n+154n - 8 = 3n + 15
Applying the distributive property removes the parentheses so like terms can be combined.
3
Subtract 3n3n from both sides.
n−8=15n - 8 = 15
This moves all variable terms to the left side of the equation to begin isolating nn.
4
Add 8 to both sides.
n=23n = 23
This isolates the variable nn to find the final solution.

Key Concept

Solving One-Variable Linear Equations
Estimated Time:1m 30s
Question 277Question →

If −4(2n−3)=2n−3(3n+1)-4(2n - 3) = 2n - 3(3n + 1), what is the value of nn?

Show answer & explanation

Answer: 15

Answer

15
The equation correctly simplifies to -8n + 12 = -7n - 3 after proper distribution and combining of like terms. Isolating the variable by adding 8n and 3 to both sides yields n = 15.

Step-by-Step Solution

1
Distribute the factors on both sides of the equation.
−8n+12=2n−9n−3-8n + 12 = 2n - 9n - 3
Multiplying the outside numbers across the terms inside the parentheses removes the grouping symbols.
2
Combine like terms on the right side.
−8n+12=−7n−3-8n + 12 = -7n - 3
Adding 2n2n and −9n-9n simplifies the right side to a single variable term.
3
Isolate the variable nn.
12+3=−7n+8n12 + 3 = -7n + 8n, which simplifies to 15=n15 = n
Moving all variable terms to one side and constant terms to the other side solves the equation.

Key Concept

Solving one-variable linear equations requiring distribution and combining like terms.
Question 278Question →

Find the value of xx that satisfies the equation:

x−34+2x+13=5x6 \frac{x - 3}{4} + \frac{2x + 1}{3} = \frac{5x}{6}

Enter the exact value of xx.

Show answer & explanation

Answer: 5

Answer

5
By finding the least common denominator (12) and multiplying every term by it, the fractions are cleanly eliminated. After applying the distributive property and combining like terms (11x−5=10x11x - 5 = 10x), isolating the variable xx yields exactly 5.

Step-by-Step Solution

1
Multiply all terms by the least common denominator (12).
12(x−34)+12(2x+13)=12(5x6)12 \left(\frac{x - 3}{4}\right) + 12 \left(\frac{2x + 1}{3}\right) = 12 \left(\frac{5x}{6}\right)
Multiplying by the LCD eliminates all fractions from the equation, simplifying further steps.
2
Simplify the resulting fractions.
3(x−3)+4(2x+1)=2(5x)3(x - 3) + 4(2x + 1) = 2(5x)
Dividing 12 by each respective denominator leaves an integer multiplier for the numerators.
3
Distribute the multipliers across the terms in the parentheses.
3x−9+8x+4=10x3x - 9 + 8x + 4 = 10x
The distributive property is required to remove the parentheses.
4
Combine like terms on the left side of the equation.
11x−5=10x11x - 5 = 10x
Grouping xx terms (3x+8x3x + 8x) and constant terms (−9+4-9 + 4) simplifies the expression.
5
Isolate the variable xx.
x=5x = 5
Subtracting 10x10x from both sides gives x−5=0x - 5 = 0. Adding 55 to both sides results in x=5x = 5.

Key Concept

Solving linear equations involving rational expressions by multiplying by a least common denominator.
Estimated Time:1m 30s
Question 279Question →

A rectangular garden has a length of (3x−2)(3x - 2) meters and a width of (2x+5)(2x + 5) meters. Inside the garden, a smaller rectangular paved patio is built with a length of (x+4)(x + 4) meters and a width of 2x2x meters. Which of the following expressions represents the total area of the garden that is NOT paved, in square meters?

Show answer & explanation

Answer: 4x2+3x−104x^2 + 3x - 10

Answer

The expression representing the unpaved area is 4x2+3x−104x^2 + 3x - 10.
The correct answer is found by calculating the area of the entire garden and subtracting the area of the patio. First, multiply the dimensions of the garden using the distributive property (FOIL): (3x−2)(2x+5)=6x2+15x−4x−10=6x2+11x−10(3x - 2)(2x + 5) = 6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10. Next, find the area of the patio by distributing 2x2x: (x+4)(2x)=2x2+8x(x + 4)(2x) = 2x^2 + 8x. Finally, subtract the patio's area from the total area: (6x2+11x−10)−(2x2+8x)(6x^2 + 11x - 10) - (2x^2 + 8x). Make sure to distribute the negative sign to both terms in the second polynomial: 6x2+11x−10−2x2−8x=4x2+3x−106x^2 + 11x - 10 - 2x^2 - 8x = 4x^2 + 3x - 10.

Step-by-Step Solution

1
Calculate the total area of the garden by multiplying its length and width.
(3x−2)(2x+5)=6x2+15x−4x−10=6x2+11x−10(3x - 2)(2x + 5) = 6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10
The area of a rectangle is found by multiplying its length by its width.
2
Calculate the area of the paved patio by multiplying its length and width.
(x+4)(2x)=2x2+8x(x + 4)(2x) = 2x^2 + 8x
We need the area of the inner rectangle in order to subtract it from the total area.
3
Subtract the patio's area from the total garden area to find the unpaved area.
(6x2+11x−10)−(2x2+8x)=6x2+11x−10−2x2−8x=4x2+3x−10(6x^2 + 11x - 10) - (2x^2 + 8x) = 6x^2 + 11x - 10 - 2x^2 - 8x = 4x^2 + 3x - 10
The unpaved region is the difference between the total area and the paved area. Note that the negative sign must be distributed to both terms in the second polynomial.

Key Concept

Multiplying Polynomials
Question 280Question →

Determine the value of the missing coefficient by evaluating the problem below.

Fill in the blanks below

A rectangular solar panel has a length of (5k−4)(5k - 4) inches and a width of (3k+7)(3k + 7) inches. The area of the panel is represented by the fully expanded polynomial Ak2+Bk+CAk^2 + Bk + C square inches. The value of the coefficient BB is .
Show answer & explanation

Answer

23
By multiplying the binomials (5k−4)(5k - 4) and (3k+7)(3k + 7), the expanded expression is 15k2+35k−12k−2815k^2 + 35k - 12k - 28. Combining the middle terms (35k−12k35k - 12k) yields 23k23k. Since the polynomial is in the standard form Ak2+Bk+CAk^2 + Bk + C, the value of BB is 23.

Step-by-Step Solution

1
Set up the area calculation by multiplying the length and width binomials.
Area = (5k−4)(3k+7)(5k - 4)(3k + 7)
The area of a rectangle is found by multiplying its length by its width.
2
Distribute the terms using the FOIL method (First, Outer, Inner, Last).
15k2+35k−12k−2815k^2 + 35k - 12k - 28
Each term in the first binomial must be multiplied by each term in the second binomial.
3
Combine the like terms in the middle to find the value of BB.
15k2+23k−2815k^2 + 23k - 28
Adding 35k35k and −12k-12k simplifies the polynomial to the standard form Ak2+Bk+CAk^2 + Bk + C.
4
Identify the coefficient BB corresponding to the kk term.
B=23B = 23
The polynomial is 15k2+23k−2815k^2 + 23k - 28, so the coefficient of the kk term is 23.

Key Concept

Multiplying binomials and combining like terms to find a specific coefficient.
PreviousPage 14 / 15Next
Algebra Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 14 | Examkin