Algebra Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB) — Sayfa 14

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

3(2x - 5) - 8x \leq 4(x - 1) - 23

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

Solving the inequality yields

x \geq 2

. Since

x

must be greater than or equal to 2, the minimum integer value that makes this statement true is exactly 2.

Adım Adım Çözüm

1

Distribute the constants into the parentheses on both sides of the inequality.

6x - 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 \leq 4x - 27

Simplifying each side makes it easier to isolate the variable.

3

Subtract

4x

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

-6x - 15 \leq -27

We need all terms with

x

on one side of the inequality to solve for it.

4

Add

15

to both sides to isolate the term with the variable.

-6x \leq -12

This leaves only the

-6x

term on the left side.

5

Divide both sides by

-6

, remembering to flip the inequality symbol.

x \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

x

must be greater than or equal to

2

. The smallest integer that meets this condition is

2

.

Anahtar Kavram

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

Soru 262Soru →

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

x^2 y - \frac{|x - z|}{y + x}

33

-31

-15

31

Cevabı ve açıklamayı göster

Cevap: 33

Cevap

33

Substituting the given values into the expression yields

(-4)^2(2) - \frac{|-4 - (-6)|}{2 + (-4)}

. Squaring

-4

gives

16

, and multiplying by

2

results in

32

. Inside the absolute value,

-4 - (-6)

becomes

-4 + 6 = 2

, so the numerator is

2

. The denominator is

2 + (-4) = -2

. The fraction simplifies to

2 / -2 = -1

. Subtracting this from the first term gives

32 - (-1) = 33

.

Adım Adım Çözüm

1

Substitute the given values for x, y, and z into the expression.

(-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

, then

16 \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

.

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) = -2

. The fraction evaluates to

\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 = 33

.

Final subtraction, ensuring that subtracting a negative value is treated as addition.

Anahtar Kavram

Evaluating Complex Algebraic Expressions with Integers

Soru 263Soru →

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

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The simplified pairs are:

3y^2 \cdot 2y^4

matches with

6y^6

,

\frac{12y^7}{4y^3}

matches with

3y^4

,

(2y^3)^3

matches with

8y^9

, and

5y^{-2}

matches with

\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.

Adım Adım Çözüm

1

Simplify the product expression

3y^2 \cdot 2y^4

.

6y^6

When multiplying terms, handle coefficients and variables separately. Multiply coefficients (

3 \times 2 = 6

) and add exponents for identical bases (

y^{2+4} = y^6

).

2

Simplify the quotient expression

\frac{12y^7}{4y^3}

.

3y^4

Divide the coefficients (

12 \div 4 = 3

) and subtract the bottom exponent from the top exponent (

y^{7-3} = y^4

).

3

Simplify the power expression

(2y^3)^3

.

8y^9

Raise both the coefficient and the variable to the power of 3. Calculate

2^3 = 8

, and multiply the exponents for the variable

3 \times 3 = 9

.

4

Simplify the negative exponent expression

5y^{-2}

.

\frac{5}{y^2}

A negative exponent creates a reciprocal. Since the exponent only applies to

y

, it moves to the denominator while the coefficient 5 stays in the numerator.

Anahtar Kavram

Handling numerical coefficients alongside basic operations with variable exponents.

Soru 264Soru →

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

57

3

-3

111

Cevabı ve açıklamayı göster

Cevap: 57

Cevap

57

The correct answer accurately substitutes the negative and positive integers into the formula and follows PEMDAS. Squaring

-3

yields

9

, which is multiplied by

3

to give

27

. The second term involves multiplying

2

,

-3

, and

5

to yield

-30

. Subtracting

-30

from

27

is equivalent to

27 + 30

, which totals

57

.

Adım Adım Çözüm

1

Substitute the assigned values for p and q into the expression.

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)

Exponents must be calculated before multiplication. A negative number squared becomes positive.

3

Perform the multiplications for both terms.

27 - (-30)

Multiply

3 \times 9

to get 27, and multiply

2 \times (-3) \times 5

to get

-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 = 57

).

Anahtar Kavram

Evaluating Algebraic Expressions

Tahmini Süre:45s

Soru 265Soru →

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

2(t + 4) - 8

8 - 2(t + 4)

2t + 4 - 8

2(5t) - 8

Cevabı ve açıklamayı göster

Cevap: $2(t + 4) - 8$

Cevap

The correct algebraic expression is

2(t + 4) - 8

.

The correct expression carefully follows the algebraic translation rules: 'sum' dictates the grouping

(t + 4)

, 'twice' multiplies that entire group by 2 to get

2(t + 4)

, and 'eight subtracted from' places the

- 8

at the end of the expression.

Adım Adım Çözüm

1

Identify and group the addition operation described in the problem.

(t + 4)

The phrase 'the sum of the baseline time,

t

, 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)

The word 'twice' means multiplying the entire group by 2.

3

Translate the subtraction phrase in the correct order.

2(t + 4) - 8

The phrase 'eight subtracted from' is a turnaround phrase, meaning 8 must be taken away from the grouped expression.

Anahtar Kavram

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.

Soru 266Soru →

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

7

-7

-23

55

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

The correct value is found by substituting

p = -2

and

q = -5

into the expression

4p - 3q

. Applying the order of operations, first multiply to get

-8

and

-15

. Finally, evaluate

-8 - (-15)

, which correctly simplifies to

-8 + 15 = 7

.

Adım Adım Çözüm

1

Substitute

-2

for

p

and

-5

for

q

into the algebraic expression.

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)

Multiplication must be performed before subtraction.

3

Simplify the resulting subtraction.

-8 + 15 = 7

Subtracting a negative number is mathematically identical to adding a positive number.

Anahtar Kavram

Evaluating Algebraic Expressions

Tahmini Süre:45s

Soru 267Soru →

Simplify the following algebraic expression completely:

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

Which of the following represents the correct equivalent form?

20x^8y^2

20x^5y^5

100x^6y^6

-20x^8y^2

Cevabı ve açıklamayı göster

Cevap: $20x^8y^2$

Cevap

The correctly simplified expression is

20x^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(x^5)^2(y^{-1})^2

, which simplifies to

4x^{10}y^{-2}

. Next, this result is multiplied by the first term:

(5x^{-2}y^4)(4x^{10}y^{-2})

. Multiplying the coefficients gives

5 \times 4 = 20

. Using the product of powers rule, adding the exponents for

x

yields

-2 + 10 = 8

, and for

y

yields

4 + (-2) = 2

. This results in the final expression

20x^8y^2

.

Adım Adım Çözüm

1

Apply the power of a product rule to the second term by squaring the coefficient and each variable.

(-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^{-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 \times 4)x^{-2+10}y^{4+(-2)} = 20x^8y^2

When multiplying exponential terms with the same base, you add the exponents.

Anahtar Kavram

Operations with Variable Exponents

Soru 268Soru →

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

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

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.

Adım Adım Çözüm

1

Evaluate

(3m^4 n^2)(4m^2 n^5)

using the Product Rule.

12m^6 n^7

Multiply coefficients (3 * 4 = 12) and add exponents of corresponding variables (

m^{4+2}

,

n^{2+5}

).

2

Evaluate

\frac{12m^8 n^3}{4m^2 n^{-4}}

using the Quotient Rule.

3m^6 n^7

Divide coefficients (12 / 4 = 3) and subtract denominator exponents from numerator exponents (

m^{8-2}

,

n^{3-(-4)}

).

3

Evaluate

(2m^3 n^2)^3

using the Power of a Product Rule.

8m^9 n^6

Cube the coefficient (

2^3 = 8

) and multiply inner exponents by the outer exponent (

m^{3 \times 3}

,

n^{2 \times 3}

).

4

Evaluate

\frac{24m^4 n^2}{3m^5 n^{-1}}

using the Quotient Rule and Negative Exponent Rule.

\frac{8n^3}{m}

Divide coefficients (24 / 3 = 8), subtract exponents (

m^{4-5} = m^{-1}

,

n^{2-(-1)} = n^3

), and rewrite the negative exponent in the denominator.

Anahtar Kavram

Applying fundamental exponent laws (product rule, quotient rule, power of a power) and properly handling numerical coefficients separately from variable exponents.

Tahmini Süre:1m 30s

Soru 269Soru →

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

18mn^8

6mn^8

18m^5n^6

\frac{18n^8}{m^9}

Cevabı ve açıklamayı göster

Cevap: $18mn^8$

Cevap

The simplified expression is

18mn^8

.

The correct answer is derived by first squaring each part of the initial term, making it

9m^{-4}n^8

. The second term simplifies to

2m^5

because

n^0 = 1

. Multiplying these together requires multiplying the coefficients (

9 \times 2 = 18

) and adding the exponents for

m

(

-4 + 5 = 1

), resulting in

18mn^8

.

Adım Adım Çözüm

1

Simplify the first factor by applying the power of 2 to the coefficient and the variable exponents.

(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.

2m^5(1) = 2m^5

Any non-zero term raised to the power of 0 equals 1, so

n^0 = 1

.

3

Multiply the two simplified factors together.

(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+5}n^8 = 18m^1n^8 = 18mn^8

Evaluating

9 \cdot 2 = 18

and combining the exponents of

m

gives the final simplified expression.

Anahtar Kavram

Operations with Variable Exponents

Soru 270Soru →

Find the numerical value of the rational expression

\frac{3a^2 - 4b}{c + 2}

using the parameter values

a = -2

,

b = -5

, and

c = -6

.

Cevabı ve açıklamayı göster

Cevap: -8

Cevap

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.

Adım Adım Çözüm

1

Substitute the given values into the expression.

\frac{3(-2)^2 - 4(-5)}{-6 + 2}

Replace variables

a

,

b

, and

c

with their assigned numerical equivalents to begin evaluation.

2

Evaluate the exponent in the numerator.

\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

.

3

Perform the multiplications in the numerator.

\frac{12 - (-20)}{-6 + 2}

Calculate the products:

3 \times 4 = 12

and

4 \times (-5) = -20

.

4

Simplify the subtraction in the numerator.

\frac{12 + 20}{-6 + 2} = \frac{32}{-6 + 2}

Subtracting a negative number is equivalent to adding a positive number:

12 - (-20) = 12 + 20 = 32

.

5

Evaluate the addition in the denominator.

\frac{32}{-4}

Combine the constant terms in the denominator:

-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 \div (-4) = -8

.

Anahtar Kavram

Evaluating Algebraic Expressions

Soru 271Soru →

Consider the following mathematical relationship:

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

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

12

14

-20

21

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

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.

Adım Adım Çözüm

1

Apply the distributive property to the left side of the equation.

\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.

\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

\frac{1}{2}m

from both sides to group the variable terms.

\frac{1}{4}m - 4 = -1

This places all terms containing the variable on one side of the equation. (Note:

\frac{3}{4}m - \frac{2}{4}m = \frac{1}{4}m

)

4

Add 4 to both sides to isolate the variable term.

\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

This clears the fractional coefficient and solves for the final value of the variable.

Anahtar Kavram

Solving one-variable linear equations with fractions and distribution

Soru 272Soru →

A fitness center offers two membership plans. Plan A requires a one-time enrollment fee of $45$ dollars and costs $30$ dollars per month. Plan B requires a one-time enrollment fee of $105$ dollars and costs $20$ dollars per month. For how many months of membership will the total cost of both plans be exactly the same?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The plans will cost the exact same amount at 6 months.

Setting the cost of Plan A (

45 + 30m

) equal to the cost of Plan B (

105 + 20m

) creates the equation

45 + 30m = 105 + 20m

. By isolating the variable, we find that

10m = 60

, which simplifies to

m = 6

. Therefore, it takes exactly

6

months for both memberships to cost the same.

Adım Adım Çözüm

1

Define the variable and formulate a one-variable linear equation.

45 + 30m = 105 + 20m

To determine when the total costs are equal, the cost expression for Plan A (

45

fee plus

30

per month) must be set equal to the cost expression for Plan B (

105

fee plus

20

per month).

2

Subtract

20m

from both sides of the equation.

45 + 10m = 105

This groups all terms containing the variable

m

on one side of the equation.

3

Subtract

45

from both sides of the equation.

10m = 60

This isolates the variable term by moving all constant values to the opposite side of the equation.

4

Divide both sides by

10

.

m = 6

Dividing by the coefficient of the variable provides the final numerical solution for the unknown.

Anahtar Kavram

Translating a real-world scenario into a one-variable linear equation and isolating the variable to solve for an unknown value.

Tahmini Süre:1m 30s

Soru 273Soru →

Which of the following values for $m$ makes the equation true?

\frac{3}{5}(10m - 15) - 4m = \frac{1}{2}(8m + 14) - 34

6

9

18

3

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

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.

Adım Adım Çözüm

1

Distribute the fractions into the parentheses on both sides of the equation.

6m - 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 - 27

Simplifying each side individually makes it easier to isolate the variable in subsequent steps.

3

Subtract 2m from both sides.

-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 = 2m

, which means

m = 9

This isolates the variable completely and provides the final solution.

Anahtar Kavram

Solving One-Variable Linear Equations

Tahmini Süre:1m 30s

Soru 274Soru →

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

-21

-11

-5

11

Cevabı ve açıklamayı göster

Cevap: -5

Cevap

The resulting load factor is -5.

Substituting

x = -4

and

y = 3

into the numerator gives

(-4)^2 - 2(3) = 16 - 6 = 10

. Substituting

x = -4

into the denominator gives

-4 + 2 = -2

. Finally, dividing the numerator by the denominator yields

10 / -2 = -5

.

Adım Adım Çözüm

1

Substitute the values

x = -4

and

y = 3

into the numerator of the expression.

(-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 = 10

Squaring the negative integer

-4

results in positive

16

. Multiplication (

2 \times 3 = 6

) must be performed before subtraction.

3

Substitute the value of

x

into the denominator and evaluate.

-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 = -5

A positive number divided by a negative number results in a negative quotient.

Anahtar Kavram

Evaluating rational algebraic expressions with negative integers

Soru 275Soru →

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

5x^3 + 8

5x^3 - 17x^2 - 26x - 8

5x^3 - 17x^2 - 14x + 8

5x^3 - 31x + 8

Cevabı ve açıklamayı göster

Cevap: $5x^3 - 17x^2 - 14x + 8$

Cevap

The correct answer is

5x^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

5x

by

(x^2 - 3x - 4)

yields

5x^3 - 15x^2 - 20x

. Next, multiplying

-2

by

(x^2 - 3x - 4)

yields

-2x^2 + 6x + 8

. Finally, combining the like terms (

-15x^2

and

-2x^2

becomes

-17x^2

;

-20x

and

6x

becomes

-14x

) results in the simplified polynomial

5x^3 - 17x^2 - 14x + 8

.

Adım Adım Çözüm

1

Distribute the first term of the binomial,

5x

, to each term in the trinomial.

5x(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

, to each term in the trinomial.

-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.

(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.

Anahtar Kavram

Multiplying Polynomials using Distribution

Soru 276Soru →

Solve the following equation for the variable $n$ :

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

What is the correct value of $n$ ?

1

7

9

23

Cevabı ve açıklamayı göster

Cevap: 23

Cevap

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.

Adım Adım Çözüm

1

Cross-multiply to eliminate the fractions.

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 + 15

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

3

Subtract

3n

from both sides.

n - 8 = 15

This moves all variable terms to the left side of the equation to begin isolating

n

.

4

Add 8 to both sides.

n = 23

This isolates the variable

n

to find the final solution.

Anahtar Kavram

Solving One-Variable Linear Equations

Tahmini Süre:1m 30s

Soru 277Soru →

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

15

-9

1

0

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

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.

Adım Adım Çözüm

1

Distribute the factors on both sides of the equation.

-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

Adding

2n

and

-9n

simplifies the right side to a single variable term.

3

Isolate the variable

n

.

12 + 3 = -7n + 8n

, which simplifies to

15 = n

Moving all variable terms to one side and constant terms to the other side solves the equation.

Anahtar Kavram

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

Soru 278Soru →

Find the value of $x$ that satisfies the equation:

\frac{x - 3}{4} + \frac{2x + 1}{3} = \frac{5x}{6}

Enter the exact value of $x$ .

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

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 = 10x

), isolating the variable

x

yields exactly 5.

Adım Adım Çözüm

1

Multiply all terms by the least common denominator (12).

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)

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 = 10x

The distributive property is required to remove the parentheses.

4

Combine like terms on the left side of the equation.

11x - 5 = 10x

Grouping

x

terms (

3x + 8x

) and constant terms (

-9 + 4

) simplifies the expression.

5

Isolate the variable

x

.

x = 5

Subtracting

10x

from both sides gives

x - 5 = 0

. Adding

5

to both sides results in

x = 5

.

Anahtar Kavram

Solving linear equations involving rational expressions by multiplying by a least common denominator.

Tahmini Süre:1m 30s

Soru 279Soru →

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

4x^2 + 3x - 10

4x^2 + 19x - 10

4x^2 - 8x - 10

4x^2 - 7x

Cevabı ve açıklamayı göster

Cevap: $4x^2 + 3x - 10$

Cevap

The expression representing the unpaved area is

4x^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) = 6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10

. Next, find the area of the patio by distributing

2x

:

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

. Finally, subtract the patio's area from the total area:

(6x^2 + 11x - 10) - (2x^2 + 8x)

. Make sure to distribute the negative sign to both terms in the second polynomial:

6x^2 + 11x - 10 - 2x^2 - 8x = 4x^2 + 3x - 10

.

Adım Adım Çözüm

1

Calculate the total area of the garden by multiplying its length and width.

(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) = 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.

(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.

Anahtar Kavram

Multiplying Polynomials

Soru 280Soru →

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

Aşağıdaki boşlukları doldurun

A rectangular solar panel has a length of

(5k - 4)

inches and a width of

(3k + 7)

inches. The area of the panel is represented by the fully expanded polynomial

Ak^2 + Bk + C

square inches. The value of the coefficient

B

is .

Cevabı ve açıklamayı göster

Cevap

23

By multiplying the binomials

(5k - 4)

and

(3k + 7)

, the expanded expression is

15k^2 + 35k - 12k - 28

. Combining the middle terms (

35k - 12k

) yields

23k

. Since the polynomial is in the standard form

Ak^2 + Bk + C

, the value of

B

is 23.

Adım Adım Çözüm

1

Set up the area calculation by multiplying the length and width binomials.

Area =

(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).

15k^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

B

.

15k^2 + 23k - 28

Adding

35k

and

-12k

simplifies the polynomial to the standard form

Ak^2 + Bk + C

.

4

Identify the coefficient

B

corresponding to the

k

term.

B = 23

The polynomial is

15k^2 + 23k - 28

, so the coefficient of the

k

term is 23.

Anahtar Kavram

Multiplying binomials and combining like terms to find a specific coefficient.

Algebra

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