Intermediate Algebra Practice Questions — ACT — Page 10

Question 181Question →

For all real numbers $x$ such that $x \neq 1$ , the function $f$ is defined by $f(x) = \frac{12}{x - 1}$ . For all real numbers $x$ , the function $g$ is defined by $g(x) = x^2 + 2$ . What is the value of the composite function $f(g(3))$ ?

Show answer & explanation

Answer: 1.2

Answer

1.2

To evaluate the composite function

f(g(3))

, we work from the inside out. First, evaluate the inner function

g(3) = 3^2 + 2 = 11

. Next, substitute this result into the outer function

f(x)

to get

f(11) = \frac{12}{11 - 1} = \frac{12}{10} = 1.2

.

Step-by-Step Solution

1

Evaluate the inner function

g(x)

at

x = 3

.

g(3) = 11

To evaluate a composite function of the form

f(g(x))

at a given value, we must first find the output of the inner function,

g(x)

, at that value.

2

Substitute the output from Step 1 as the input for the outer function

f(x)

and evaluate.

f(g(3)) = 1.2

Using the result

g(3) = 11

as the input for

f(x)

gives

f(11) = \frac{12}{11 - 1} = 1.2

.

Key Concept

Function Composition and Evaluation

Question 182Question →

For a certain quadratic equation $0.25x^2 - kx + 4.5 = 0$ , where $k$ is a constant, the difference between the two real solutions is exactly $3$ . What is the positive value of $k$ ?

Show answer & explanation

Answer: 2.25

Answer

The positive value of

k

is

2.25

.

By applying the root difference formula

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

with

a = 0.25

,

b = -k

, and

c = 4.5

, we get the equation

\frac{\sqrt{k^2 - 4.5}}{0.25} = 3

. This simplifies to

4\sqrt{k^2 - 4.5} = 3

. Dividing by

4

and squaring both sides gives

k^2 - 4.5 = 0.5625

, which means

k^2 = 5.0625

. Taking the positive square root yields

k = 2.25

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation and express the formula for the difference of the roots.

The coefficients are

a = 0.25

,

b = -k

, and

c = 4.5

. The difference between the roots is

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

.

This sets up the algebraic relationship between the root difference and the coefficients of the quadratic equation.

2

Substitute the coefficients and the given root difference of

3

into the difference formula.

\frac{\sqrt{(-k)^2 - 4(0.25)(4.5)}}{0.25} = 3

This creates an equation in terms of the variable

k

using the given problem constraints.

3

Simplify the equation and isolate the radical term.

4\sqrt{k^2 - 4.5} = 3 \implies \sqrt{k^2 - 4.5} = 0.75

Simplifying the fraction makes it easier to isolate the radical prior to squaring.

4

Square both sides of the equation to eliminate the radical and solve for

k^2

.

k^2 - 4.5 = 0.5625 \implies k^2 = 5.0625

Squaring is the inverse operation of the square root, allowing us to solve for

k^2

.

5

Calculate the positive square root of

5.0625

to find the value of

k

.

k = 2.25

Since the question asks for the positive value of

k

, we choose the positive square root.

Key Concept

Relating the difference of the roots of a quadratic equation to its coefficients using the discriminant and the quadratic formula.

Question 183Question →

For what value of the constant $k$ does the quadratic equation $0.5x^2 - 1.5x + k = 0$ have two complex roots of the form $a \pm 2i$ , where $a$ is a real number?

0.875

1.5625

2.125

3.125

6.25

Show answer & explanation

Answer: 3.125

Answer

3.125

Applying the quadratic formula to

0.5x^2 - 1.5x + k = 0

yields

x = 1.5 \pm \sqrt{2.25 - 2k}

. For the roots to be of the form

a \pm 2i

, the imaginary component must be

2i

, which requires the radicand

2.25 - 2k

to equal

(2i)^2 = -4

. Solving the equation

2.25 - 2k = -4

yields

k = 3.125

. This corresponds to the option containing

3.125

.

Step-by-Step Solution

1

Identify the coefficients

a

,

b

, and

c

of the given quadratic equation

0.5x^2 - 1.5x + k = 0

.

a = 0.5

,

b = -1.5

, and

c = k

.

These coefficients are required to substitute into the quadratic formula.

2

Apply the quadratic formula,

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

, to express the roots of the equation.

x = \frac{-(-1.5) \pm \sqrt{(-1.5)^2 - 4(0.5)k}}{2(0.5)} = \frac{1.5 \pm \sqrt{2.25 - 2k}}{1} = 1.5 \pm \sqrt{2.25 - 2k}

.

Substituting the coefficients simplifies the expression for the roots of the quadratic equation.

3

Set the radical term equal to the imaginary component

2i

to find the value of

k

that yields complex roots of the form

a \pm 2i

.

\sqrt{2.25 - 2k} = 2i \Rightarrow 2.25 - 2k = (2i)^2 \Rightarrow 2.25 - 2k = -4

.

For the roots to have an imaginary part of

2i

, the term under the square root must equal

(2i)^2

, which is

-4

.

4

Solve the linear equation

2.25 - 2k = -4

for

k

.

-2k = -4 - 2.25 \Rightarrow -2k = -6.25 \Rightarrow k = 3.125

.

Isolating

k

gives the constant value that satisfies the root condition.

Key Concept

Using the quadratic formula and discriminant to determine the value of a parameter for a specified type of complex roots.

Question 184Question →

A line is defined by the equation $y = 3x + k$ , where $k$ is a constant. This line intersects the parabola $y = x^2 - x + 2$ at two distinct points, $P$ and $Q$ . If the midpoint of the line segment $PQ$ lies on the line $y = 2x + 7$ , what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of

k

is

5

.

Equating the equations of the line and the parabola gives a quadratic equation

x^2 - 4x + (2 - k) = 0

. The average of the roots of this quadratic equation gives the

x

-coordinate of the midpoint,

x_m = 2

. Substituting this into the first line's equation gives the

y

-coordinate of the midpoint,

y_m = 6 + k

. Since the midpoint

(2, 6+k)

lies on the line

y = 2x + 7

, we substitute these coordinates to get

6 + k = 11

, which simplifies to

k = 5

.

Step-by-Step Solution

1

Equate the equations of the line and the parabola.

x^2 - 4x + (2 - k) = 0

To find the

x

-coordinates of the intersection points

P

and

Q

.

2

Determine the sum of the

x

-coordinates and find the midpoint's

x

-coordinate.

x_m = 2

By Vieta's formulas, the sum of the roots of the quadratic equation is

4

. The

x

-coordinate of the midpoint is the average of these roots:

4 / 2 = 2

.

3

Find the

y

-coordinate of the midpoint.

y_m = 6 + k

Because the midpoint lies on the line segment

PQ

, its coordinates must satisfy the equation of the line passing through

P

and

Q

, which is

y = 3x + k

.

4

Substitute the midpoint coordinates into the second line's equation and solve for

k

.

k = 5

We are given that the midpoint lies on the line

y = 2x + 7

.

Key Concept

Systems of Linear and Quadratic Equations and Midpoint Properties

Question 185Question →

Two functions are defined as $f(x) = (x - 3)^2 - 16$ and $g(x) = |x - 1| - 8$ . If $g(f(x)) = 0$ , what is the sum of all positive values of $x$ ?

4

8

10

14

24

Show answer & explanation

Answer: $14$

Answer

The correct sum of all positive values of

x

is 14.

To find the sum of all positive values of

x

that satisfy

g(f(x)) = 0

, first substitute

f(x)

into

g(x)

to get the equation

|f(x) - 1| - 8 = 0

. This simplifies to

|f(x) - 1| = 8

. Solving this absolute value equation yields two cases:

f(x) - 1 = 8

or

f(x) - 1 = -8

.

For the first case,

f(x) = 9

. Substituting

f(x) = (x - 3)^2 - 16

gives

(x - 3)^2 - 16 = 9 \Rightarrow (x - 3)^2 = 25

. Taking the square root of both sides gives

x - 3 = 5

or

x - 3 = -5

, which results in

x = 8

or

x = -2

. The only positive solution from this case is

8

.

For the second case,

f(x) = -7

. Substituting

f(x) = (x - 3)^2 - 16

gives

(x - 3)^2 - 16 = -7 \Rightarrow (x - 3)^2 = 9

. Taking the square root of both sides gives

x - 3 = 3

or

x - 3 = -3

, which results in

x = 6

or

x = 0

. The only positive solution from this case is

6

(since

0

is not positive).

Adding the positive solutions gives

8 + 6 = 14

.

Step-by-Step Solution

1

Set up the composite equation

g(f(x)) = 0

.

|f(x) - 1| - 8 = 0 \Rightarrow |f(x) - 1| = 8

Substitute the expression for

f(x)

into

g(x)

to establish the relationship.

2

Solve the absolute value equation by separating it into two distinct cases.

f(x) - 1 = 8 \Rightarrow f(x) = 9

or

f(x) - 1 = -8 \Rightarrow f(x) = -7

An absolute value equation

|u| = c

splits into

u = c

and

u = -c

when

c \geq 0

.

3

Substitute the definition of

f(x) = (x - 3)^2 - 16

into the first case and solve for

x

.

(x - 3)^2 - 16 = 9 \Rightarrow (x - 3)^2 = 25 \Rightarrow x - 3 = \pm 5 \Rightarrow x = 8

or

x = -2

Isolate the squared binomial and take the square root of both sides to find all real solutions for this case.

4

Substitute the definition of

f(x) = (x - 3)^2 - 16

into the second case and solve for

x

.

(x - 3)^2 - 16 = -7 \Rightarrow (x - 3)^2 = 9 \Rightarrow x - 3 = \pm 3 \Rightarrow x = 6

or

x = 0

Isolate the squared binomial and take the square root of both sides to find all real solutions for this case.

5

Identify the positive solutions and calculate their sum.

The positive solutions are

8

and

6

. The sum is

8 + 6 = 14

.

Exclude non-positive values (

-2

is negative, and

0

is neither positive nor negative) and add the remaining values.

Key Concept

Function composition involves substituting one function into another, and evaluating the resulting composite equation requires solving multi-step equations including absolute value and quadratic relations.

Question 186Question →

An arithmetic sequence $a_1, a_2, a_3, \dots$ has a first term $a_1$ and a common difference $d$ , where both $a_1$ and $d$ are non-zero. Let $S_n$ represent the sum of the first $n$ terms of this sequence. If the ratio $\frac{S_{3n}}{S_n}$ is equal to a constant value $C$ for all positive integers $n$ , what is the ratio of the tenth term, $a_{10}$ , to the first term, $a_1$ ?

10

21

19

9

46

Show answer & explanation

Answer: 19

Answer

The ratio of the tenth term to the first term is 19.

The sum of the first

n

terms of an arithmetic sequence is

S_n = \frac{n}{2}[2a_1 + (n-1)d]

. For

n=1

, the ratio is

\frac{S_3}{S_1} = 3 + \frac{3d}{a_1}

. For

n=2

, the ratio is

\frac{S_6}{S_2} = \frac{6a_1 + 15d}{2a_1 + d}

. Setting these equal because the ratio is constant for all positive integers

n

gives

3 + \frac{3d}{a_1} = \frac{6a_1 + 15d}{2a_1 + d}

, which simplifies to

\frac{3a_1+3d}{a_1} = \frac{6a_1+15d}{2a_1+d}

. Cross-multiplying and simplifying yields

3d^2 = 6a_1 d

. Since

d \neq 0

, we have

d = 2a_1

. The tenth term is

a_{10} = a_1 + 9d = a_1 + 9(2a_1) = 19a_1

. Thus, the ratio of the tenth term to the first term is

19

.

Step-by-Step Solution

1

Write the formula for the sum of the first

n

terms of an arithmetic sequence,

S_n = \frac{n}{2}[2a_1 + (n-1)d]

, and find the expressions for

S_1

and

S_3

.

S_1 = a_1

and

S_3 = 3a_1 + 3d

.

To evaluate the ratio

\frac{S_{3n}}{S_n}

for the case where

n = 1

.

2

Write the expressions for

S_2

and

S_6

using the arithmetic sum formula.

S_2 = 2a_1 + d

and

S_6 = 6a_1 + 15d

.

To evaluate the ratio

\frac{S_{3n}}{S_n}

for the case where

n = 2

.

3

Equate the ratio for

n = 1

to the ratio for

n = 2

since the ratio

\frac{S_{3n}}{S_n}

must be constant for all

n

.

\frac{3a_1 + 3d}{a_1} = \frac{6a_1 + 15d}{2a_1 + d}

.

To set up an algebraic equation to find the relationship between the first term

a_1

and the common difference

d

.

4

Solve the equation for

d

in terms of

a_1

by cross-multiplying and simplifying.

d = 2a_1

.

Cross-multiplying gives

(3a_1 + 3d)(2a_1 + d) = a_1(6a_1 + 15d)

, which expands to

6a_1^2 + 9a_1 d + 3d^2 = 6a_1^2 + 15a_1 d

. Subtracting

6a_1^2

and

9a_1 d

from both sides yields

3d^2 = 6a_1 d

. Since

d \neq 0

, dividing by

3d

gives

d = 2a_1

.

5

Substitute

d = 2a_1

into the formula for the tenth term,

a_{10} = a_1 + 9d

, and calculate the ratio

\frac{a_{10}}{a_1}

.

a_{10} = 19a_1

, so the ratio is

19

.

To determine the final value of the requested ratio.

Key Concept

Relating arithmetic sequence term and sum formulas through systems of algebraic equations.

Question 187Question →

If $8^{2x - 1} = \left(\frac{1}{4}\right)^{x - 3}$ , what is the value of $x$ ?

-\frac{3}{8}

\frac{1}{2}

\frac{9}{8}

1

-\frac{3}{4}

Show answer & explanation

Answer: $\frac{9}{8}$

Answer

The correct answer is

\frac{9}{8}

.

By writing both sides of the equation with a common base of

2

, we get

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

. Applying the power-to-a-power exponent rule, this simplifies to

2^{6x-3} = 2^{-2x+6}

. Since the bases are now identical, their exponents must be equal:

6x-3 = -2x+6

. Adding

2x

and

3

to both sides results in

8x = 9

, which gives

x = \frac{9}{8}

.

Step-by-Step Solution

1

Express both bases as powers of

2

.

8 = 2^3

and

\frac{1}{4} = 2^{-2}

, so the equation becomes

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

.

Finding a common base allows us to equate the exponents directly.

2

Apply the power-to-a-power rule

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

to simplify the exponents.

2^{3(2x-1)} = 2^{-2(x-3)} \Rightarrow 2^{6x-3} = 2^{-2x+6}

.

This simplifies the exponential expressions on both sides of the equation.

3

Equate the exponents and solve for

x

.

6x - 3 = -2x + 6 \Rightarrow 8x = 9 \Rightarrow x = \frac{9}{8}

.

Since the bases are equal, their exponents must be equal.

Key Concept

Solving exponential equations by finding a common base and applying exponent properties.

Estimated Time:1m 30s

Question 188Question →

In the standard $(x, y)$ coordinate plane, a parabola is defined by the equation $y = x^2 + 2x + 7$ and a line is defined by the equation $y = mx + 3$ , where $m$ is a constant. If the system of equations consisting of this parabola and line has exactly one real solution, and this solution lies in the first quadrant, what is the value of $m$ ?

-2

-6

6

2\sqrt{3}

2 + 2\sqrt{7}

Show answer & explanation

Answer: 6

Answer

6

The correct answer is

6

. When setting the equations of the parabola and line equal, we get

x^2 + (2-m)x + 4 = 0

. Setting the discriminant to zero yields

(2-m)^2 - 16 = 0

, which gives

m = 6

or

m = -2

. Substituting

m = 6

back gives a single intersection point of

(2, 15)

, which is in the first quadrant since both coordinates are positive. The other value,

m = -2

, gives an intersection point of

(-2, 7)

, which is in the second quadrant.

Step-by-Step Solution

1

Set the equations of the parabola and the line equal to each other to find their intersection points.

x^2 + 2x + 7 = mx + 3

To find the coordinates where the two graphs intersect.

2

Rearrange the equation into standard quadratic form:

ax^2 + bx + c = 0

.

x^2 + (2 - m)x + 4 = 0

To express the intersection condition as a single quadratic equation where we can analyze the number of solutions.

3

Set the discriminant of the quadratic equation to zero.

\Delta = (2 - m)^2 - 4(1)(4) = 0 \implies (2 - m)^2 - 16 = 0

For the system to have exactly one real solution, the quadratic equation must have a discriminant of zero (tangency).

4

Solve for the possible values of

m

.

(2 - m)^2 = 16 \implies 2 - m = \pm 4

, giving

m = -2

or

m = 6

.

To find all values of

m

that result in exactly one intersection point.

5

Find the intersection point for each value of

m

and determine which lies in the first quadrant.

For

m = -2

,

(x+2)^2 = 0 \implies x = -2

and

y = -2(-2)+3 = 7

, yielding

(-2, 7)

(Quadrant II). For

m = 6

,

(x-2)^2 = 0 \implies x = 2

and

y = 6(2)+3 = 15

, yielding

(2, 15)

(Quadrant I).

To satisfy the condition that the single solution must lie in the first quadrant (where both

x > 0

and

y > 0

).

Key Concept

Solving systems of linear and quadratic equations and applying the discriminant to find conditions for tangency.

Estimated Time:2m 0s

Question 189Question →

A thermometer is considered accurate if its temperature reading, $T$ degrees Fahrenheit, differs from the actual temperature by less than $1.5^\circ\text{F}$ . If the actual temperature is $72.0^\circ\text{F}$ , which of the following inequalities represents the range of reading temperatures, $T$ , that are NOT considered accurate?

|T - 1.5| \ge 72.0

T - 72.0 \ge 1.5

|T - 72.0| \ge 1.5

|T - 72.0| < 1.5

|T - 72.0| > 1.5

Show answer & explanation

Answer: $|T - 72.0| \ge 1.5$

Answer

|T - 72.0| \ge 1.5

The difference between the temperature reading,

T

, and the actual temperature of

72.0^\circ\text{F}

is represented by

|T - 72.0|

. A thermometer is accurate when this difference is less than

1.5^\circ\text{F}

(

|T - 72.0| < 1.5

). Therefore, the thermometer is not accurate when the difference is greater than or equal to

1.5^\circ\text{F}

, which is written as

|T - 72.0| \ge 1.5

.

Step-by-Step Solution

1

Represent the difference between the reading and the actual temperature.

|T - 72.0|

The difference between the reading temperature and the actual temperature of

72.0^\circ\text{F}

is given by the absolute value expression, representing the distance between the two temperatures on a thermometer.

2

Formulate the condition for an accurate reading.

|T - 72.0| < 1.5

An accurate reading differs from the actual temperature by less than

1.5^\circ\text{F}

.

3

Determine the complement condition for a reading that is NOT accurate.

|T - 72.0| \ge 1.5

To find the range of readings that are NOT accurate, we take the complement of the accurate condition. The opposite of 'less than

1.5

' is 'greater than or equal to

1.5

'.

Key Concept

Absolute Value Inequalities in Real-World Contexts

Estimated Time:1m 0s

Question 190Question →

What is the product of all real values of $x$ that satisfy the equation $\log_3(x) - 6\log_x(3) = 1$ ?

Show answer & explanation

Answer: 3

Answer

The product of all real values of

x

that satisfy the equation is 3.

By applying the change-of-base formula, the equation becomes

\log_3(x) - \frac{6}{\log_3(x)} = 1

. Substituting

y = \log_3(x)

leads to

y^2 - y - 6 = 0

, which has roots

y = 3

and

y = -2

. These roots correspond to

x = 3^3 = 27

and

x = 3^{-2} = \frac{1}{9}

. Both solutions are valid because they are positive and do not equal 1. The product of these solutions is

27 \times \frac{1}{9} = 3

. Alternatively, using Vieta's formulas, the sum of the roots of the quadratic equation is

y_1 + y_2 = 1

. The product of the solutions is

x_1 x_2 = 3^{y_1} \cdot 3^{y_2} = 3^{y_1 + y_2} = 3^1 = 3

.

Step-by-Step Solution

1

Apply the change-of-base formula to rewrite the variable base term.

\log_3(x) - \frac{6}{\log_3(x)} = 1

This expresses the equation in terms of logarithms with the same base.

2

Use substitution to convert the equation into a quadratic form.

y - \frac{6}{y} = 1 \Rightarrow y^2 - y - 6 = 0

where

y = \log_3(x)

Substitution simplifies the logarithmic equation into a polynomial equation.

3

Solve the quadratic equation by factoring.

(y-3)(y+2) = 0 \Rightarrow y = 3

or

y = -2

Finding the roots for

y

is the intermediate step to solving for

x

.

4

Back-substitute to find the values of

x

.

x = 3^3 = 27

and

x = 3^{-2} = \frac{1}{9}

Converting from logarithmic form back to exponential form yields the values of

x

.

5

Multiply the solutions together.

27 \times \frac{1}{9} = 3

The question asks for the product of all real solutions.

Key Concept

Solving logarithmic equations using the change-of-base formula and quadratic substitution.

Alternative Method

Instead of solving for individual values of

x

, note that if

y_1

and

y_2

are the roots of the quadratic equation

y^2 - y - 6 = 0

, then

y_1 + y_2 = 1

by Vieta's formulas. Since

x_1 = 3^{y_1}

and

x_2 = 3^{y_2}

, the product of the solutions is

x_1 x_2 = 3^{y_1} \cdot 3^{y_2} = 3^{y_1 + y_2} = 3^1 = 3

.

Estimated Time:2m 0s

Question 191Question →

A model rocket is launched vertically upward from a platform. Its height, $h$ , in meters, $t$ seconds after launch is modeled by the equation $h = -4.9t^2 + v_0 t + 2$ , where $v_0$ is the constant initial upward velocity in meters per second. If the rocket never reaches a height of $12$ meters, which of the following inequalities represents all possible values of $v_0$ ?

0 < v_0 < 7

0 < v_0 < 9.8

0 < v_0 < 14

0 < v_0 < 19.6

0 < v_0 < 28

Show answer & explanation

Answer: $0 < v_0 < 14$

Answer

0 < v_0 < 14

The correct answer is the range of values where the velocity is between 0 and 14. Setting the rocket's height equal to 12 meters gives the equation

-4.9t^2 + v_0 t + 2 = 12

. Subtracting 12 from both sides results in

-4.9t^2 + v_0 t - 10 = 0

. Since the rocket never reaches 12 meters, this quadratic equation has no real solutions, meaning its discriminant must be negative. Calculating the discriminant yields

v_0^2 - 4(-4.9)(-10) < 0

, which simplifies to

v_0^2 - 196 < 0

. Solving for a positive velocity gives

0 < v_0 < 14

.

Step-by-Step Solution

1

Set up the equation for the rocket reaching a height of 12 meters.

-4.9t^2 + v_0 t + 2 = 12

To find when the rocket reaches 12 meters, we set the height equation equal to 12.

2

Rearrange the equation into standard quadratic form.

-4.9t^2 + v_0 t - 10 = 0

Subtracting 12 from both sides of the equation sets it to 0, which is the standard form

at^2 + bt + c = 0

.

3

Apply the condition for no real solutions using the discriminant.

\Delta = v_0^2 - 4(-4.9)(-10) < 0

For the rocket to never reach 12 meters, the quadratic equation must have no real solutions. This requires the discriminant

\Delta = b^2 - 4ac

to be strictly negative.

4

Solve the inequality for the positive velocity

v_0

.

v_0^2 - 196 < 0 \implies v_0^2 < 196 \implies v_0 < 14

Multiplying the constants yields

4 \times 4.9 \times 10 = 196

. Taking the square root gives

v_0 < 14

. Since initial upward velocity must be positive,

0 < v_0 < 14

.

Key Concept

Using the discriminant of a quadratic equation to determine the number of real solutions in a physical context.

Question 192Question →

If $w$ is a real number such that $|3 - 2w| \le 9$ , what is the complete range of possible values for $w$ ?

w \ge -3

w \le -3

or

w \ge 6

-3 \le w \le 6

-9 \le w \le 9

-6 \le w \le 3

Show answer & explanation

Answer: $-3 \le w \le 6$

Answer

-3 \le w \le 6

The inequality

|3 - 2w| \le 9

is equivalent to the compound inequality

-9 \le 3 - 2w \le 9

. Subtracting

3

from all parts yields

-12 \le -2w \le 6

. Dividing all parts by

-2

and reversing the inequality signs results in

6 \ge w \ge -3

, which can be rewritten as

-3 \le w \le 6

.

Step-by-Step Solution

1

Set up the compound inequality representing the absolute value inequality.

-9 \le 3 - 2w \le 9

An absolute value inequality of the form

|f(x)| \le c

is equivalent to

-c \le f(x) \le c

.

2

Subtract

3

from all three parts of the inequality to isolate the term containing

w

.

-12 \le -2w \le 6

To solve for

w

, we must isolate the variable term by performing inverse operations on all parts of the inequality.

3

Divide all three parts of the inequality by

-2

and reverse the inequality signs.

6 \ge w \ge -3

, which is equivalent to

-3 \le w \le 6

Dividing an inequality by a negative number requires reversing the direction of the inequality signs to preserve the truth of the statement.

Key Concept

Solving absolute value inequalities of the form

|ax + b| \le c

by converting them into compound inequalities and solving for the variable while reversing the inequality signs when dividing by a negative number.

Estimated Time:1m 30s

Question 193Question →

Given the functions $f(x) = \sqrt{3x + 4}$ and $g(x) = x^2 - 5$ , what is the value of $g(f(7))$ ?

Show answer & explanation

Answer: 20

Answer

20

Evaluating the inner function first gives

f(7) = \sqrt{3(7) + 4} = 5

. Substituting this value into the outer function yields

g(5) = 5^2 - 5 = 20

.

Step-by-Step Solution

1

Evaluate the inner function

f(x)

at

x = 7

.

f(7) = 5

To evaluate a composite function of the form

g(f(x))

, we first calculate the value of the inner function

f(x)

at the given input.

2

Substitute the output of the inner function as the input for the outer function

g(x)

.

g(5) = 20

Since

f(7) = 5

, evaluating

g(f(7))

is equivalent to evaluating

g(5)

.

Key Concept

Function composition involves evaluating an inner function and then using that result as the input for an outer function.

Question 194Question →

The functions $f$ and $g$ are defined for the integers $-2 \leq x \leq 3$ by the table below.

$x$	$f(x)$	$g(x)$
$-2$	$3$	$1$
$-1$	$2$	$3$
$0$	$-2$	$-1$
$1$	$0$	$2$
$2$	$-1$	$0$
$3$	$1$	$2$

What is the value of $x$ for which $g(f(x)) = -1$ ?

-2

-1

0

1

3

Show answer & explanation

Answer: $1$

Answer

The value of

x

is

1

.

To solve

g(f(x)) = -1

, we work from the outside in. First, we find the input to the outer function

g

that results in an output of

-1

. Looking at the table,

g(0) = -1

, which means the inner expression

f(x)

must equal

0

. Next, we find the input to

f

that results in an output of

0

. Looking at the table,

f(1) = 0

. Therefore, the value of

x

that satisfies the equation is

1

.

Step-by-Step Solution

1

Identify the outer function in the composition

g(f(x)) = -1

.

The outer function is

g

, so we must first find an input

u

such that

g(u) = -1

.

This allows us to break down the composition into two simpler function evaluation steps.

2

Look at the column for

g(x)

in the table to find where the output is

-1

.

The table shows

g(0) = -1

, meaning the input to

g

must be

0

. Therefore,

f(x) = 0

.

This establishes the required output value for the inner function

f(x)

.

3

Look at the column for

f(x)

in the table to find where the output is

0

.

The table shows

f(1) = 0

, meaning the input

x

must be

1

.

This provides the final value of the variable

x

that satisfies the original equation.

Key Concept

Evaluating composite functions using a table of values

Estimated Time:1m 0s

Question 195Question →

Which of the following inequality expressions represents the complete solution set for $p$ in the inequality $15 - 3|2p + 5| < 6$ ?

p > -1

-4 < p < -1

p < -4

or

p > -1

-\frac{11}{4} < p < -\frac{9}{4}

p < -1

Show answer & explanation

Answer: $p < -4$ or $p > -1$

Answer

p < -4

or

p > -1

The correct solution is obtained by first isolating the absolute value expression. Subtracting 15 from both sides of the inequality

15 - 3|2p + 5| < 6

yields

-3|2p + 5| < -9

. Dividing both sides by

-3

and reversing the inequality sign gives

|2p + 5| > 3

. This absolute value inequality splits into two cases:

2p + 5 > 3

(which simplifies to

p > -1

) or

2p + 5 < -3

(which simplifies to

p < -4

). Combining these yields the complete solution set

p < -4

or

p > -1

.

Step-by-Step Solution

1

Subtract 15 from both sides of the inequality

15 - 3|2p + 5| < 6

.

-3|2p + 5| < -9

To isolate the absolute value term, first subtract the constant term from both sides.

2

Divide both sides of

-3|2p + 5| < -9

by

-3

and reverse the inequality sign.

|2p + 5| > 3

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

3

Solve the absolute value inequality

|2p + 5| > 3

by setting up two separate inequalities.

2p + 5 > 3

or

2p + 5 < -3

An absolute value inequality of the form

|u| > c

splits into

u > c

or

u < -c

.

4

Solve each linear inequality for

p

.

p > -1

or

p < -4

Subtract 5 from both sides and then divide by 2 for both inequalities to isolate

p

.

Key Concept

Solving absolute value inequalities involving algebraic manipulation and reversing the inequality sign when multiplying or dividing by a negative number.

Question 196Question →

For all positive real numbers $x$ and $y$ , which of the following expressions is equivalent to $\log_3(9x^4 y^{-2})$ ?

9 + 4\log_3(x) - 2\log_3(y)

2 + 4\log_3(x) + 2\log_3(y)

2 + 4\log_3(x) - 2\log_3(y)

\frac{2 + 4\log_3(x)}{2\log_3(y)}

2 + \log_3(4x) - \log_3(2y)

Show answer & explanation

Answer: $2 + 4\log_3(x) - 2\log_3(y)$

Answer

2 + 4\log_3(x) - 2\log_3(y)

The correct answer is correct because applying the product and power properties of logarithms allows us to expand the argument

9x^4 y^{-2}

. Specifically, the expression can be written as

\log_3(9) + \log_3(x^4) + \log_3(y^{-2})

. Since

3^2 = 9

, the term

\log_3(9)

evaluates to

2

. Applying the power rule

\log_b(M^k) = k\log_b(M)

to the remaining terms yields

4\log_3(x)

and

-2\log_3(y)

. Combining these terms results in the equivalent expression.

Step-by-Step Solution

1

Use the product rule of logarithms,

\log_b(MN) = \log_b(M) + \log_b(N)

, to separate the factors in the argument.

\log_3(9x^4 y^{-2}) = \log_3(9) + \log_3(x^4) + \log_3(y^{-2})

This allows the expression to be broken down into individual terms for each base and variable.

2

Evaluate the constant logarithm

\log_3(9)

. Since

3^2 = 9

, this simplifies to

2

.

\log_3(9) = 2

To simplify the numerical term to its integer value.

3

Use the power rule of logarithms,

\log_b(M^k) = k\log_b(M)

, to bring the exponents of the variables to the front of each logarithm.

\log_3(x^4) = 4\log_3(x)

and

\log_3(y^{-2}) = -2\log_3(y)

To isolate the variable terms inside simpler logarithmic expressions.

4

Combine all the simplified parts into a single expression.

2 + 4\log_3(x) - 2\log_3(y)

To write the final expanded equivalent expression.

Key Concept

Logarithmic properties, including product, power, and evaluation of basic log terms.

Alternative Method

Substitute test values for the variables. For example, let

x = 3

and

y = 3

. The original expression evaluates to

\log_3(9 \cdot 3^4 \cdot 3^{-2}) = \log_3(81) = 4

. Substituting

x = 3

and

y = 3

into the correct expression yields

2 + 4\log_3(3) - 2\log_3(3) = 2 + 4(1) - 2(1) = 4

, which matches the original expression's value.

Estimated Time:1m 0s

Question 197Question →

A certain radioactive isotope decays according to the formula $N(t) = N_0 \cdot 2^{-t/8}$ , where $N_0$ is the initial amount of the isotope and $t$ is the time in years. If a sample initially contains $120$ grams of the isotope, how many years will it take for the amount of the isotope to decay to $15$ grams?

Show answer & explanation

Answer: 24

Answer

24

The correct answer is 24 because substituting the initial value of 120 and the final value of 15 into the equation yields

15 = 120 \cdot 2^{-t/8}

. Dividing both sides by 120 gives

\frac{1}{8} = 2^{-t/8}

, which can be rewritten as

2^{-3} = 2^{-t/8}

. Setting the exponents equal to each other gives

-3 = -t/8

, and solving for

t

yields 24.

Step-by-Step Solution

1

Substitute the given values into the decay formula.

15 = 120 \cdot 2^{-t/8}

The initial amount

N_0

is

120

grams and the final amount

N(t)

is

15

grams.

2

Isolate the exponential term.

\frac{1}{8} = 2^{-t/8}

Divide both sides by

120

. Since

\frac{15}{120}

reduces to

\frac{1}{8}

, this isolates the base

2

term.

3

Write the fraction as a power with base 2.

2^{-3} = 2^{-t/8}

Using exponent rules,

\frac{1}{8} = \frac{1}{2^3} = 2^{-3}

.

4

Equate the exponents and solve for

t

.

t = 24

Since the bases are equal, the exponents must be equal, so

-3 = -\frac{t}{8}

which gives

t = 24

.

Key Concept

Solving exponential equations using a common base.

Question 198Question →

The daily revenue $R$ , in dollars, of a manufacturing company is modeled by the quadratic function $R(x) = -0.2x^2 + kx - 1,200$ , where $x$ is the number of units produced and sold, and $k$ is a positive constant. If the maximum daily revenue the company can achieve is $800$ dollars, what is the value of $k$ ?

Show answer & explanation

Answer: 40

Answer

The value of the constant

k

is 40.

Setting the daily revenue function equal to 800 and rewriting it in standard form yields

-0.2x^2 + kx - 2000 = 0

. For a quadratic equation to have exactly one real solution, which represents the maximum vertex of the parabola, the discriminant must be equal to 0. Setting the discriminant

b^2 - 4ac = 0

gives

k^2 - 4(-0.2)(-2000) = 0

, which simplifies to

k^2 - 1600 = 0

. Solving for the positive constant

k

gives

k = 40

.

Step-by-Step Solution

1

Set the revenue function equal to the maximum daily revenue of 800 dollars.

-0.2x^2 + kx - 1,200 = 800

The maximum revenue is the highest point (vertex) on the parabola, where the line

y = 800

is tangent to the curve.

2

Convert the equation into standard quadratic form

ax^2 + bx + c = 0

.

-0.2x^2 + kx - 2,000 = 0

Standard form is required to identify the coefficients

a

,

b

, and

c

for the discriminant formula.

3

Set the discriminant equal to zero.

k^2 - 4(-0.2)(-2,000) = 0

Since the maximum daily revenue is achieved at exactly one point, the quadratic equation must have exactly one real solution, meaning its discriminant (

b^2 - 4ac

) must be zero.

4

Solve for the positive constant

k

.

k^2 - 1,600 = 0 \implies k = 40

Solving the equation yields

k = \pm 40

. Since the problem specifies that

k

is a positive constant, we select

k = 40

.

Key Concept

Using the discriminant of a quadratic equation to find the value of a parameter when there is exactly one real solution.

Estimated Time:2m 0s

Question 199Question →

What is the complete solution set for the inequality $|7 - 3n| < 11$ ?

n < -\frac{4}{3}

or

n > 6

-\frac{4}{3} < n < 6

-6 < n < \frac{4}{3}

n > -\frac{4}{3}

n < -\frac{4}{3}

Show answer & explanation

Answer: $-\frac{4}{3} < n < 6$

Answer

-\frac{4}{3} < n < 6

To solve

|7 - 3n| < 11

, rewrite it as the compound inequality

-11 < 7 - 3n < 11

. Subtracting

7

from all parts yields

-18 < -3n < 4

. Finally, dividing by

-3

and reversing the inequality signs gives

6 > n > -\frac{4}{3}

, which simplifies to the interval

-\frac{4}{3} < n < 6

. This matches the correct option.

Step-by-Step Solution

1

Express the absolute value inequality as a compound inequality.

-11 < 7 - 3n < 11

An absolute value inequality of the form

|u| < c

is equivalent to the compound inequality

-c < u < c

.

2

Subtract

7

from all three parts of the inequality.

-18 < -3n < 4

To isolate the term containing

n

, we subtract

7

from all parts of the inequality.

3

Divide all three parts by

-3

and reverse the inequality signs.

6 > n > -\frac{4}{3}

, which is equivalent to

-\frac{4}{3} < n < 6

Dividing an inequality by a negative number requires reversing the direction of the inequality signs to maintain a true statement.

Key Concept

Absolute Value Equations and Inequalities

Question 200Question →

For the functions $f(x) = |4x - 17|$ and $g(x) = 3 - 2x$ , what is the value of $f(g(5))$ ?

Show answer & explanation

Answer: 45

Answer

The correct answer is 45.

Evaluating the inner function first yields

g(5) = 3 - 2(5) = -7

. Substituting this result into the outer function gives

f(-7) = |4(-7) - 17| = |-45| = 45

.

Step-by-Step Solution

1

Evaluate the inner function

g(5)

g(5) = 3 - 2(5) = -7

In a composite function of the form

f(g(x))

, the inner function

g(x)

must be evaluated first at the given input.

2

Evaluate the outer function

f(x)

at the output of the inner function

f(-7) = |4(-7) - 17| = |-28 - 17| = |-45| = 45

The output of the inner function,

-7

, becomes the input for the outer function

f(x)

.

Key Concept

Function Evaluation and Composition

Intermediate Algebra

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer