2. Equations and Inequalities Flashcards
In the rectangular coordinate system, the horizontal number line is called the ______.
x-axis
In the rectangular coordinate system, the vertical number line is called the ______.
y-axis
In the rectangular coordinate system, the point of intersection of the horizontal axis and the vertical axis is called the ______.
origin
The axes of the rectangular coordinate system divide
the plane into regions, called ______. There are ______ of these regions.
quadrants
four
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 _______.
x-coordinate
y-coordinate
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.
solution
satisfies
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 ______.
x-intercept
zero
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 ______.
y-intercept
zero
An equation in the form ax + b = 0, a ≠ 0, such as 3x + 17 = 0, is called a/an ______ equation in one variable.
linear
Two or more equations that have the same solution set are called ______ equations.
equivalent
The first step in solving the 7x + 3(x - 2) = 2 x + 10 is to ______.
apply the distributive property
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 ______.
least common denominator
12
We reject any proposed solution of a rational equation that causes a denominator to equal ______.
0
The first step in solving 4/x + 1/2 = 5/x is to multiply both sides by _______.
2x
An equation that is true for all real numbers for which both sides are defined is called a/an _______.
identity
An equation that is not true for even one real number is called a/an _______ equation.
inconsistent
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 ______.
x + 658.6
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 _______.
31 + 2.4x
A text message plan costs $4 per month plus $0.15
per text. The monthly cost for x text messages can be
represented by ______.
4 + 0.15x
purchased a computer after a 15% price reduction. If
x represents the computer’s original price, the reduced
price can be represented by ______.
x - 0.15x
or
0.85x
The combined yearly interest for x dollars invested
at 12% and 30,000 – x dollars invested at 9% is _______.
0.12x + 0.09(30,000 - x)
Solving a formula for a variable means rewriting the
formula so that the variable is ______.
isolated on one side
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.
factoring
The imaginary unit i is defined as i = _______, where
i² = _______.
√–1
–1
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.
complex
imaginary
real
–9i + 3i = ______.
–6i
10i – (–4i) = _______
14i
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 _______.
18
–15i
12i
–10i²
10
The conjugate of 2 – 9i is ______.
2 + 9i
The division (7 + 4i) / (2 – 5i) is performed by multiplying the numerator and denominator by _______.
2 + 5i
(√–20) = _____√20 = _____(√4∙5) = _______
i
i
2i√5
An equation can be written in the general form ax² + bx + c = 0, a ≠ 0, is called a/an _______ equation.
quadratic
The zero-product principle states that if AB = 0, then _______.
A = 0 or B + 0
The solution of ax² + bx + c = 0 correspond to the _______ for the graph of y = ax² + bx + c.
x-intercepts
The square root property states that if u² = d, then u = _______.
± √d
if x² = 7, then x = _______.
± √7
To complete the square on x² – 3x, add _______.
9/4
To complete the square on x² – (4/5)x, add ______.
4/25
To solve x² + 6x = 7 by completing the square, add _______ to both sides of the equation.
9
To solve x² – (2/3)x = 4/9 by completing the square, add _______ to both sides of the equation.
1/9
The solutions of a quadratic equation in the general form ax² + bx + c = 0, a ≠ 0, are given by the quadratic formula x = _______.
[–b ± (√b² – 4ac)] / (2a)
In order to solve 2x² + 9x – 5 = 0 by the quadratic formula, we use a = ______, b = ______, and c = ______.
2
9
–5
In order to solve x² - 4x + 1 = 0 by the quadratic formula, we use a = ______, b = ______, and c = ______.
1
–4
–1
The discriminant of ax² + bx + c = 0 is defined by ______.
b² – 4ac
If the discriminant of ax² + bx + c = 0 is negative, the quadratic equation has ______ real solutions.
no
If the discriminant of ax² + bx + c = 0 is positive, the quadratic equation has ______ real solutions.
two
The most efficient technique for solving (2x + 7)² = 25 is by using ______.
the square root property
The most efficient technique for solving x² + 5x – 10 = 0 is by using ________.
the quadratic formula of the hypotenuse
The most efficient technique for solving x² + 8x + 15 = 0 is by using _______.
factoring and the zero-product principle
A triangle with one angle measuring 90° angle is called the ______. The other sides are called ______.
right
hypotenus
legs
The Pythagorean Theorem states that in any _______ triangle, the sum of the squares of the lengths of the ______ equals ________.
right
legs
the square of the length of the hypotenuse
The first step in solving the polynomial equation 2x^3 + 3x² = 8x + 12 is to _______.
subtract 8x and subtract 12 from both sides
An equation in which the variable occurs in a square root, cube root, or any higher root is called a/an _______ equation.
radical
Solutions of a squared equation that are not solutions of the original equation are called _______ solutions.
extraneous
Consider the equation (√2x + 1) = x – 7
Squaring the left side and simplifying results in ______.
Squaring the rightside and simplifying results in _______.
2x + 1
x² – 14x + 49
Consider the equation (√x + 2) = 3 – (√x – 1)
Squaring the left side and simplifying results in ______.
Squaring the rightside and simplifying results in _______.
x + 2
8 – 6(√x – 1)
if x^(3/4) = 5, then x = ______.
5^(4/3)
if x^(2/3) = 5, then x = ______.
± 5^(3/2)
We solve x^4 – 13x² + 36 = 0 by letting u = _______. We then rewrite the equation in terms of u as _______.
x²
u² – 13u + 36 = 0
We solve x^(2/3) + 2x^(1/3) – 3 = 0 by letting u = _______. We then rewrite the equation in terms of u as _______.
x^(1/3)
u² +2u – 3 = 0
if c > 0, |u| = c is equivalent to u = _______ or u = _______.
c
–c
|3x – 1| = 7 is equivalent to _______ or _______.
3x – 1 = 7
3x – 1 = –7
In interval notation, [2, 5) represents the set of real numbers between _______ and _______, including _______ but not including _______.
2
5
2
5
In interval notation, (–2, ∞) represents the set of real numbers _______ –2.
greater than
In interval notation, (–∞, –1] represents the set of real numbers _______ –1.
less than or equal to
The set of elements common to both (–∞, 9) or (–∞, 12) is _______. This represents the _______ of these intervals.
(–∞, 9)
intersection
The set of elements in (–∞, 9) or (–∞, 12) or in both sets is _______. This represents the ______ of these intervals.
(–∞, 12)
union
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 ______.
adding 4
dividing
–3
direction
>
In solving an inequality, if you eliminate the variable and obtain a false statement such as 7 < –2, the solution set is ______.
∅
In solving an inequality, if you eliminate the variable and obtain a true statement such as 8 > 3, the solution set is ______.
(–∞, ∞)
The way to solve –7 < 3x – 4 ≤ 5 is to isolate x in the ______.
middle
if c > 0, |u| < c is equivalent to _____ < u < ______.
–c
c
If c > 0, |u| > c is equivalent to u < ______ or u > ______.
–c
c
|x – 7| < 2 can be rewritten without absolute value bars as ______.
–2 < x – 7 < 2
|x – 7| > 2 can be rewritten without absolute value bars as ______.
x – 7 < –2
or
x – 7 > 2
