Deck 1: The Real Numbers

Question 1

Find all numbers x satisfying the following equation.
$|x-5|=0$

Answer 1

5

Question 2

Find all numbers x satisfying the equation.
$|17 x+3|=0$

A) $\frac{17}{3}$
B) $-\frac{17}{3}$
C) $\frac{3}{17}$
D) $-\frac{3}{17}$

Answer 2

$-\frac{3}{17}$

Question 3

Find all numbers x satisfying the equation. If there is more than one answer, enter the largest one first and separate them by commas.
$|x-8|=2$

Answer 3

10, 6

Question 4

Find all numbers x satisfying the equation.
$\left|\frac{x+7}{x-7}\right|=1$

Question 5

Find all numbers x satisfying the equation. If there is more than one answer, enter the largest one first and separate them by commas.
$\left|\frac{8 x+9}{x-11}\right|=10$

Question 6

Write the union as a single interval.
$[7,10) \cup(8,12]$

Question 7

Write the union as a single interval.
$(-\infty, 9) \cup(0,8]$

Question 8

Write the union as a single interval.

Question 9

Write the union as a single interval.
$(-\infty, 6] \cup[-12, \infty)$

Question 10

Write the union as a single interval. $(-9,14] \cup[10,23)$

A) $(-9,14]$
B) $[14,23)$
C) $(-9,23]$
D) $(-9,23)$

Question 11

A solution of the simultaneous equations $|x+y|=2$ And $|x|+|y|=4$ Is an ordered pair of the form (a, b). How many solutions have these simultaneous equations?

A) None
B) 4
C) 2
D) 1

Question 12

Write the set as an interval or as the union of two intervals.
$\{x:|x|<4\}$

Question 13

Write the set as an interval or as the union of two intervals.
$\{x:|x-13| \leq 8\}$

Question 14

Write the set as an interval or as the union of two intervals.
$\{x:|x| \geq 5\}$

Question 15

Write the set as an interval or as the union of two intervals.
$\{x:|15 x-17|<17\}$

Question 16

Which of the following sets corresponds to the interval [1, 9]?
I . $\{x:|x-5| \leq 4\}$
II . $\{x:|x-3| \leq 6\}$
III . $\{x:|5-x| \leq 4\}$

A) I
B) II
C) I and III
D) I, II, and III

Question 17

Write the intersection as a single interval.
$[2,7) \cap[5,11]$

Question 18

Write the intersection as a single interval.
$(-1,21] \cap(9,23]$

Question 19

Which of the following alternatives corresponds to the interval (2, 9]?

A) $(2, \infty] \cap[-\infty, 9)$
B) $(2, \infty) \cap(-\infty, 10]$
C) $(2, \infty) \cap(-\infty, 4]$
D) $(2, \infty) \cap(-\infty, 9]$

Question 20

Which of the following intersections corresponds to the interval [-1, 5)?
I . $(-\infty, 5) \cap[-1,9]$
II . $(-\infty, 5) \cap[-1, \infty)$
III . $(-\infty, 4] \cap[-1,9]$

A) I
B) II
C) I and II
D) II and III

Question 21

Find all numbers with absolute value 6.

Question 22

Evaluate
$|-4|-|-9|$

Question 23

Find all numbers x satisfying the equation. If there is more than one answer, enter the largest one first and separate them by commas.
$|x-8|=x-8$

Question 24

Find all numbers x satisfying the equation. If there is more than one answer, enter the largest one first and separate them by commas.
$|x|=x+5$

Question 25

Find all numbers x satisfying the equation. If there is more than one answer, enter the largest one first and separate them by commas.
$|x-7|=x+9$

Question 26

Find all numbers x satisfying the inequality. Give your solution in interval notation.
$\frac{5 x+1}{x-4}<7$

Question 27

Find all numbers x satisfying the inequality. Give your solution in interval notation.
$\left|\frac{7 x-5}{x+4}\right|<1$

Question 28

Perform the following operations.
$1+2 \cdot 7+1 \cdot 1-1$

Question 29

Perform the following operations.
$5+6 \cdot 9+12 \cdot 2$

Question 30

Expand the given expression.
(4x - 2y)(4z + 2w - t)

Question 31

Expand the given expression.
(x + 4y - r)(2z - w + 4t)

Question 32

Expand the given expression.
(x - 2y + 3z)²

Question 33

Expand the given expression.
(2x + 9y + 4z)²_.

Question 34

Evaluate: $6 \cdot 8 \cdot 4+3 \cdot 9$

A) 45
B) 219
C) 567
D) 165

Question 35

Simplify the given expression as much as possible.
7(m + 7n) + 6m

Question 36

Simplify the given expression as much as possible.
2(5N + 3(M - 2N)) + 4N

Question 37

Which of the following is true?

A) $\frac{16}{8}=\frac{1}{2}$
B) 3x - x = 0
C) $\frac{3}{8}-\frac{c}{16}=\frac{6-c}{16}$
D) $\frac{3 x+1}{x}=3+1$

Question 38

Which of the following expressions equals 0? (Assume x is not zero.)
I. 3x - x
II. $\frac{3}{x}-\frac{3 x}{x^{2}}$
III. $\frac{3 x-b}{b}-3 x+1$

A) I
B) II
C) III
D) II and III

Question 39

Which of the following expressions equals 7x - y?
I. 7(x - 2y) + 2(7 - 1)y + y
II. 2y + 7(x - 1) + (7 - 3y)
III. 2(7x - y) + (y - 7x)

A) I
B) II
C) III
D) I, II, and III

Question 40

Which alternative equals the fraction $\frac{b}{c}$ ?

A) $\frac{b+5}{c+5}$
B) $\frac{b-5}{c+5}$
C) $\frac{b-5}{c-5}$
D) $\frac{5 b-1}{5 c-1}$

Question 41

Perform the indicated operation.
$\frac{2}{5}+\frac{3}{10}$

Question 42

Perform the indicated operation.
$\frac{3}{9} \cdot \frac{18}{9}$

Question 43

Simplify the given expression as much as possible.
$\frac{N}{5}+\frac{3}{M-6}$

Question 44

Simplify the given expression as much as possible.
$\frac{x-5}{y+2} \div \frac{y-2}{x+5}$

Question 45

Simplify the given expression as much as possible.
$\frac{(7 x+b)^{2}-49 x^{2}}{b}$

Question 46

Simplify the given expression as much as possible.
$\frac{x-7}{y} \div \frac{z}{x+7}$

Question 47

Simplify the given expression as much as possible.
$\frac{1}{81(x-y)} \cdot\left(\frac{9 x}{y}-\frac{81 y}{9 x}\right)$

Question 48

Expand the given expression.
(6x - 5)(6x³ + 7x² + 11x).

Question 49

Simplify the given expression as much as possible.
$\frac{w-2}{w^{5}}-\frac{2}{w(w-1)}$

Question 50

Simplify the given expression as much as possible.
$\frac{\frac{x-3}{y+4}}{\frac{y-4}{x+3}}$

Question 51

The sum of any irrational number with any rational number is always irrational.

Question 52

The division of an irrational number and a non-zero rational number is irrational.

Question 53

The square of an irrational number is always rational.

Question 54

Which of the following alternatives corresponds to an irrational number?

A) $\sqrt{10,000}$
B) $\sqrt{361}$
C) $\sqrt{19.36}$
D) $\sqrt{10}$

Question 55

Which of the following expressions always represents an irrational number?

A) $\sqrt{a}$ , where a is an odd number
B) $\sqrt{p}$ , where p is a prime number
C) b², where b is an irrational number
D) c + d, where c and d are irrational numbers

Question 56

Given the irrational number $A=4+\sqrt{5}$ , find a number B such that A + B = 6.

Question 57

Given the irrational number $A=5+\sqrt{5}$ , find a number B such that A + B = $10+2 \sqrt{5}$

Question 58

The number $\frac{3 \sqrt{7}}{4}$ is an irrational number.

Question 59

Fill in the blank. $(\ldots+\sqrt{5})+(1+3 \sqrt{5})=4 \sqrt{5}$

Question 60

Fill in the blanks.
$(4.7+\sqrt{7})+(\ldots+ \ldots 39 \sqrt{7} ) = 18 .3 + 39.1 \sqrt{7}$

Question 61

Let $A=\sqrt{19}+\sqrt{43}$ and $B=5+\sqrt{43}$ . Find a number C such that A + B + C equals 10.

Question 62

Let $A=\sqrt{11}+\sqrt{29}$ and $B=5+\sqrt{11}$ . Find a number C such that A + B + C equals
$2+10 \sqrt{29}$

Question 63

Let $A=6+\sqrt{3}$ . Find a number B such that A + B is irrational.

A) 4
B) $6-2 \sqrt{3}$
C) $6+\sqrt{29}$
D) All of them

Question 64

Let $A=1.78+\sqrt{7}$ . Find a number B such that A + B is rational.

A) $1.78+\sqrt{7}$
B) $1.78-\sqrt{7}$
C) $\frac{1.78}{\sqrt{7}}$
D) $\sqrt{7}$

Question 65

Find an irrational number A such that $\frac{\sqrt{7}}{A}$ s a positive rational number.

A) $\frac{\sqrt{7}}{A}$
B) $\sqrt{21}$
C) $-\sqrt{7}$
D) None of them

Question 66

Find the smallest positive number b such that
$(10+b \sqrt{2})^{2}$ is rational.

Question 67

For which values of a does the expression $\sqrt{13 a}$ Correspond to a rational number?

A) 0, 1
B) -13, 13
C) 0, 13
D) 1, 13

Question 68

Let $A=5+\sqrt{23}$ . Find an irrational number B such that the product AB is rational

A) $\sqrt{23}$
B) 5
C) $5-\sqrt{23}$
D) $6+\sqrt{23}$

Question 69

Let $A=2.6+\sqrt{7}$ . Find an irrational number B such that the product AB is rational

A) $2.6-\sqrt{7}$
B) $2.6-2 \sqrt{7}$
C) $2.6+2 \sqrt{7}$
D) $\sqrt{7}$