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1. College Algebra > 2. Equations and Inequalities > Flashcards

2. Equations and Inequalities Flashcards

1
Q

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

A

x-axis

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2
Q

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

A

y-axis

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3
Q

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

A

origin

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4
Q

The axes of the rectangular coordinate system divide

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

A

quadrants

four

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5
Q

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 _______.

A

x-coordinate

y-coordinate

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6
Q

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.

A

solution

satisfies

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7
Q

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 ______.

A

x-intercept

zero

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8
Q

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 ______.

A

y-intercept

zero

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9
Q

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

A

linear

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10
Q

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

A

equivalent

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11
Q

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

A

apply the distributive property

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12
Q

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 ______.

A

least common denominator

12

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13
Q

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

A

0

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14
Q

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

A

2x

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15
Q

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

A

identity

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16
Q

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

A

inconsistent

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17
Q

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 ______.

A

x + 658.6

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18
Q

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 _______.

A

31 + 2.4x

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19
Q

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

A

4 + 0.15x

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20
Q

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

A

x - 0.15x

or

0.85x

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21
Q

The combined yearly interest for x dollars invested

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

A

0.12x + 0.09(30,000 - x)

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22
Q

Solving a formula for a variable means rewriting the

formula so that the variable is ______.

A

isolated on one side

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23
Q

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.

A

factoring

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24
Q

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

i² = _______.

A

√–1

–1

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25
Q

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.

A

complex

imaginary

real

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26
Q

–9i + 3i = ______.

A

–6i

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27
Q

10i – (–4i) = _______

A

14i

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28
Q

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 _______.

A

18

–15i

12i

–10i²

10

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29
Q

The conjugate of 2 – 9i is ______.

A

2 + 9i

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30
Q

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

A

2 + 5i

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31
Q

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

A

i

i

2i√5

32
Q

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

A

quadratic

33
Q

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

A

A = 0 or B + 0

34
Q

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

A

x-intercepts

35
Q

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

A

± √d

36
Q

if x² = 7, then x = _______.

A

± √7

37
Q

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

A

9/4

38
Q

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

A

4/25

39
Q

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

A

9

40
Q

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

A

1/9

41
Q

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

A

[–b ± (√b² – 4ac)] / (2a)

42
Q

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

A

2

9

–5

43
Q

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

A

1

–4

–1

44
Q

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

A

b² – 4ac

45
Q

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

A

no

46
Q

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

A

two

47
Q

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

A

the square root property

48
Q

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

A

the quadratic formula of the hypotenuse

49
Q

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

A

factoring and the zero-product principle

50
Q

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

A

right

hypotenus

legs

51
Q

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

A

right

legs

the square of the length of the hypotenuse

52
Q

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

A

subtract 8x and subtract 12 from both sides

53
Q

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

A

radical

54
Q

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

A

extraneous

55
Q

Consider the equation (√2x + 1) = x – 7

Squaring the left side and simplifying results in ______.

Squaring the rightside and simplifying results in _______.

A

2x + 1

x² – 14x + 49

56
Q

Consider the equation (√x + 2) = 3 – (√x – 1)

Squaring the left side and simplifying results in ______.

Squaring the rightside and simplifying results in _______.

A

x + 2

8 – 6(√x – 1)

57
Q

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

A

5^(4/3)

58
Q

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

A

± 5^(3/2)

59
Q

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

A

x²

u² – 13u + 36 = 0

60
Q

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

A

x^(1/3)

u² +2u – 3 = 0

61
Q

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

A

c

–c

62
Q

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

A

3x – 1 = 7

3x – 1 = –7

63
Q

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

A

2

5

2

5

64
Q

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

A

greater than

65
Q

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

A

less than or equal to

66
Q

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

A

(–∞, 9)

intersection

67
Q

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

A

(–∞, 12)

union

68
Q

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 ______.

A

adding 4

dividing

–3

direction

>

69
Q

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

A

∅

70
Q

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

A

(–∞, ∞)

71
Q

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

A

middle

72
Q

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

A

–c

c

73
Q

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

A

–c

c

74
Q

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

A

–2 < x – 7 < 2

75
Q

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

A

x – 7 < –2
or
x – 7 > 2

1. College Algebra (8 decks)

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