Intermediate Algebra Practice Questions — ACT — Page 3

Question 41Question →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = 2x - 3$ and $g(x) = x + 5$ . What is the value of $f(g(4))$ ?

Show answer & explanation

Answer: 15

Answer

The value of

f(g(4))

is

15

.

To evaluate the composite function

f(g(4))

, the inner function must be evaluated first. Substituting

4

into

g(x) = x + 5

gives

g(4) = 9

. Next, substitute this output of

9

into the outer function

f(x) = 2x - 3

, which gives

f(9) = 2(9) - 3 = 15

.

Step-by-Step Solution

1

Evaluate the inner function

g(4)

g(4) = 9

Substitute

4

for

x

in the definition of

g(x) = x + 5

.

2

Evaluate the outer function

f(x)

at the output of the inner function

f(9) = 15

Substitute

9

(the value of

g(4)

) for

x

in the definition of

f(x) = 2x - 3

.

Key Concept

Function Composition and Evaluation

Estimated Time:45s

Question 42Question →

For all real numbers $x$ such that $x \neq 0$ , the expression $\left(\frac{1}{x} + 1\right)^2 - \frac{1}{x^2}$ is equivalent to which of the following?

\frac{x+2}{x}

1

\frac{3}{x^2}

\frac{x+2}{2x}

2x

Show answer & explanation

Answer: $\frac{x+2}{x}$

Answer

\frac{x+2}{x}

Expanding the squared term yields

\frac{1}{x^2} + \frac{2}{x} + 1

. Subtracting

\frac{1}{x^2}

from this leaves

\frac{2}{x} + 1

. Finding a common denominator of

x

to add these terms results in

\frac{x+2}{x}

.

Step-by-Step Solution

1

Expand the binomial term

\left(\frac{1}{x} + 1\right)^2

.

\frac{1}{x^2} + \frac{2}{x} + 1

Using the algebraic identity

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

where

a = \frac{1}{x}

and

b = 1

.

2

Substitute this expansion back into the original expression and subtract

\frac{1}{x^2}

.

\frac{2}{x} + 1

The positive and negative

\frac{1}{x^2}

terms cancel each other out.

3

Write the expression as a single fraction over the common denominator

x

.

\frac{x+2}{x}

Convert

1

to

\frac{x}{x}

and add the numerators.

Key Concept

Simplifying rational expressions by expanding binomials and finding common denominators.

Alternative Method

Use the difference of squares factorization:

A^2 - B^2 = (A-B)(A+B)

. Let

A = \frac{1}{x} + 1

and

B = \frac{1}{x}

. Then the expression simplifies to

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

.

Estimated Time:1m 0s

Question 43Question →

The table below gives values for the functions $f$ and $g$ at selected values of $x$ :

$x$	$f(x)$	$g(x)$
1	3	5
2	5	4
3	2	1
4	1	2
5	4	3

Based on the table, what is the value of $f(g(3))$ ?

1

2

3

4

5

Show answer & explanation

Answer: 3

Answer

3

To evaluate the composite function

f(g(3))

, we first evaluate the inner function,

g(3)

. Looking at the table, when

x = 3

,

g(3) = 1

. We then substitute this output into the outer function to evaluate

f(1)

. According to the table, when

x = 1

,

f(1) = 3

. Therefore, the value of the composite function is 3.

Step-by-Step Solution

1

Locate the input value of 3 in the table to evaluate the inner function

g(3)

.

g(3) = 1

We must evaluate the inner function first in the composition

f(g(x))

.

2

Substitute the result of

g(3)

into the outer function, giving

f(1)

, and find its value in the table.

f(1) = 3

Evaluating

f(x)

at

x = 1

completes the composition

f(g(3))

.

Key Concept

Evaluating a composite function from a table of values by finding the output of the inner function and using it as the input for the outer function.

Estimated Time:45s

Question 44Question →

For the functions $f(x) = (x - 3)^2 - 4$ and $g(x) = |x + 1|$ , what is the sum of all real values of $x$ for which $f(g(x)) = 5$ ?

-3

-2

4

5

6

Show answer & explanation

Answer: $-3$

Answer

The sum of all real values of

x

for which

f(g(x)) = 5

is

-3

.

To find the sum of all real values of

x

for which

f(g(x)) = 5

, we set up the composite function:

(g(x) - 3)^2 - 4 = 5

. Adding

4

to both sides gives

(g(x) - 3)^2 = 9

. Taking the square root yields two possible cases:

g(x) - 3 = 3

(which means

g(x) = 6

) or

g(x) - 3 = -3

(which means

g(x) = 0

). Next, we substitute

g(x) = |x + 1|

. For the first case,

|x + 1| = 6

gives

x + 1 = 6 \implies x = 5

or

x + 1 = -6 \implies x = -7

. For the second case,

|x + 1| = 0

gives

x + 1 = 0 \implies x = -1

. Summing these three real solutions gives

5 + (-7) + (-1) = -3

.

Step-by-Step Solution

1

Substitute

g(x)

into

f(x)

and set the expression equal to

5

.

f(g(x)) = (g(x) - 3)^2 - 4 = 5

This sets up the composition equation to be solved.

2

Isolate the squared term and solve for

g(x)

by taking the square root of both sides.

(g(x) - 3)^2 = 9

, which gives two cases:

g(x) - 3 = 3 \implies g(x) = 6

or

g(x) - 3 = -3 \implies g(x) = 0

Solving a quadratic equation of the form

u^2 = k

yields two possibilities:

u = \pm\sqrt{k}

.

3

Substitute the expression for

g(x) = |x + 1|

into both cases and solve for

x

.

For

g(x) = 6

,

|x + 1| = 6 \implies x + 1 = 6

or

x + 1 = -6

, yielding

x = 5

or

x = -7

. For

g(x) = 0

,

|x + 1| = 0 \implies x + 1 = 0

, yielding

x = -1

.

Absolute value equations of the form

|v| = c

yield

v = \pm c

if

c > 0

, and

v = 0

if

c = 0

.

4

Calculate the sum of all the real solutions.

5 + (-7) + (-1) = -3

The question asks for the sum of all real values of

x

that satisfy the equation.

Key Concept

Function composition involves substituting one function into another, and solving the resulting equation requires accounting for multiple cases when dealing with quadratic and absolute value expressions.

Question 45Question →

Determine the values of $x$ for which the given rational expression is undefined.

Fill in the blanks below

The rational expression

\frac{x + 5}{x^2 - 4x - 12}

is undefined for two real values of

x

. The smaller of these values is and the larger of these values is .

Show answer & explanation

Answer

The rational expression is undefined for

x = -2

and

x = 6

.

The expression is undefined when the denominator is zero. Solving

x^2 - 4x - 12 = 0

yields

(x-6)(x+2) = 0

, which gives

x = -2

and

x = 6

. The smaller value is

-2

and the larger value is

6

.

Step-by-Step Solution

1

Identify the condition that makes a rational expression undefined.

The denominator must equal zero:

x^2 - 4x - 12 = 0

Division by zero is undefined in the set of real numbers.

2

Factor the quadratic equation

x^2 - 4x - 12 = 0

.

(x - 6)(x + 2) = 0

We need to find two numbers that multiply to

-12

and add to

-4

. These numbers are

-6

and

2

.

3

Solve for

x

by setting each factor to zero.

x = 6

or

x = -2

By the zero product property, if

(x - 6)(x + 2) = 0

, then

x - 6 = 0

or

x + 2 = 0

.

4

Determine the smaller and larger values to place in the blanks.

Smaller value is

-2

and larger value is

6

.

Comparing the two numbers,

-2

is less than

6

.

Key Concept

A rational expression is undefined when its denominator equals zero.

Question 46Question →

For all real values of $x$ where the expression is defined, which of the following inequality expressions represents the complete set of values that satisfy the inequality below?

\frac{|x - 2| - 3}{|x + 1| - 2} \leq 0

-3 \leq x \leq -1

or

1 \leq x \leq 5

-1 \leq x \leq 5

-3 < x \leq -1

or

1 < x \leq 5

1 < x \leq 5

x < -3

or

-1 \leq x < 1

or

x \geq 5

Show answer & explanation

Answer: $-3 < x \leq -1$ or $1 < x \leq 5$

Answer

-3 < x \leq -1

or

1 < x \leq 5

The correct answer represents the union of the two valid cases where the numerator and denominator have opposite signs. Specifically, the numerator

|x - 2| - 3

is non-positive when

-1 \leq x \leq 5

, and the denominator

|x + 1| - 2

is positive when

x < -3

or

x > 1

. Their intersection is the interval

1 < x \leq 5

. Alternatively, the numerator is non-negative when

x \leq -1

or

x \geq 5

, and the denominator is negative when

-3 < x < 1

. Their intersection is the interval

-3 < x \leq -1

. Combining these yields the solution set

-3 < x \leq -1

or

1 < x \leq 5

.

Step-by-Step Solution

1

Analyze the conditions required for a rational expression to be non-positive.

The fraction

\frac{N}{D} \leq 0

is satisfied when the numerator

N

and denominator

D

have opposite signs, while ensuring the denominator is not equal to zero (

D \neq 0

). Thus, we must evaluate two cases: Case 1 where

N \leq 0

and

D > 0

, and Case 2 where

N \geq 0

and

D < 0

.

This establishes the logical framework for solving the rational inequality.

2

Determine the critical intervals for the numerator,

N = |x - 2| - 3

.

Solve

|x - 2| - 3 \leq 0 \implies |x - 2| \leq 3 \implies -3 \leq x - 2 \leq 3 \implies -1 \leq x \leq 5

. Thus, the numerator is non-positive on

[-1, 5]

and non-negative on

(-\infty, -1] \cup [5, \infty)

.

This identifies the values of

x

where the numerator changes sign.

3

Determine the critical intervals for the denominator,

D = |x + 1| - 2

, and identify restrictions.

Solve

|x + 1| - 2 < 0 \implies |x + 1| < 2 \implies -2 < x + 1 < 2 \implies -3 < x < 1

. Thus, the denominator is negative on

(-3, 1)

and positive on

(-\infty, -3) \cup (1, \infty)

. The boundary points

x = -3

and

x = 1

make the denominator zero and must be excluded from the domain.

This identifies the values of

x

where the denominator changes sign or makes the expression undefined.

4

Evaluate Case 1 and Case 2, then combine the solutions.

For Case 1 (

N \leq 0

and

D > 0

), find the intersection of

[-1, 5]

and

(-\infty, -3) \cup (1, \infty)

, which yields

1 < x \leq 5

. For Case 2 (

N \geq 0

and

D < 0

), find the intersection of

(-\infty, -1] \cup [5, \infty)

and

(-3, 1)

, which yields

-3 < x \leq -1

. Combining these intervals gives the final solution set:

-3 < x \leq -1

or

1 < x \leq 5

.

This determines the overall solution set that satisfies the starting inequality.

Key Concept

Solving rational inequalities containing absolute value expressions by finding critical intervals.

Estimated Time:3m 0s

Question 47Question →

For the complex number $z = \frac{4 + 2i}{3 - i}$ , where $i$ is the imaginary unit such that $i^2 = -1$ , what is the imaginary part of $z$ ?

\frac{5}{4}

\frac{6}{5}

\frac{1}{5}

1

-2

Show answer & explanation

Answer: $1$

Answer

The imaginary part of

z

is

1

.

To find the imaginary part of

z

, multiply the numerator and denominator of

\frac{4 + 2i}{3 - i}

by the complex conjugate of the denominator, which is

3 + i

. This simplifies the denominator to

9 - i^2 = 10

and the numerator to

12 + 10i + 2i^2 = 10 + 10i

. Dividing each term by

10

yields

1 + i

, which has an imaginary part of

1

.

Step-by-Step Solution

1

Identify the complex conjugate of the denominator.

The complex conjugate of the denominator,

3 - i

, is

3 + i

.

To divide complex numbers, we multiply both the numerator and denominator by the conjugate of the denominator to make the denominator a real number.

2

Multiply the numerator and denominator of the fraction by the complex conjugate.

z = \frac{(4 + 2i)(3 + i)}{(3 - i)(3 + i)}

Multiplying by

\frac{3 + i}{3 + i}

is equivalent to multiplying by

1

, which preserves the value of the complex number.

3

Expand and simplify the numerator and denominator.

Numerator:

(4 + 2i)(3 + i) = 12 + 4i + 6i + 2i^2 = 12 + 10i + 2(-1) = 10 + 10i

. Denominator:

(3 - i)(3 + i) = 9 - i^2 = 9 - (-1) = 10

.

Apply the distributive property (FOIL method) and substitute

i^2 = -1

to simplify the expression.

4

Divide each term in the numerator by the denominator to find the imaginary part.

z = \frac{10 + 10i}{10} = 1 + i

, so the imaginary part is

1

.

Rewrite the fraction in standard form

a + bi

, where the imaginary part is

b

(the coefficient of

i

).

Key Concept

Simplifying a quotient of complex numbers by multiplying both the numerator and the denominator by the complex conjugate of the denominator.

Question 48Question →

When solving the radical equation $\sqrt{3x+10} + \sqrt{x+2} = 2$ for all real values of $x$ , squaring both sides yields two potential solutions. What is the numerical value of the extraneous solution?

Fill in the blanks below

The numerical value of the extraneous solution is .

Show answer & explanation

Answer

The extraneous solution is

x = 2

.

The value

x = 2

is the extraneous solution because substituting it into the original equation yields

\sqrt{16} + \sqrt{4} = 6 \neq 2

, showing it does not satisfy the original equation despite being generated by the algebraic squaring process.

Step-by-Step Solution

1

Isolate one of the radical terms in the equation.

\sqrt{3x+10} = 2 - \sqrt{x+2}

Isolating one radical simplifies the squaring process.

2

Square both sides of the equation.

3x+10 = 4 - 4\sqrt{x+2} + (x+2)

, which simplifies to

2x+4 = -4\sqrt{x+2}

.

Squaring eliminates the isolated radical.

3

Divide both sides by 2 and square both sides again.

x+2 = -2\sqrt{x+2} \Rightarrow (x+2)^2 = 4(x+2)

, which simplifies to

x^2 - 4 = 0

.

This eliminates the remaining radical term.

4

Solve the quadratic equation for

x

.

x = 2

or

x = -2

Factoring

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

gives the potential solutions.

5

Check both potential solutions in the original equation.

For

x = -2

:

\sqrt{4} + \sqrt{0} = 2

, which is true. For

x = 2

:

\sqrt{16} + \sqrt{4} = 6 \neq 2

, which is false.

This identifies which solution is extraneous.

Key Concept

Solving radical equations and verifying solutions to identify extraneous roots

Estimated Time:2m 0s

Question 49Question →

For all real numbers $x$ such that $x \neq 0$ and $x \neq -1$ , which of the following is equivalent to the expression $\frac{1}{x} + \frac{1}{x+1}$ ?

\frac{2}{2x+1}

\frac{2}{x^2+x}

\frac{2x+1}{x^2+1}

\frac{2x+1}{x^2+x}

\frac{3}{x+1}

Show answer & explanation

Answer: $\frac{2x+1}{x^2+x}$

Answer

The correct expression is

\frac{2x+1}{x^2+x}

.

The correct expression is obtained by finding a common denominator of

x(x+1)

, rewriting the numerators as

x+1

and

x

, and then adding them to get

2x+1

over the expanded denominator

x^2+x

.

Step-by-Step Solution

1

Find a common denominator for the two rational terms.

The common denominator for

\frac{1}{x}

and

\frac{1}{x+1}

is

x(x+1)

, which expands to

x^2+x

.

To add fractions with different denominators, they must share a common denominator.

2

Rewrite each fraction with the common denominator by multiplying their numerators and denominators by the appropriate factor.

\frac{1}{x} = \frac{x+1}{x(x+1)}

and

\frac{1}{x+1} = \frac{x}{x(x+1)}

Multiplying the numerator and denominator of each fraction by the missing factor of the common denominator preserves the value of the fraction.

3

Add the numerators together over the common denominator and simplify.

\frac{(x+1) + x}{x(x+1)} = \frac{2x+1}{x^2+x}

Combining the numerators and expanding the denominator gives the simplified equivalent expression.

Key Concept

Adding rational expressions by finding a common denominator

Question 50Question →

If $f(x) = 3x - 2$ and $g(x) = x^2 + 1$ , what is the value of $g(f(2))$ ?

Show answer & explanation

Answer: 17

Answer

The value of

g(f(2))

is 17.

To evaluate the composite function

g(f(2))

, we first find the value of the inner function

f(2) = 3(2) - 2 = 4

. Then, we substitute this output as the input for the outer function to get

g(4) = 4^2 + 1 = 17

.

Step-by-Step Solution

1

Substitute

x = 2

into the expression for

f(x)

to evaluate the inner function.

f(2) = 4

In the composition

g(f(2))

, the inner function

f(x)

must be evaluated first at the input value of 2.

2

Substitute the result from Step 1, which is 4, into the expression for

g(x)

.

g(4) = 17

The output of the inner function becomes the input for the outer function, so

g(f(2)) = g(4) = 4^2 + 1 = 17

.

Key Concept

Function Composition

Estimated Time:45s

Question 51Question →

Which of the following is the solution set for the inequality $|5 - 2x| < 9$ ?

-2 < x < 7

x < -2

or

x > 7

-7 < x < 2

x > -2

x < 7

Show answer & explanation

Answer: $-2 < x < 7$

Answer

-2 < x < 7

The correct solution is found by setting up the compound inequality

-9 < 5 - 2x < 9

. Subtracting

5

from all parts gives

-14 < -2x < 4

. Finally, dividing by

-2

and reversing the inequality signs yields the interval

-2 < x < 7

.

Step-by-Step Solution

1

Rewrite the absolute value inequality as a compound inequality.

-9 < 5 - 2x < 9

An absolute value inequality of the form

|u| < c

represents all points within distance

c

from

0

, which translates to

-c < u < c

.

2

Subtract

5

from all three parts of the inequality.

-14 < -2x < 4

To isolate the variable term, we perform the inverse operation of adding

5

, which is subtracting

5

.

3

Divide all three parts by

-2

and reverse the inequality signs.

-2 < x < 7

Dividing by a negative number reverses the inequality direction. Doing so gives

7 > x > -2

, which is conventionally written as

-2 < x < 7

.

Key Concept

Solving absolute value inequalities of the form

|ax + b| < c

Question 52Question →

A company's daily profit is modeled by the function $P(x) = 3x - 50$ , where $x$ represents the number of items sold. The number of items sold depends on the number of hours the store is open, $h$ , according to the function $x(h) = 5h$ . What is the daily profit, in dollars, when the store is open for $8$ hours?

-26

40

70

120

170

Show answer & explanation

Answer: 70

Answer

70

Evaluating the composite function in the correct order,

P(x(8))

, yields

70

. First, find the number of items sold when the store is open for 8 hours:

x(8) = 5(8) = 40

. Then, calculate the daily profit by substituting this output into the profit function:

P(40) = 3(40) - 50 = 120 - 50 = 70

.

Step-by-Step Solution

1

Determine the number of items sold,

x

, when the store is open for

8

hours by evaluating the function

x(h) = 5h

at

h = 8

.

x(8) = 5 \times 8 = 40

items.

The number of items sold is the input for the profit function.

2

Substitute the number of items sold (

x = 40

) into the profit function

P(x) = 3x - 50

.

P(40) = 3(40) - 50 = 120 - 50 = 70

dollars.

Evaluating the profit function at the given number of items sold determines the final daily profit.

Key Concept

Evaluating a composite function by finding the output of the inner function and using it as the input for the outer function.

Estimated Time:1m 0s

Question 53Question →

What is the positive real solution to the equation $\frac{3}{x-2} - \frac{2}{x} = 1$ ?

Show answer & explanation

Answer: 4

Answer

The positive real solution to the equation is 4.

By multiplying the entire equation by the least common denominator,

x(x-2)

, the rational equation is cleared of fractions, resulting in

3x - 2(x-2) = x(x-2)

. Simplifying this yields the quadratic equation

x^2 - 3x - 4 = 0

. Factoring the quadratic equation gives

(x - 4)(x + 1) = 0

, which has roots of

x = 4

and

x = -1

. The positive solution is 4.

Step-by-Step Solution

1

Multiply both sides of the equation by the least common denominator,

x(x-2)

, to eliminate the fractions.

3x - 2(x-2) = x(x-2)

Eliminating the denominators simplifies the rational equation into a polynomial equation.

2

Distribute and combine like terms to simplify both sides of the equation.

x + 4 = x^2 - 2x

Simplification is necessary before rearranging the equation into a solvable form.

3

Rearrange the terms into standard quadratic form:

ax^2 + bx + c = 0

.

x^2 - 3x - 4 = 0

Setting the quadratic expression equal to zero allows for factoring and solving.

4

Factor the quadratic equation.

(x - 4)(x + 1) = 0

Finding the factors reveals the potential values for

x

.

5

Solve for the roots and select the positive real solution.

x = 4

The equation yields two solutions,

x = 4

and

x = -1

, and the question explicitly requests the positive solution.

Key Concept

Solving rational equations by converting them to quadratic equations using the least common denominator.

Question 54Question →

For the imaginary unit $i$ , where $i^2 = -1$ , what is the value of the expression $(3 + 2i)^2 - (3 - 2i)^2$ ?

0

10

24

24i

10 + 24i

Show answer & explanation

Answer: $24i$

Answer

The correct answer is

24i

.

The expression can be simplified by expanding each binomial term first. The first term,

(3 + 2i)^2

, expands to

9 + 12i + 4i^2

. Since

i^2 = -1

, this simplifies to

9 + 12i - 4 = 5 + 12i

. The second term,

(3 - 2i)^2

, expands to

9 - 12i + 4i^2

, which simplifies to

9 - 12i - 4 = 5 - 12i

. Subtracting the second simplified term from the first gives

(5 + 12i) - (5 - 12i) = 5 - 5 + 12i - (-12i) = 24i

.

Step-by-Step Solution

1

Expand the first squared binomial expression,

(3 + 2i)^2

.

(3 + 2i)^2 = 9 + 12i + 4i^2 = 9 + 12i - 4 = 5 + 12i

Apply the binomial squaring formula

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

and use the property of the imaginary unit where

i^2 = -1

.

2

Expand the second squared binomial expression,

(3 - 2i)^2

.

(3 - 2i)^2 = 9 - 12i + 4i^2 = 9 - 12i - 4 = 5 - 12i

Apply the binomial squaring formula

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

and use the property of the imaginary unit where

i^2 = -1

.

3

Subtract the second expanded expression from the first.

(5 + 12i) - (5 - 12i) = 5 + 12i - 5 + 12i = 24i

Distribute the negative sign to both terms of the second complex number and combine the real and imaginary parts.

Key Concept

Complex Numbers and Operations

Question 55Question →

For the functions $f(x) = x^2 - 3x - 1$ and $g(x) = |x - 1|$ , what is the product of all real values of $x$ for which $f(g(x)) = 3$ ?

5

0

-15

-5

-6

Show answer & explanation

Answer: -15

Answer

The product of all real values of

x

is

-15

.

The correct answer is

-15

. By substituting

g(x) = |x - 1|

into

f(x) = x^2 - 3x - 1

, we get

|x - 1|^2 - 3|x - 1| - 1 = 3

. Letting

u = |x - 1|

yields the quadratic

u^2 - 3u - 4 = 0

, which factors into

(u - 4)(u + 1) = 0

. Since

u \geq 0

, we have

u = 4

. Solving

|x - 1| = 4

yields

x = 5

and

x = -3

. The product of these solutions is

5 \times (-3) = -15

.

Step-by-Step Solution

1

Substitute the function

g(x) = |x - 1|

into

f(x) = x^2 - 3x - 1

to form the composite function

f(g(x))

.

f(g(x)) = |x - 1|^2 - 3|x - 1| - 1

To evaluate the composite function before setting it equal to

3

.

2

Set the composite function equal to

3

and substitute

u = |x - 1|

(where

u \geq 0

) to rewrite the equation as a quadratic in terms of

u

.

u^2 - 3u - 1 = 3 \Rightarrow u^2 - 3u - 4 = 0

Using substitution simplifies the absolute value equation into a standard quadratic equation.

3

Factor the quadratic equation

(u - 4)(u + 1) = 0

to solve for

u

, and discard the extraneous solution.

u = 4

or

u = -1

. Since

u = |x - 1| \geq 0

, we discard

u = -1

, leaving

u = 4

.

The absolute value of a real expression must be non-negative, so

u

cannot be

-1

.

4

Solve the absolute value equation

|x - 1| = 4

for all possible real values of

x

.

x - 1 = 4 \Rightarrow x = 5

, and

x - 1 = -4 \Rightarrow x = -3

.

An absolute value equation

|A| = B

(where

B > 0

) yields two scenarios:

A = B

and

A = -B

.

5

Calculate the product of the two real solutions found.

5 \times (-3) = -15

To find the final product of all real values of

x

that satisfy the equation.

Key Concept

Function composition and solving absolute value equations using quadratic substitution

Estimated Time:2m 0s

Question 56Question →

For all real numbers $x$ such that $x \neq 1$ and $x \neq -1$ , which of the following is equivalent to the expression $\left(\frac{1}{x-1} + \frac{1}{x+1}\right)^2$ ?

\frac{1}{x^2}

\frac{2x^2}{(x^2-1)^2}

\frac{2x^2+2}{(x^2-1)^2}

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

\frac{4x^2}{(x^2-1)^2}

Show answer & explanation

Answer: $\frac{4x^2}{(x^2-1)^2}$

Answer

The expression is equivalent to the fraction with numerator four times x squared and denominator the square of the difference x squared minus one.

The correct answer is found by first rewriting the terms inside the parentheses with the common denominator

(x^2-1)

, which simplifies the sum to

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

. Squaring the numerator

(2x)^2

yields

4x^2

, and squaring the denominator yields

(x^2-1)^2

.

Step-by-Step Solution

1

Find a common denominator for the two fractions inside the parentheses.

The common denominator is

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

. The expression inside the parentheses becomes

\frac{x+1}{x^2-1} + \frac{x-1}{x^2-1}

.

To add rational expressions, they must have a common denominator.

2

Combine the numerators over the common denominator.

The sum is

\frac{(x+1) + (x-1)}{x^2-1} = \frac{2x}{x^2-1}

.

Combine the like terms in the numerator to simplify the expression before squaring.

3

Square the simplified rational expression by squaring the numerator and the denominator separately.

\left(\frac{2x}{x^2-1}\right)^2 = \frac{(2x)^2}{(x^2-1)^2} = \frac{4x^2}{(x^2-1)^2}.

Apply the power of a quotient property to obtain the final simplified expression.

Key Concept

Simplifying rational expressions and applying the power of a quotient rule.

Estimated Time:1m 30s

Question 57Question →

Let the function $f$ be defined by $f(x) = \frac{x - 1}{x + 1}$ for all real numbers $x \neq -1$ . Let $f^1(x) = f(x)$ , and let $f^n(x) = f(f^{n-1}(x))$ for all integers $n \geq 2$ . What is the value of $30 \cdot f^{2026}(3)$ ?

Show answer & explanation

Answer: -10

Answer

The value of the expression is -10.

Evaluating successive iterations of

f(3)

reveals a repeating sequence:

f^1(3) = 1/2

,

f^2(3) = -1/3

,

f^3(3) = -2

,

f^4(3) = 3

, and then

f^5(3) = 1/2

. This indicates a cycle of period 4. Dividing the power 2026 by the period 4 gives a remainder of 2, meaning

f^{2026}(3)

is equal to

f^2(3) = -1/3

. Multiplying this by 30 yields the final answer of -10.

Step-by-Step Solution

1

Calculate the first few compositions of the function evaluated at the given input

x = 3

.

f^1(3) = 1/2

,

f^2(3) = -1/3

,

f^3(3) = -2

, and

f^4(3) = 3

.

To look for a repeating pattern or periodic behavior in the iterated function composition.

2

Identify the period of the repeating cycle.

The cycle has a length of 4, repeating the values

[1/2, -1/3, -2, 3]

.

Since

f^4(3) = 3

, evaluating further iterations will yield the same sequence of values.

3

Use modular arithmetic to find the value of the 2026th composition.

2026 \equiv 2 \pmod 4

, meaning

f^{2026}(3) = f^2(3) = -1/3

.

Since the cycle repeats every 4 iterations, dividing 2026 by 4 yields a remainder of 2, indicating the second value in the sequence.

4

Multiply the computed composition value by the given coefficient.

30 \cdot (-1/3) = -10

.

To find the final value of the expression

30 \cdot f^{2026}(3)

.

Key Concept

Evaluating repeated function compositions using periodicity and modular arithmetic.

Question 58Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined as $z = \frac{a + bi}{1 - 2i}$ , where $a$ and $b$ are real numbers. If $z = 4 + 3i$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a + b

is 5.

To find the value of

a + b

, we start with the equation

\frac{a + bi}{1 - 2i} = 4 + 3i

. Multiplying both sides by the denominator gives

a + bi = (4 + 3i)(1 - 2i)

. Expanding the right side using the distributive property, we get

a + bi = 4(1) + 4(-2i) + 3i(1) + 3i(-2i) = 4 - 8i + 3i - 6i^2

. Substituting

i^2 = -1

simplifies the expression to

4 - 5i - 6(-1) = 4 - 5i + 6 = 10 - 5i

. By comparing the real and imaginary parts of both sides, we find that

a = 10

and

b = -5

. The sum of these two values is

a + b = 10 + (-5) = 5

.

Step-by-Step Solution

1

Isolate the numerator by multiplying both sides by the denominator.

a + bi = (4 + 3i)(1 - 2i)

To solve for the variables

a

and

b

in the numerator, we clear the fraction by multiplying by the denominator.

2

Expand the product of the two complex numbers.

a + bi = 4 - 8i + 3i - 6i^2

Distribute each term of the first binomial to each term of the second binomial.

3

Simplify the expression using the definition of

i^2

.

a + bi = 10 - 5i

Since

i^2 = -1

, the term

-6i^2

becomes

+6

. Combine the real parts (

4 + 6 = 10

) and imaginary parts (

-8i + 3i = -5i

).

4

Equate the components and calculate

a + b

.

a = 10

,

b = -5

, and

a + b = 5

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Therefore,

a = 10

and

b = -5

. Summing these yields

10 + (-5) = 5

.

Key Concept

Equality and multiplication of complex numbers

Question 59Question →

A weather balloon's altitude in meters after $t$ minutes is given by the function $A(t) = 100 + 50t$ . The air temperature in degrees Celsius at an altitude of $a$ meters is modeled by the function $T(a) = 25 - 0.02a$ . What is the temperature of the air surrounding the balloon, in degrees Celsius, after $2$ minutes?

Show answer & explanation

Answer: 21

Answer

The temperature of the air surrounding the balloon after 2 minutes is 21 degrees Celsius.

The temperature of the air surrounding the balloon after 2 minutes is found by first calculating the balloon's altitude,

A(2) = 100 + 50(2) = 200

meters, and then using this altitude to evaluate the temperature function,

T(200) = 25 - 0.02(200) = 21

.

Step-by-Step Solution

1

Calculate the altitude of the weather balloon at

t = 2

minutes using the function

A(t)

.

A(2) = 200

meters

To find the temperature surrounding the balloon, we must first determine its altitude at the given time of 2 minutes.

2

Substitute the altitude of 200 meters into the temperature function

T(a)

.

T(200) = 21

degrees Celsius

The temperature function depends on the altitude, so evaluating

T(200)

yields the temperature at that height.

Key Concept

Evaluating composite functions in a real-world context.

Question 60Question →

Which of the following inequality expressions represents the complete set of real values of $p$ that satisfy the inequality $|7 - 2p| < 13$ ?

p < -3

or

p > 10

p > -3

-3 < p < 10

-\frac{13}{5} < p < \frac{13}{5}

-10 < p < 3

Show answer & explanation

Answer: $-3 < p < 10$

Answer

-3 < p < 10

To solve

|7 - 2p| < 13

, write it as the compound inequality

-13 < 7 - 2p < 13

. Subtracting

7

from all parts gives

-20 < -2p < 6

. Dividing by

-2

and reversing the inequality signs yields

-3 < p < 10

.

Step-by-Step Solution

1

Write the absolute value inequality as a compound inequality.

-13 < 7 - 2p < 13

An absolute value inequality of the form

|x| < c

is equivalent to

-c < x < c

.

2

Subtract

7

from all three parts of the compound inequality.

-20 < -2p < 6

To isolate the variable term, we perform the inverse operation of adding

7

by subtracting

7

from each part.

3

Divide all parts by

-2

and reverse the inequality signs.

10 > p > -3

, which simplifies to

-3 < p < 10

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| < c

Intermediate Algebra

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