Download Inequalities Handout

 An inequality is a statement that contains one or more of the following symbols.
 < is less than
 ≤ is less than or equal to
 > is greater than
 ≥ is greater than or equal to
 ≠ is not equal to
 An inequality can be true, false, or neither true nor false.
 An inequality that contains a variable can be made true or false depending on the
number that is substituted for the variable.
 A number that makes an inequality true is called a solution of the inequality, and
we say that the number satisfies the inequality.
 A linear inequality in one variable can be written in one of the following forms where
a, b, and c are real numbers and a ≠ 0.
 ax + b > c
 ax + b ≥ c
 ax + b < c
 ax + b ≤ c
 Checking if a number is a solution for an inequality:
Is 9 a solution of 2x + 4 ≤ 21?
 Is 2(9) + 4 ≤ 21 true?
 Is 18 + 4 ≤ 21 true?
 Is 22 ≤ 21 true?
 No, 22 21, therefore 9 is NOT a solution
 A solution set of an inequality is the set of all numbers that make the inequality true. You
can write solution sets in 2 ways:

Graphing Inequalities:

 Two methods for graphing < or >: you can use an open circle or a parenthesis
that opens in the direction where the inequality is true.
 Two methods for graphing ≤ or ≥ : you can use a closed circle or a bracket
that opens in the direction where the inequality is true.
 If students get used to graphing with parentheses or brackets, it is a bit easier
to teach them interval notation.
Interval notation
 Interval notation is always written from left to right as it appears on a number
line. If the interval you are graphing extends indefinitely to the left, that is
designated −∞ . If the interval you are graphing extends indefinitely to the
right, that is designated ∞.
 In interval notation, −∞ and ∞ always have round brackets since, infinity is
approached but never reached.
 In interval notation the smaller number appears first separated by a comma
from the larger number.
Example:
 Solving linear inequalities:

Addition and Subtraction Properties of Inequality (this one is parallel to the
property for equations):
 If a, b, c are real numbers, and a < b, then
o a + c < b + c and
o a−c<b−c.
Examples: Solve and write answers in interval notation:
(1) x – 5 <7
(2) 3 + x > 4
(3) 5 ≥ 2 + x
(4) -3 < x + 2 ≤ 5

Multiplication and Division Properties of Inequality:
 If a, b, c are real numbers, where c is positive (greater than 0), if a < b, then
o a·c < b·c
o a/c < b/c
 If a, b, c are real numbers, where c is negative (greater than 0), if a < b, then
o a·c > b·c.
o a/c > b/c
Examples: (solve and write answers in interval notation)
1) 5x < 25
2) 8 >-2x
4) -x ≤ 5
5) -3 ≤ 3x < 12
6) 2x – 1 > 8
7) 4 – x ≤ 8
Problem:
 A student has scores of 72%, 74%, and 78% on three exams. What percent score does he
need on the last exam to earn a grade of no less than B (80%)?
Additional exercises to wrap your mind around it.
Complete this table:
Chapter 3
3.1 Graphing Paired Data:
Vocabulary:
• rectangular coordinate graph, rectangular coordinate system Cartesian coordinate system
• x-axis, y-axis , origin, coordinate plane, quadrants, ordered pair, x-coordinate, y-coordinate
graphing, plotting
• interpolation, extrapolation
Exercise:
Example 1
Example 2: Gas mileage
In 2007, the most fuel-efficient mid-sized car for city driving was the Toyota Prius, which
was rated at 60 miles per gallon. The least efficient mid-sized car for city driving was
the Ferrari 612 Scaglietti, rated at 10 miles per gallon. (www.fueleconomy.gov)
Prius
Gallons (x)
Mileage (y)
Ferrari
Gallons (x)
Mileage (y)
3.2 Graphing linear equations:
Vocabulary
• Linear equation in one variable, linear equation in two variables (when power is 1 and if
variable is not multiplied or divided by another variable)
• Solution of an equation in two variables, table of solutions (or table of values)
• Graph of the equation, infinitely many solutions
• Standard form (general form)
Exercises:
1) Creating a table of values using a linear equation:
2) Solving for y
3) Application Example:
A group is going to hold a scrapbooking party as a fundraiser for a trip. They estimate that it will
cost them $100 to create posters and flyers to advertise the event, and $300 to rent a large room
for the event. They estimate that they will spend $10 on food for each participant. The organizer
writes an equation for their costs: C = 400 +10n and she can spend a total of $1,000 on the event.
a) Assuming that the room is large enough, how many participants could be invited?
b) If each participant pays $25 for the event, will they make any money? If so, how much?
4)
5) Fill in the missing values in the table of solutions for the linear equation, y = -x + 4, plot
the points, and then draw the line through the points
x
y
3
3
0
-2
5
6) A student has created a table of solutions for each linear equation below:
 Plot the points that the student has found.
 Use the graphed points to identify which points (if any) are most likely incorrect.
 Circle the incorrect points in the table of values.
 Find the correct ordered pairs for any that are circled.
 Plot the corrected ordered pairs.
(A)
(B)