2. Equations and Inequalities

Question 1

In the rectangular coordinate system, the horizontal number line is called the ______.

Answer 1

x-axis

Question 2

In the rectangular coordinate system, the vertical number line is called the ______.

Answer 2

y-axis

Question 3

In the rectangular coordinate system, the point of intersection of the horizontal axis and the vertical axis is called the ______.

Answer 3

origin

Question 4

The axes of the rectangular coordinate system divide

the plane into regions, called ______. There are ______ of these regions.

Answer 4

quadrants

four

Question 5

The first number in an ordered pair such as (8, 3) is called the ______. The second number in such an ordered pair is called the _______.

Answer 5

x-coordinate

y-coordinate

Question 6

The ordered pair (4, 19) is a/an ______ of the
equation y = x² + 3 because when 4 is substituted
for x and 19 is substituted for y, we obtain a true statement. We also say that (4, 19) ______ the
equation.

Answer 6

solution

satisfies

Question 7

The x-coordinate of a point where a graph crosses the
x-axis is called a/an ______. The y-coordinate of such
a point is always ______.

Answer 7

x-intercept

zero

Question 8

The y-coordinate of a point where a graph crosses the
y-axis is called a/an ______. The x-coordinate of such
a point is always ______.

Answer 8

y-intercept

zero

Question 9

An equation in the form ax + b = 0, a ≠ 0, such as 3x + 17 = 0, is called a/an ______ equation in one variable.

Answer 9

linear

Question 10

Two or more equations that have the same solution set are called ______ equations.

Answer 10

equivalent

Question 11

The first step in solving the 7x + 3(x - 2) = 2 x + 10 is to ______.

Answer 11

apply the distributive property

Question 12

The fractions in the equation x/4 = 2 + (x - 3)/3 can be eliminated by multiplying both sides by the ______ of x/4 and (x-3)/3, which is ______.

Answer 12

least common denominator

12

Question 13

We reject any proposed solution of a rational equation that causes a denominator to equal ______.

Answer 13

0

Question 14

The first step in solving 4/x + 1/2 = 5/x is to multiply both sides by _______.

Answer 14

2x

Question 15

An equation that is true for all real numbers for which both sides are defined is called a/an _______.

Answer 15

identity

Question 16

An equation that is not true for even one real number is called a/an _______ equation.

Answer 16

inconsistent

Question 17

According to the U.S. Office of Management and Budget, the 2011 budget for defense exceeded the budget for education by $658.6 billion. If x represents the budget for education, in billions of dollars, the budget for defense can be represented by ______.

Answer 17

x + 658.6

Question 18

In 2000, 31% of U.S. adults viewed a college education as essential for success. For the period from 2000 through 2010, this percentage increased by approximately 2.4 each year. The percentage of U.S. adults who viewed a college education as essential for
success x years after 2000 can be represented by _______.

Answer 18

31 + 2.4x

Question 19

A text message plan costs $4 per month plus $0.15
per text. The monthly cost for x text messages can be
represented by ______.

Answer 19

4 + 0.15x

Question 20

purchased a computer after a 15% price reduction. If
x represents the computer’s original price, the reduced
price can be represented by ______.

Answer 20

x - 0.15x

or

0.85x

Question 21

The combined yearly interest for x dollars invested

at 12% and 30,000 – x dollars invested at 9% is _______.

Answer 21

0.12x + 0.09(30,000 - x)

Question 22

Solving a formula for a variable means rewriting the

formula so that the variable is ______.

Answer 22

isolated on one side

Question 23

The first step in solving IR + Ir = E for I is to obtain

a single occurrence of I by ______ I from the two terms on the left.

Answer 23

factoring

Question 24

The imaginary unit i is defined as i = _______, where

i² = _______.

Answer 24

√–1

–1

Question 25

The set of all numbers in the form a + bi is called the set of ______ numbers. If b ≠ 0, then the number is also called a/an ______ number. If b = 0, then the number is also called a/an ______ number.

Answer 25

complex imaginary real

Question 26

–9i + 3i = ______.

Answer 26

–6i

Question 27

10i – (–4i) = _______

Answer 27

14i

Question 28

Consider the following multiplication problem: (3 + 2i)(6 – 5i) Using the FOIL method, the product of the first terms is _______, the product of the outside terms is _______. The product of the last terms in terms of i² is _______, which simplifies to _______.

Answer 28

18 –15i 12i –10i² 10

Question 29

The conjugate of 2 – 9i is ______.

Answer 29

2 + 9i

Question 30

The division (7 + 4i) / (2 – 5i) is performed by multiplying the numerator and denominator by _______.

Answer 30

2 + 5i

Question 31

(√–20) = _____√20 = _____(√4∙5) = _______

Answer 31

i i 2i√5

Question 32

An equation can be written in the general form ax² + bx + c = 0, a ≠ 0, is called a/an _______ equation.

Answer 32

quadratic

Question 33

The zero-product principle states that if AB = 0, then _______.

Answer 33

A = 0 or B + 0

Question 34

The solution of ax² + bx + c = 0 correspond to the _______ for the graph of y = ax² + bx + c.

Answer 34

x-intercepts

Question 35

The square root property states that if u² = d, then u = _______.

Answer 35

± √d

Question 36

if x² = 7, then x = _______.

Answer 36

± √7

Question 37

To complete the square on x² – 3x, add _______.

Answer 37

9/4

Question 38

To complete the square on x² – (4/5)x, add ______.

Answer 38

4/25

Question 39

To solve x² + 6x = 7 by completing the square, add _______ to both sides of the equation.

Answer 39

9

Question 40

To solve x² – (2/3)x = 4/9 by completing the square, add _______ to both sides of the equation.

Answer 40

1/9

Question 41

The solutions of a quadratic equation in the general form ax² + bx + c = 0, a ≠ 0, are given by the quadratic formula x = _______.

Answer 41

[–b ± (√b² – 4ac)] / (2a)

Question 42

In order to solve 2x² + 9x – 5 = 0 by the quadratic formula, we use a = ______, b = ______, and c = ______.

Answer 42

2 9 –5

Question 43

In order to solve x² - 4x + 1 = 0 by the quadratic formula, we use a = ______, b = ______, and c = ______.

Answer 43

1 –4 –1

Question 44

The discriminant of ax² + bx + c = 0 is defined by ______.

Answer 44

b² – 4ac

Question 45

If the discriminant of ax² + bx + c = 0 is negative, the quadratic equation has ______ real solutions.

Answer 45

no

Question 46

If the discriminant of ax² + bx + c = 0 is positive, the quadratic equation has ______ real solutions.

Answer 46

two

Question 47

The most efficient technique for solving (2x + 7)² = 25 is by using ______.

Answer 47

the square root property

Question 48

The most efficient technique for solving x² + 5x – 10 = 0 is by using ________.

Answer 48

the quadratic formula of the hypotenuse

Question 49

The most efficient technique for solving x² + 8x + 15 = 0 is by using _______.

Answer 49

factoring and the zero-product principle

Question 50

A triangle with one angle measuring 90° angle is called the ______. The other sides are called ______.

Answer 50

right hypotenus legs

Question 51

The Pythagorean Theorem states that in any _______ triangle, the sum of the squares of the lengths of the ______ equals ________.

Answer 51

right legs the square of the length of the hypotenuse

Question 52

The first step in solving the polynomial equation 2x^3 + 3x² = 8x + 12 is to _______.

Answer 52

subtract 8x and subtract 12 from both sides

Question 53

An equation in which the variable occurs in a square root, cube root, or any higher root is called a/an _______ equation.

Answer 53

radical

Question 54

Solutions of a squared equation that are not solutions of the original equation are called _______ solutions.

Answer 54

extraneous

Question 55

Consider the equation (√2x + 1) = x – 7 Squaring the left side and simplifying results in ______. Squaring the rightside and simplifying results in _______.

Answer 55

2x + 1 x² – 14x + 49

Question 56

Consider the equation (√x + 2) = 3 – (√x – 1) Squaring the left side and simplifying results in ______. Squaring the rightside and simplifying results in _______.

Answer 56

x + 2 8 – 6(√x – 1)

Question 57

if x^(3/4) = 5, then x = ______.

Answer 57

5^(4/3)

Question 58

if x^(2/3) = 5, then x = ______.

Answer 58

± 5^(3/2)

Question 59

We solve x^4 – 13x² + 36 = 0 by letting u = _______. We then rewrite the equation in terms of u as _______.

Answer 59

x² u² – 13u + 36 = 0

Question 60

We solve x^(2/3) + 2x^(1/3) – 3 = 0 by letting u = _______. We then rewrite the equation in terms of u as _______.

Answer 60

x^(1/3) u² +2u – 3 = 0

Question 61

if c > 0, |u| = c is equivalent to u = _______ or u = _______.

Answer 61

c –c

Question 62

|3x – 1| = 7 is equivalent to _______ or _______.

Answer 62

3x – 1 = 7 3x – 1 = –7

Question 63

In interval notation, [2, 5) represents the set of real numbers between _______ and _______, including _______ but not including _______.

Answer 63

2 5 2 5

Question 64

In interval notation, (–2, ∞) represents the set of real numbers _______ –2.

Answer 64

greater than

Question 65

In interval notation, (–∞, –1] represents the set of real numbers _______ –1.

Answer 65

less than or equal to

Question 66

The set of elements common to both (–∞, 9) or (–∞, 12) is _______. This represents the _______ of these intervals.

Answer 66

(–∞, 9) intersection

Question 67

The set of elements in (–∞, 9) or (–∞, 12) or in both sets is _______. This represents the ______ of these intervals.

Answer 67

(–∞, 12) union

Question 68

The linear inequality –3x –4 > 5 can be solved by first ______ to both sides and then _______ both sides by _______, which changes the _______ of the inequality symbol from _____ to ______.

Answer 68

adding 4 dividing –3 direction >

Question 69

In solving an inequality, if you eliminate the variable and obtain a false statement such as 7 < –2, the solution set is ______.

Answer 69

∅

Question 70

In solving an inequality, if you eliminate the variable and obtain a true statement such as 8 > 3, the solution set is ______.

Answer 70

(–∞, ∞)

Question 71

The way to solve –7 < 3x – 4 ≤ 5 is to isolate x in the ______.

Answer 71

middle

Question 72

if c > 0, |u| < c is equivalent to _____ < u < ______.

Answer 72

–c c

Question 73

If c > 0, |u| > c is equivalent to u < ______ or u > ______.

Answer 73

–c c

Question 74

|x – 7| < 2 can be rewritten without absolute value bars as ______.

Answer 74

–2 < x – 7 < 2

Question 75

|x – 7| > 2 can be rewritten without absolute value bars as ______.

Answer 75

x – 7 < –2 or x – 7 > 2