Download Algebra 1 Chapter 2 Note-Taking Guide Name Inequalities Per ____

Algebra 1 Chapter 2 Note-Taking Guide
Inequalities
2.1
Name ________________________
Per ____ Date __________________
An inequality is a statement that two quantities are not equal. The quantities are compared by using
the following signs:
> greater than
≥ greater than or equal to
< less than
≤ less than or equal to
A solution of an inequality is any value of the variable that makes the inequality true.
Is x = 3 a solution to x - 6 >4?
Is x = 10?
Is x = 12?
Is x = 13?
2.1
Since there are many solutions to an inequality, we can graph all possible solutions on a number line.
All inequalities can be written in three forms:
algebraically,
in words, and
graphically.
2.1
Write the equation for each graph:
2.1
2.1
Graph each inequality:
A.) x<-5
B.) x ≤ -20
C.) 5 > x
D.) 9 ≤ x
Reading word problems:
“No more than” means “less than or equal to.”
“At least” means “greater than or equal to”.
Ray’s dad told him not to turn on the air conditioner unless the temperature is at least 85°F. Define a
variable and write an inequality for the temperatures at which Ray can turn on the air conditioner.
Graph the solutions.
A cell phone plan offers free minutes for no more than 250 minutes per month. Define a variable and
write an inequality for the possible number of free minutes. Graph the solution.
2.2/2.3 Is each statement true?
1>0
-2 < 5
Multiply each statement by -2. Is it still true?
What would we have to do to the inequality symbol to make it true?
5 > -1
2.2/2.3 Rule: When solving inequalities, the only thing different from equations is if you multiply or divide by
a negative number the inequality symbol ___________________.
Solve and graph each inequality:
A.) x  2  6
D.)
3x  12
G.)
2
x8
3
B.)
x68
C.)
2  x  5
E.)
4 x  24
F.)
x
 7
2
F.)
3
8
x
4
3
2.2/2.3 What happens if the variable is on the wrong side?
A.)
2.4A
8 x7
Solving 2-step inequalities
A) 45 + 2b > 61
D) –4(2 – x) ≤ 8
B.)
3
x
x
4
8
B) 8 – 3y ≥ 29
E.)
3
x57
4
C) –12 ≥ 3x + 6
F.)
2
1 1
x 
3
2 3
2.4B
2.4B
A) 6b + 6 - 11b ≤ -4
B) 2x + 8 - 6x - 11 > 5
C) 3 + 2(x + 4) > 3
D) 5x - 2(x + 3) ≤ -12
E) 3 (x + 2) - 6x + 6 ≤ 0
F) -18 > -(2x + 9) - 4 + x
To rent a certain vehicle, Rent-A-Ride charges $55.00 per day with unlimited miles. The cost of
renting a similar vehicle at We Got Wheels is $38.00 per day plus $0.20 per mile. For what number of
miles is the cost at Rent-A-Ride less than the cost at We Got Wheels?
A video store has two movie rental plans. Plan A includes a $25 membership fee plus $1.25 for each
movie rental. Plan B costs $40 for unlimited movie rentals. For what number of movie rentals is plan B
less than plan A?
2.5
A) 5y + 1 < -2y - 6
B) 4m – 3 < 2m + 6
C) x ≥ 3x + 8
D) 4x ≤ 7x + 6
E) 5(2 - y) ≥ 3(r - 2)
F) -4(3 - p) > 5(p + 1)
2.5
Special Cases:
G) x + 5 ≤ x + 3
2.5
The Home Cleaning Company charges $312 to power-wash the siding of a house plus $12 for each
window. Power Clean charges $36 per window, and the price includes power-washing the siding. How
many windows must a house have to make the total cost from The Home Cleaning Company less
expensive than Power Clean?
H) 3x - 2 - x > 2x – 5
I) 2(k – 3) > 6 + 3k – 3
2.6A
Compound Inequalities
When two simple inequalities are combined into one statement by the words AND or OR, the result is
called a compound inequality.
2.6A
You can graph the solutions of a compound inequality involving AND by using the idea of an
overlapping region. The overlapping region is called the intersection and shows the numbers that are
solutions of both inequalities.
“AND”
A)
x + 1 > -5 AND x + 1 < 2
B)
3x - 1 > 8 AND 3x - 1 ≤ 11
C)
-9 < x - 10 < -5
D)
–4 ≤ 3n + 5 < 11
2.6A
OR
You can graph the solutions of a compound inequality involving OR by using the idea of combining
regions. The combined regions are called the union and show the numbers that are solutions of
either inequality.
A)
2.6A
2.6B
8 + t ≥ 7 OR 8 + t < 2
B.) 4x +1 ≤ 21 OR 3x - 2 > 19
The target heart rate during exercise for a 15 year-old is between 154 and 174 beats per minute
inclusive. Write a compound inequality to show the heart rates that are within the target range. Graph
the solutions.
2.6B
2.6B
A)
11 < 2x + 3 < 21
B) 5 ≤ 4x + 1 ≤ 13
C.)
3x + 2 < -1 OR 4x - 1 > 15
D) 2x + 3 ≤ 7 OR 3x + 5 > 26
Special Case: Graph the solution for x < 3 AND x > 7.
Every solution of a compound inequality involving AND must be a solution of both parts of the
compound inequality.
If no numbers are solutions of both simple inequalities, then the compound inequality has
2.6B
2.6B
_________________________.
Special Case: The solutions of a compound inequality involving OR are not always two separate sets
of numbers. There may be numbers that are solutions of both parts of the compound inequality.
Graph x < 5 OR x > 3
Special Case: Graph x > 5 OR x > 3
2.6B
Write each inequality
2.7A
When an inequality contains an absolute-value expression, it can be written as a compound
inequality.
The inequality |x| < 5 describes all real numbers whose distance from 0 is less than 5 units.
The solutions are all numbers between –5 and 5,
so |x|< 5 can be rewritten as –5 < x < 5, or as x > –5 AND x < 5.
All absolute value inequalities that have a < or ≤ symbol can be rewritten as an "AND"
problem.
A.) |x + 4| ≤ 2
B.) |2x - 1| < 5
2.7A
The inequality |x| > 5 describes all real numbers whose distance from 0 is greater than 5 units.
The solutions are all numbers less than –5 or greater than 5. The inequality |x| > 5 can be rewritten as
the compound inequality x < –5 OR x > 5.
All absolute value inequalities that have a > or ≥ symbol can be rewritten as an "OR" problem.
A.) |x - 8| > 2
B.) |2x + 3| ≥ 7
2.7A
2.7A
Just as you do when solving absolute-value equations, you first isolate the absolute-value
expression when solving absolute-value inequalities.
A.) 4|x| + 12 < 48
B.) 3|x - 2| - 9 ≤ 6
C.) 7 |x| - 5 ≥ 16
D.) 6 |2x - 3| + 7 > 19
Special Case: Remember absolute value represents a distance, and distance cannot be less than 0.
When you get an absolute value expression

Less than, less than or equal to a negative number, there is ______________________.

greater than, greater than or equal to a negative number, the solution is _________________.
A.) |x| – 9 ≥ –11
2.7B
B.) -4|x – 3| ≥ 8
A pediatrician recommends that a baby’s bath water be 95°F, but it is acceptable for the temperature
to vary from this amount by as much as 3°F. Write an absolute-value inequality to find the range of
acceptable temperatures.
A dry-chemical fire extinguisher should be pressurized to 125 psi, but it is acceptable for the pressure
to differ from this value by at most 75 psi. Write an absolute-value inequality to find the range of
acceptable pressures.
A number, n, is no more than 7 units away from 5. Write and solve an inequality to show the range of
possible values for n.
2.7B
A.)
3|x - 2| > 15
B.) |x + 3| + 1 < 3
B.)
|3x| + 4 < 1
D.) |x + 2| – 3 ≥ – 8