Elementary Algebra Practice Questions — ACT — Page 9

Question 161Question →

A square garden plot has a side length of $3s - 2$ yards. A walkway of uniform width $s + 3$ yards is built around the entire garden. Which of the following expressions represents the area, in square yards, of the walkway?

7s^2 + 20s - 3

16s^2 + 28s + 20

16s^2 + 52s + 12

16s^2 + 12

4s^2 + 52s + 12

Show answer & explanation

Answer: 16s^2 + 52s + 12

Answer

16s^2 + 52s + 12

The correct answer is found by first calculating the outer side length, which is the inner side length plus twice the walkway width:

(3s - 2) + 2(s + 3) = 5s + 4

. Squaring this yields the outer area of

25s^2 + 40s + 16

. The inner area is

(3s - 2)^2 = 9s^2 - 12s + 4

. Subtracting the inner area from the outer area and distributing the negative sign properly gives

(25s^2 + 40s + 16) - (9s^2 - 12s + 4) = 16s^2 + 52s + 12

.

Step-by-Step Solution

1

Determine the outer side length of the square including the walkway.

The outer side length is

(3s - 2) + 2(s + 3) = 3s - 2 + 2s + 6 = 5s + 4

yards.

Since the walkway surrounds the garden on all sides, the width of the walkway must be added to both ends of the garden's side length.

2

Calculate the area of the outer square and the inner square garden by squaring their respective side lengths.

Outer Area =

(5s + 4)^2 = 25s^2 + 40s + 16

and Inner Area =

(3s - 2)^2 = 9s^2 - 12s + 4

.

The area of a square is equal to the square of its side length.

3

Subtract the inner garden area from the outer area to find the walkway area, distributing the negative sign to all terms of the inner area.

Walkway Area =

(25s^2 + 40s + 16) - (9s^2 - 12s + 4) = 25s^2 + 40s + 16 - 9s^2 + 12s - 4 = 16s^2 + 52s + 12

square yards.

The area of the walkway is the difference between the total outer area and the inner garden area.

Key Concept

Operations on Polynomials

Question 162Question →

If the equation $3x^2 + 5x = 2$ is solved for $x$ , what is the positive difference between the two solutions?

\frac{1}{3}

\frac{4}{3}

\frac{5}{3}

\frac{7}{3}

3

Show answer & explanation

Answer: $\frac{7}{3}$

Answer

The positive difference between the two solutions is

\frac{7}{3}

.

Rearranging the equation

3x^2 + 5x = 2

by subtracting

2

from both sides gives the standard quadratic equation

3x^2 + 5x - 2 = 0

. Factoring this expression yields

(3x - 1)(x + 2) = 0

. Setting each factor to zero gives the solutions

x = \frac{1}{3}

and

x = -2

. The positive difference between these solutions is

\frac{1}{3} - (-2) = \frac{1}{3} + 2 = \frac{7}{3}

.

Step-by-Step Solution

1

Rearrange the quadratic equation into standard form

ax^2 + bx + c = 0

.

3x^2 + 5x - 2 = 0

Before factoring a quadratic equation, all terms must be moved to one side so that the equation is set to zero.

2

Factor the quadratic expression by finding two binomials that multiply to

3x^2 + 5x - 2

.

(3x - 1)(x + 2) = 0

Since the product of

3

and

-2

is

-6

, we look for factors of

-6

that sum to

5

, which are

6

and

-1

. Splitting the middle term yields

3x^2 + 6x - x - 2 = 3x(x + 2) - 1(x + 2) = (3x - 1)(x + 2)

.

3

Use the zero-product property to solve for

x

by setting each binomial factor to zero.

3x - 1 = 0 \Rightarrow x = \frac{1}{3}

and

x + 2 = 0 \Rightarrow x = -2

If the product of two quantities is zero, at least one of the quantities must be zero.

4

Calculate the positive difference between the two solutions.

|\frac{1}{3} - (-2)| = |\frac{1}{3} + 2| = \frac{7}{3}

The positive difference is the absolute value of the subtraction of one root from the other.

Key Concept

Solving a quadratic equation by factoring over the integers after setting the equation equal to zero.

Alternative Method

Alternatively, the quadratic formula can be used. Once the equation is rewritten as

3x^2 + 5x - 2 = 0

, substitute

a=3

,

b=5

, and

c=-2

into

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

to get

x = \frac{-5 \pm \sqrt{25 - 4(3)(-2)}}{6} = \frac{-5 \pm 7}{6}

. This simplifies to

x = \frac{1}{3}

and

x = -2

. The positive difference between these values is

\frac{1}{3} - (-2) = \frac{7}{3}

.

Estimated Time:1m 30s

Question 163Question →

For what greatest integer value of $y$ is the inequality $9 - 2y \geq 16$ true?

Show answer & explanation

Answer: -4

Answer

The greatest integer value of

y

that satisfies the inequality is

-4

.

Subtracting 9 from both sides of

9 - 2y \geq 16

gives

-2y \geq 7

. Dividing both sides by

-2

and reversing the inequality sign yields

y \leq -3.5

. The greatest integer less than or equal to

-3.5

is

-4

.

Step-by-Step Solution

1

Subtract 9 from both sides of the inequality.

-2y \geq 7

This isolates the term containing

y

on the left side.

2

Divide both sides of the inequality by

-2

and reverse the direction of the inequality sign.

y \leq -3.5

Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.

3

Identify the greatest integer that is less than or equal to

-3.5

.

-4

The value of

y

must be less than or equal to

-3.5

. The integers satisfying this condition are

-4, -5, -6, \dots

, and the greatest of these is

-4

.

Key Concept

Solving linear inequalities by isolating the variable and reversing the inequality sign when dividing by a negative number.

Question 164Question →

An agricultural cooperative packages a premium seed mixture containing rye grass, fescue, and bluegrass. The weight of the fescue in the mixture is $10$ pounds less than twice the weight of the rye grass. The weight of the bluegrass is $15$ pounds more than half the weight of the fescue. If the total weight of the mixture is $120$ pounds, how many pounds of bluegrass are in the mixture?

Show answer & explanation

Answer: 40

Answer

The weight of the bluegrass in the mixture is 40 pounds.

By translating the given relationships into equations (

f = 2r - 10

,

b = \frac{1}{2}f + 15

, and

r + f + b = 120

), we can express all variables in terms of

r

, yielding

r + (2r - 10) + (r + 10) = 120

. Solving this gives

r = 30

. Substituting this back gives the weight of bluegrass as

30 + 10 = 40

pounds.

Step-by-Step Solution

1

Define variables for each type of grass in the mixture.

Let

r

represent the weight of rye grass,

f

represent the weight of fescue, and

b

represent the weight of bluegrass.

Establishing variables is necessary to translate the verbal descriptions into algebraic terms.

2

Translate the given relationships into equations.

f = 2r - 10

and

b = \frac{1}{2}f + 15

The problem states the fescue is 10 pounds less than twice the rye grass, and the bluegrass is 15 pounds more than half the fescue.

3

Substitute the expression for f into the equation for b to express b solely in terms of r.

b = \frac{1}{2}(2r - 10) + 15 = r - 5 + 15 = r + 10

Reducing the number of variables simplifies the system of equations.

4

Set up the total weight equation and solve for r.

r + (2r - 10) + (r + 10) = 120 \implies 4r = 120 \implies r = 30

The sum of the three grass weights is given as 120 pounds.

5

Calculate the weight of the bluegrass using the value of r.

b = 30 + 10 = 40

The question asks specifically for the weight of the bluegrass.

Key Concept

Translating verbal relationships into linear equations and solving a system of equations

Question 165Question →

A manufacturing company determines that its weekly profit from producing $x$ batches of a product is constrained by resource availability. To meet these resource constraints, the number of batches $x$ must satisfy the inequality:

\frac{5(6 - x)}{3} - \frac{3(x + 2)}{4} > 2x - 11

What is the greatest number of whole batches the company can produce while satisfying this constraint?

Show answer & explanation

Answer: 4

Answer

The greatest number of whole batches the company can produce is 4.

Solving the inequality yields

x < \frac{234}{53} \approx 4.415

. The greatest integer value that satisfies this condition is 4.

Step-by-Step Solution

1

Multiply both sides of the inequality by the least common multiple of the denominators, which is 12.

20(6 - x) - 9(x + 2) > 24x - 132

Multiplying by 12 eliminates the fractions, making the inequality easier to solve.

2

Distribute the constants on the left side of the inequality.

120 - 20x - 9x - 18 > 24x - 132

Distributing 20 to

(6 - x)

yields

120 - 20x

, and distributing

-9

to

(x + 2)

yields

-9x - 18

.

3

Combine like terms on the left side of the inequality.

102 - 29x > 24x - 132

Combining

120 - 18

gives

102

, and combining

-20x - 9x

gives

-29x

.

4

Subtract

24x

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

102 - 53x > -132

This groups all terms containing the variable

x

on one side of the inequality.

5

Subtract 102 from both sides to isolate the variable term.

-53x > -234

This isolates the term containing

x

on the left side of the inequality.

6

Divide both sides by

-53

and reverse the inequality sign.

x < \frac{234}{53}

Dividing by a negative number requires reversing the direction of the inequality sign.

7

Evaluate the fraction as a decimal and determine the greatest integer value of

x

that satisfies the inequality.

x < 4.415

, which means the greatest integer is 4.

Since the company must produce a whole number of batches, we find the largest integer less than 4.415.

Key Concept

Solving multi-step linear inequalities involving fractional coefficients, distributing negative numbers, and reversing the inequality sign when multiplying or dividing by a negative number.

Alternative Method

Instead of solving algebraically, you can test integer values for

x

directly in the inequality. Testing

x = 4

gives

\frac{10}{3} - \frac{18}{4} = 3.33 - 4.5 = -1.17

, which is greater than

2(4) - 11 = -3

(True). Testing

x = 5

gives

\frac{5}{3} - \frac{21}{4} = 1.67 - 5.25 = -3.58

, which is not greater than

2(5) - 11 = -1

(False). This confirms 4 is the largest integer satisfying the inequality.

Estimated Time:2m 30s

Question 166Question →

Which of the following is the completely factored form of the expression $3x(x - 2)^2 - 12x$ ?

3x(x^2 - 8)

-9x(x - 2)^2

3x^2(x - 4)

3x(x - 4)

12x(x - 1)(x - 3)

Show answer & explanation

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

Answer

The completely factored form of the expression is

3x^2(x - 4)

.

The correct answer is

3x^2(x-4)

. Factoring out the greatest common factor

3x

from the terms

3x(x-2)^2

and

-12x

gives

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

. The expression within the brackets is a difference of squares that can be factored as

[(x-2)-2][(x-2)+2]

, which simplifies to

x(x-4)

. Multiplying this by the GCF

3x

yields

3x^2(x-4)

.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF),

3x

, from both terms of the expression

3x(x - 2)^2 - 12x

.

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

Both terms

3x(x-2)^2

and

12x

share the common factors

3

and

x

.

2

Factor the difference of squares inside the bracket,

(x-2)^2 - 4

, using the formula

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

where

a = x-2

and

b = 2

.

[(x-2)-2][(x-2)+2] = (x-4)(x)

Since

4 = 2^2

, the terms inside the brackets form a difference of squares.

3

Combine the factored parts and simplify the expression by multiplying the variable terms.

3x \cdot x(x-4) = 3x^2(x-4)

Multiplying

3x

by

x

requires adding their exponents (

1 + 1 = 2

).

Key Concept

Factoring polynomials using the greatest common factor (GCF) and difference of squares.

Question 167Question →

When the expression $4x(x - 2y) - (2x - 3y)^2 + 5y(2x - y)$ is simplified to the form $Ax^2 + Bxy + Cy^2$ , where $A$ , $B$ , and $C$ are constants, what is the value of $B$ ?

Show answer & explanation

Answer: 14

Answer

The value of the coefficient

B

is 14.

Expanding the entire expression yields

4x^2 - 8xy - 4x^2 + 12xy - 9y^2 + 10xy - 5y^2

. Grouping and combining the

xy

terms gives

(-8 + 12 + 10)xy = 14xy

. Therefore, the coefficient

B

is 14.

Step-by-Step Solution

1

Expand the first term

4x^2 - 8xy

Distribute

4x

to both terms inside the parentheses:

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

.

2

Expand the squared binomial and apply the negative sign

-4x^2 + 12xy - 9y^2

Use the binomial expansion formula

(2x - 3y)^2 = 4x^2 - 12xy + 9y^2

, then multiply each term by

-1

.

3

Expand the third term

10xy - 5y^2

Distribute

5y

to both terms inside the parentheses:

5y(2x) - 5y(y) = 10xy - 5y^2

.

4

Combine the coefficients of the like terms

0x^2 + 14xy - 14y^2

Sum the coefficients for each corresponding variable group:

(4 - 4)x^2 + (-8 + 12 + 10)xy + (-9 - 5)y^2

.

Key Concept

Simplifying Algebraic Expressions and Combining Like Terms

Question 168Question →

For a real number $x$ , $\frac{1}{2}$ minus $\frac{2}{3}$ of $x$ is greater than $\frac{5}{6}$ . Which of the following is the complete set of solutions for $x$ ?

x > -\frac{1}{2}

x > 2

x < -\frac{1}{2}

x < -\frac{3}{2}

x < -5

Show answer & explanation

Answer: $x < -\frac{1}{2}$

Answer

x < -\frac{1}{2}

Subtracting

\frac{1}{2}

from both sides of the inequality

\frac{1}{2} - \frac{2}{3}x > \frac{5}{6}

gives

-\frac{2}{3}x > \frac{2}{6}

, which simplifies to

-\frac{2}{3}x > \frac{1}{3}

. Multiplying both sides by the negative fraction

-\frac{3}{2}

isolates

x

on the left and reverses the inequality sign from greater than (

>

) to less than (

<

). Performing the multiplication on the right yields

\frac{1}{3} \cdot \left(-\frac{3}{2}\right) = -\frac{1}{2}

. Thus, the solution set is

x < -\frac{1}{2}

.

Step-by-Step Solution

1

Translate the verbal description into an algebraic inequality.

\frac{1}{2} - \frac{2}{3}x > \frac{5}{6}

The phrase '

\frac{1}{2}

minus

\frac{2}{3}

of

x

' represents the expression

\frac{1}{2} - \frac{2}{3}x

, and 'is greater than' translates to the inequality symbol

>

.

2

Subtract

\frac{1}{2}

from both sides of the inequality.

-\frac{2}{3}x > \frac{1}{3}

To isolate the variable term on the left, we subtract

\frac{1}{2}

(which is equivalent to

\frac{3}{6}

) from

\frac{5}{6}

to get

\frac{2}{6} = \frac{1}{3}

.

3

Multiply both sides of the inequality by

-\frac{3}{2}

and reverse the inequality sign.

x < -\frac{1}{2}

Multiplying both sides of an inequality by a negative number requires reversing the direction of the inequality sign from

>

to

<

to keep the inequality true.

Key Concept

Solving multi-step linear inequalities involving multiplication or division by a negative number.

Question 169Question →

The polynomial $P(x)$ is defined by $P(x) = (2x^2 - 3x + 5)^2 - 4x(x^3 - 2x^2 - 7x + 1)$ . When $P(x)$ is written in standard form, what is the coefficient of the $x^2$ term?

Show answer & explanation

Answer: 57

Answer

The coefficient of the

x^2

term is 57.

Expanding

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

yields

(2x^2 - 3x + 5)(2x^2 - 3x + 5) = 4x^4 - 12x^3 + 29x^2 - 30x + 25

. Distributing the

-4x

term yields

-4x(x^3 - 2x^2 - 7x + 1) = -4x^4 + 8x^3 + 28x^2 - 4x

. Combining the

x^2

terms from both expressions gives

29x^2 + 28x^2 = 57x^2

. Thus, the coefficient of the

x^2

term is 57.

Step-by-Step Solution

1

Expand the squared trinomial

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

4x^4 - 12x^3 + 29x^2 - 30x + 25

Expanding the first part of the expression by multiplying the trinomial by itself.

2

Distribute the term

-4x

to the trinomial

(x^3 - 2x^2 - 7x + 1)

-4x^4 + 8x^3 + 28x^2 - 4x

Expanding the second part of the polynomial expression while distributing the negative sign to all terms.

3

Combine the expanded expressions and isolate the

x^2

terms

29x^2 + 28x^2 = 57x^2

Adding the coefficients of the terms of degree 2 to find the combined coefficient.

Key Concept

Operations on Polynomials

Question 170Question →

For all non-zero real numbers

y

, the expression

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

is equivalent to

y^{-7}

. What is the value of

a

?

2

-3

-4

\frac{1}{12}

\frac{3}{2}

Show answer & explanation

Answer: $2$

Answer

The correct value of

a

is 2.

Applying the exponent rules systematically allows us to simplify the expression. First,

(y^3)^2

becomes

y^6

and

(y^{-4})^a

becomes

y^{-4a}

using the power of a power rule. Second, we combine the terms in the numerator using the product rule to get

y^{6-4a}

. Third, we divide by

y^5

using the quotient rule to obtain

y^{6-4a-5} = y^{1-4a}

. Setting this equal to the target expression

y^{-7}

gives the equation

1-4a = -7

. Solving for

a

gives

-4a = -8

, which simplifies to

2

.

Step-by-Step Solution

1

Apply the power of a power rule,

(x^m)^n = x^{mn}

, to the exponential terms in the numerator.

(y^3)^2 = y^6

and

(y^{-4})^a = y^{-4a}

This simplifies nested exponent terms into single base terms.

2

Apply the product rule of exponents,

x^m \cdot x^n = x^{m+n}

, to combine the numerator terms.

y^6 \cdot y^{-4a} = y^{6-4a}

This simplifies the numerator to a single power of

y

.

3

Apply the quotient rule of exponents,

\frac{x^m}{x^n} = x^{m-n}

, to divide by the denominator.

\frac{y^{6-4a}}{y^5} = y^{(6-4a) - 5} = y^{1-4a}

This simplifies the entire rational expression into a single exponential expression.

4

Equate the simplified exponent to the exponent of the equivalent expression and solve the linear equation for

a

.

1-4a = -7 \implies -4a = -8 \implies a = 2

Since the bases are equal and non-zero, their exponents must be equal for the expressions to be equivalent.

Key Concept

Properties of Exponents in Algebraic Expressions

Estimated Time:1m 30s

Question 171Question →

If the quadratic expression $6x^2 - 7x - 5$ is factored completely into the product of two linear binomials of the form $(ax + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are integers such that $a > c > 0$ , what is the value of the expression $ad - bc$ ?

Show answer & explanation

Answer: 13

Answer

The value of the expression

ad - bc

is 13.

Step-by-Step Solution

1

Factor the quadratic expression

6x^2 - 7x - 5

.

(3x - 5)(2x + 1)

Find two numbers that multiply to

6 \times (-5) = -30

and add to

-7

, which are

-10

and

3

. Rewrite the middle term and factor by grouping:

6x^2 - 10x + 3x - 5 = 2x(3x - 5) + 1(3x - 5) = (3x - 5)(2x + 1)

.

2

Determine the values of the coefficients

a

,

b

,

c

, and

d

.

a = 3

,

b = -5

,

c = 2

, and

d = 1

The expression is factored into the form

(ax + b)(cx + d)

where

a > c > 0

. Comparing the factors

(3x - 5)

and

(2x + 1)

, we see the coefficients of

x

are

3

and

2

. Since

3 > 2 > 0

, we have

a = 3

and

c = 2

. This leaves

b = -5

and

d = 1

.

3

Calculate the value of

ad - bc

.

13

Substitute

a = 3

,

b = -5

,

c = 2

, and

d = 1

into the expression:

ad - bc = (3)(1) - (-5)(2) = 3 + 10 = 13

.

Key Concept

Factoring quadratic polynomials of the form

ax^2 + bx + c

with

a > 1

.

Question 172Question →

If

x

and

y

are positive real numbers such that

(x^3 y^{-2})^k \cdot (x^{-1} y^4)^3 = x^6 y^6

what is the value of the exponent

k

?

Show answer & explanation

Answer: 3

Answer

The value of the exponent

k

is 3.

Applying the exponent rules simplifies the left side of the equation to

x^{3k-3}y^{12-2k}

. Equating the exponent of

x

to the right side gives

3k - 3 = 6

, which yields

k = 3

. This value is confirmed by equating the exponent of

y

, since

12 - 2(3) = 6

.

Step-by-Step Solution

1

Apply the power of a power rule

(a^m)^n = a^{mn}

to expand the terms in the expression.

(x^3 y^{-2})^k = x^{3k} y^{-2k}

and

(x^{-1} y^4)^3 = x^{-3} y^{12}

To remove the outer parentheses by multiplying the internal exponents of each variable by the outer exponent.

2

Multiply the terms together by applying the product rule for exponents,

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

.

(x^{3k} y^{-2k})(x^{-3} y^{12}) = x^{3k-3} y^{12-2k}

To combine the like bases of

x

and

y

into a single simplified expression.

3

Set the exponents of like bases equal to the exponents on the right-hand side of the equation,

x^6 y^6

.

3k - 3 = 6

and

12 - 2k = 6

Since the bases are equal and non-zero, their respective exponents must also be equal.

4

Solve the linear equation

3k - 3 = 6

for

k

.

3k = 9 \implies k = 3

To determine the numerical value of the variable

k

.

5

Verify the solution by solving the second linear equation,

12 - 2k = 6

.

-2k = -6 \implies k = 3

To ensure consistency across both variable exponents in the expression.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 173Question →

A manufacturing plant operates two machines, Machine X and Machine Y, to produce identical components. Machine Y produces 12 fewer components per hour than Machine X. On a certain day, Machine X operated for 5 hours and Machine Y operated for 7 hours. If the two machines produced a combined total of 636 components, how many components did Machine X produce on that day?

46

60

230

270

300

Show answer & explanation

Answer: 300

Answer

300

To find the number of components Machine X produced, let

x

represent Machine X's hourly production rate. This makes Machine Y's hourly rate

x - 12

. Since Machine X operated for 5 hours and Machine Y operated for 7 hours to produce a total of 636 components, we can write the equation:

5x + 7(x - 12) = 636

. Distributing the 7 yields

5x + 7x - 84 = 636

, which simplifies to

12x - 84 = 636

. Adding 84 to both sides gives

12x = 720

, and dividing by 12 yields

x = 60

components per hour for Machine X. To find the total produced by Machine X, we multiply its hourly rate by the 5 hours it operated:

5 \times 60 = 300

. This matches the correct option.

Step-by-Step Solution

1

Define variables for the hourly rates of both machines.

Let

x

be the number of components Machine X produces per hour. Since Machine Y produces 12 fewer components per hour, Machine Y's hourly rate is

x - 12

.

Establishing algebraic expressions for each machine's rate allows us to set up a linear equation.

2

Set up an equation representing the total combined production.

The total production is the sum of the components produced by Machine X in 5 hours and Machine Y in 7 hours:

5x + 7(x - 12) = 636

.

The sum of the products of each machine's rate and its operating time equals the total combined production of 636 components.

3

Solve the equation for the hourly rate of Machine X,

x

.

Distribute the 7:

5x + 7x - 84 = 636

. Combine like terms:

12x - 84 = 636

. Add 84 to both sides:

12x = 720

. Divide by 12:

x = 60

.

Solving for

x

gives the hourly rate of Machine X.

4

Calculate the total components produced by Machine X.

Multiply the hourly rate of Machine X by its hours of operation:

5 \text{ hours} \times 60 \text{ components/hour} = 300

.

The question asks for the total components produced by Machine X, not its hourly rate.

Key Concept

Setting up and solving single-variable linear equations from verbal descriptions

Alternative Method

Instead of defining the variable as Machine X's rate, we could define

y

as Machine Y's hourly rate. Then Machine X's rate is

y + 12

. The total equation becomes

5(y + 12) + 7y = 636 \implies 5y + 60 + 7y = 636 \implies 12y = 576 \implies y = 48

. Since Machine X's hourly rate is

y + 12 = 60

, it produced

5 \times 60 = 300

components.

Estimated Time:1m 30s

Question 174Question →

When $5$ is subtracted from $2$ times a number $x$ , the result is at least $9$ . Which of the following inequalities represents all possible values of $x$ ?

x \leq -2

x \geq -2

x \geq 3

x \geq 7

x \leq 7

Show answer & explanation

Answer: $x \geq 7$

Answer

The inequality

x \geq 7

represents all possible values of

x

.

The verbal statement translates to the inequality

2x - 5 \geq 9

. Adding

5

to both sides gives

2x \geq 14

, and dividing both sides by the positive number

2

results in

x \geq 7

. Since we divide by a positive number, the direction of the inequality does not change.

Step-by-Step Solution

1

Translate the verbal phrase into an algebraic inequality.

2x - 5 \geq 9

'5 subtracted from 2 times a number

x

' translates to

2x - 5

, and 'at least 9' means greater than or equal to 9.

2

Add 5 to both sides of the inequality to isolate the variable term.

2x \geq 14

Adding 5 to both sides maintains the inequality and simplifies the left side.

3

Divide both sides by 2.

x \geq 7

Dividing by a positive number does not change the direction of the inequality sign.

Key Concept

Solving linear inequalities by translating verbal statements into algebraic forms and applying inverse operations.

Question 175Question →

What is the smallest integer value of $x$ that satisfies the inequality $11 - 3x < 2$ ?

Show answer & explanation

Answer: 4

Answer

The smallest integer value of

x

that satisfies the inequality is 4.

Solving the inequality

11 - 3x < 2

leads to

-3x < -9

. Dividing by

-3

and reversing the inequality sign gives

x > 3

. The smallest integer that is strictly greater than 3 is 4.

Step-by-Step Solution

1

Isolate the variable term on one side of the inequality by subtracting 11 from both sides.

-3x < -9

Subtracting 11 from both sides of the inequality keeps the relationship balanced.

2

Divide both sides of the inequality by the coefficient of

x

, which is

-3

.

x > 3

Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.

3

Determine the smallest integer that is strictly greater than 3.

4

Since the inequality is strict (

x > 3

), 3 is not included in the solution set. The smallest integer greater than 3 is 4.

Key Concept

Solving linear inequalities and reversing the inequality sign when dividing by a negative number.

Question 176Question →

A storage tank contains $18$ gallons of water and is being filled at a constant rate of $\frac{2}{3}$ gallons per minute. A second storage tank contains $33$ gallons of water and is being drained at a constant rate of $\frac{5}{6}$ gallons per minute. If both tanks begin their processes at the same time, they will contain the same amount of water after $m$ minutes. What is the value of $2m + 3$ ?

23

26

33

71

183

Show answer & explanation

Answer: 23

Answer

23

The correct answer is 23. By representing the volume of the first tank as

18 + \frac{2}{3}m

and the second tank as

33 - \frac{5}{6}m

, setting them equal gives

18 + \frac{2}{3}m = 33 - \frac{5}{6}m

. Solving this equation by finding a common denominator for the fraction coefficients yields

\frac{9}{6}m = 15

, which simplifies to

\frac{3}{2}m = 15

, and thus

m = 10

. Substituting

m = 10

into the expression

2m + 3

results in

2(10) + 3 = 23

.

Step-by-Step Solution

1

Set up the equation representing the water volume in both tanks over time.

18 + \frac{2}{3}m = 33 - \frac{5}{6}m

The first tank starts with

18

gallons and increases by

\frac{2}{3}

gallons per minute, while the second starts with

33

gallons and decreases by

\frac{5}{6}

gallons per minute. We set their volumes equal to find when they contain the same amount.

2

Isolate the variable terms on one side and the constant terms on the other side.

\frac{2}{3}m + \frac{5}{6}m = 33 - 18

Adding

\frac{5}{6}m

to both sides and subtracting

18

from both sides groups like terms together.

3

Find a common denominator to add the fraction coefficients.

\frac{4}{6}m + \frac{5}{6}m = 15 \Rightarrow \frac{9}{6}m = 15 \Rightarrow \frac{3}{2}m = 15

A common denominator of

6

is used to add the fractions

\frac{2}{3}

and

\frac{5}{6}

.

4

Solve for

m

by multiplying both sides by the reciprocal of the coefficient.

m = 15 \times \frac{2}{3} = 10

Multiplying by

\frac{2}{3}

isolates

m

on the left side.

5

Evaluate the expression

2m + 3

using the value of

m

.

2(10) + 3 = 20 + 3 = 23

Substitute

m = 10

into the expression and follow the correct order of operations (multiply first, then add).

Key Concept

Solving linear equations with fractional coefficients and translating word problems into algebraic equations.

Question 177Question →

A triangle has a base of $2x + 4$ inches and a height of $x - 3$ inches. What is the coefficient of $x$ when the expression representing the area of the triangle, in square inches, is written in standard form?

Show answer & explanation

Answer: -1

Answer

The coefficient of

x

is

-1

.

The area of a triangle is calculated using

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

. Substituting the given expressions, we get

\text{Area} = \frac{1}{2}(2x + 4)(x - 3)

. First, we can multiply the

\frac{1}{2}

by

(2x + 4)

, which simplifies to

x + 2

. Next, we expand

(x + 2)(x - 3)

using FOIL to get

x^2 - 3x + 2x - 6

. Combining like terms yields

x^2 - x - 6

. The coefficient of the

x

term in this simplified expression is

-1

.

Step-by-Step Solution

1

State the formula for the area of a triangle.

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

To establish the mathematical relationship.

2

Substitute the given values into the formula.

\text{Area} = \frac{1}{2}(2x + 4)(x - 3)

To express the area in terms of the variable

x

.

3

Distribute the fraction to the first binomial.

x + 2

Distributing

\frac{1}{2}

to

(2x + 4)

simplifies the expression before multiplying.

4

Expand the product of the binomials.

x^2 - 3x + 2x - 6

Using the FOIL method to multiply

(x + 2)

and

(x - 3)

.

5

Combine the linear terms.

x^2 - x - 6

To simplify the polynomial and write it in standard form.

6

Identify the coefficient of

x

.

-1

The coefficient of the

x

term in

x^2 - x - 6

is

-1

.

Key Concept

Multiplying binomials and applying formulas in geometric contexts

Estimated Time:1m 0s

Question 178Question →

If the expression $\frac{(x^2 y^3)^4}{(x^{-1} y^2)^3}$ is written in the equivalent form $x^a y^b$ , what is the value of $a - b$ ?

Show answer & explanation

Answer: 5

Answer

5

Simplifying the numerator yields

x^8 y^{12}

and simplifying the denominator yields

x^{-3} y^6

. Dividing these expressions by subtracting the exponents of like bases results in

x^{8 - (-3)} y^{12 - 6} = x^{11} y^6

. Comparing this to the expression

x^a y^b

shows that

a = 11

and

b = 6

. The value of

a - b

is

11 - 6 = 5

.

Step-by-Step Solution

1

Simplify the numerator of the expression.

x^8 y^{12}

Apply the power of a power and power of a product properties of exponents:

(x^2 y^3)^4 = x^{2 \cdot 4} y^{3 \cdot 4}

.

2

Simplify the denominator of the expression.

x^{-3} y^6

Apply the power of a power and power of a product properties of exponents:

(x^{-1} y^2)^3 = x^{-1 \cdot 3} y^{2 \cdot 3}

.

3

Simplify the quotient by dividing the simplified numerator by the simplified denominator.

x^{11} y^6

Use the quotient property of exponents,

\frac{x^m}{x^n} = x^{m-n}

, to subtract the exponents:

8 - (-3) = 11

and

12 - 6 = 6

.

4

Identify the values of

a

and

b

and compute the difference

a - b

.

5

By comparing

x^{11} y^6

to the target form

x^a y^b

, we find

a = 11

and

b = 6

. Subtracting

b

from

a

yields

11 - 6 = 5

.

Key Concept

Properties of exponents including power of a power, power of a product, and quotient rules.

Question 179Question →

For all real values of $x$ , which of the following inequalities represents the complete solution set to the inequality $\frac{5 - 2x}{3} - \frac{3x - 1}{4} \leq \frac{x + 2}{2}$ ?

x \leq \frac{11}{23}

x \geq \frac{5}{23}

x \geq -\frac{2}{17}

x \geq \frac{11}{23}

x \leq \frac{5}{23}

Show answer & explanation

Answer: $x \geq \frac{11}{23}$

Answer

The complete solution set is the set of all real numbers greater than or equal to 11/23.

The correct answer is found by clearing the denominators with the least common multiple of 12, carefully expanding the terms to get

20 - 8x - 9x + 3 \leq 6x + 12

, simplifying to

23 - 17x \leq 6x + 12

, grouping terms to get

-23x \leq -11

, and dividing by

-23

which flips the sign to yield all real values greater than or equal to 11/23.

Step-by-Step Solution

1

Multiply all terms of the inequality by the least common multiple of the denominators (3, 4, and 2), which is 12, to clear the fractions.

4(5 - 2x) - 3(3x - 1) \leq 6(x + 2)

Multiplying by a positive number allows us to eliminate denominators without changing the direction of the inequality.

2

Distribute the coefficients on both sides of the inequality, paying close attention to the distribution of the negative sign over the second term.

20 - 8x - 9x + 3 \leq 6x + 12

Distributing

-3

to both

3x

and

-1

yields

-9x

and

+3

respectively.

3

Combine the constant terms and the variable terms on the left side of the inequality.

23 - 17x \leq 6x + 12

Simplifying the expressions on each side makes the inequality easier to isolate.

4

Isolate the variable terms on the left and the constant terms on the right by subtracting

6x

and 23 from both sides.

-23x \leq -11

Grouping like terms together is necessary to solve for the variable.

5

Divide both sides of the inequality by

-23

and reverse the direction of the inequality sign.

x \geq \frac{11}{23}

Dividing both sides of an inequality by a negative number reverses the direction of the inequality sign from

\leq

to

\geq

.

Key Concept

Solving linear inequalities involving fractions and distributing negative coefficients, specifically applying the rule that multiplying or dividing by a negative number reverses the inequality direction.

Alternative Method

Instead of clearing the fractions first, write each fraction as separate terms:

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

. Then, collect the constant terms on one side and the variable terms on the other side using decimal or fractional conversions, and isolate the variable.

Estimated Time:2m 0s

Question 180Question →

A square playground has a side length of $2x^3$ meters. A square sandbox with a side length of $x^3 - 2x$ meters is built inside the playground. Which of the following expressions represents the area, in square meters, of the playground that is NOT covered by the sandbox?

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

3x^6 - 4x^2

3x^6 + 4x^4 + 4x^2

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

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

Show answer & explanation

Answer: $3x^6 + 4x^4 - 4x^2$

Answer

The correct expression is

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

.

The expression

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

is correct because the area of the playground is

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

and the area of the sandbox is

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

. Subtracting the sandbox area from the playground area requires distributing the negative sign, resulting in

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

.

Step-by-Step Solution

1

Calculate the area of the square playground.

\text{Area}_{\text{playground}} = (2x^3)^2 = 4x^6

The area of a square is equal to the square of its side length, and applying the exponent rules yields

(2x^3)^2 = 2^2 \cdot (x^3)^2 = 4x^6

.

2

Calculate the area of the square sandbox.

\text{Area}_{\text{sandbox}} = (x^3 - 2x)^2 = x^6 - 4x^4 + 4x^2

Using the binomial squaring formula

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

, we square each term and compute the middle product, adding exponents for

x^3 \cdot x^1 = x^4

.

3

Subtract the sandbox's area from the playground's area.

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

Distribute the negative sign to all three terms inside the parentheses and combine the like terms of

x^6

.

Key Concept

Polynomial subtraction and squaring binomials with variables containing exponents.

Alternative Method

Evaluate the expression for a small integer value of

x

. If

x = 2

, the playground side length is

2(2)^3 = 16

, giving an area of

16^2 = 256

. The sandbox side length is

2^3 - 2(2) = 4

, giving an area of

4^2 = 16

. The remaining area is

256 - 16 = 240

. Substituting

x = 2

into the correct expression

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

yields

3(64) + 4(16) - 4(4) = 192 + 64 - 16 = 240

, confirming its correctness.

Estimated Time:2m 0s

Elementary Algebra

Answer

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