Solving and Graphing Inequalities
Chapter 6
Rules for graphing your answers
 If the letter is on the left then we can follow the direction of
the arrow
 We must mark the numbers
 < or > we use an open circle
 < or > we use a closed circle
Our answers will look like this and we
will graph the answers on a number
line
X<2
2. X > -2
3. Z > 1
4. 0 < d
1.
Now you practice:
T>1
2. X >-1
3. N < 0
4. 4 >y
1.
Our problems will look like this:
1.
X+5>3
We will use our Chapter 3 rules to solve:
1. Draw tracks
2. Count variables
1. If one, numbers jump tracks
2. If 2 or more where are they
1. Same side, family members
2. Different sides, letters jump
3. Add or subtract the numbers
4. Is there a number on the letters?
1. If its an integer divide each side
2. If it’s a fraction, flip and multiply
3. if the integer or fraction is negative
I must turn the arrow around to
the opposite direction
5. Graph the solution
More examples:
1.
2.
3.
4.
5.
6.
7.
X+4<7
N+6>2
5>a+5
-2 > n – 4
X–5>2
P – 1 < -4
-3 < y - 2
Pg 327 #’s 42 – 54 even
Sec 6.2
Solving Equations using Multi or Div
 We must be able to identify if there is an integer on the letter
or a fraction on the letter.
 If the integer or the fraction is negative then we MUST
change the direction of the arrow in our answer
Examples
1.
2.
3.
4.
5.
6.
a/4 < 4
4x > 20
k/4 < ½
18 < 2k
6 < t/5
-21 < 3y
Examples: Negative numbers
1.
2.
3.
4.
5.
6.
-1/2 y < 5
-12m > 18
-8x < 20
-1/5 p > 1
-2/3 x < -5
-24 < 6t
Word Problems
Kayla wants to buy some posters for her dorm room.
Posters are on sale for $6 each. Write and solve an
inequality to determine how many posters she can buy and
spend no more than $25.
2. Crandell plans to take figure skating lessons. He can rent
skates for $5 per lesson. He can buy skates for $75. For
what number of lessons is it cheaper for him to buy rather
than rent skates?
1.
Pg 334 #’s 36 – 46 even
Quiz
Sections 6.1 and 6.2
1. X – 4 < -5
2. -11 > y + 4
3. -1/2 x > -5
4. -3x < -27
5. -3/4 x < -1/4
Sec 6.3; Solving Multi-step Inequalities
1. 2y – 5 < 7
2. 5 – x > 4
3. 3(x + 2) < 7
4. -2(x + 1) < 2
5. 2x – 3 > 4x – 1
Sec 6.3; Solving Multi-step Inequalities
1. 5n – 21 < 8n
2. -3z + 15 > 2z
3. X + 3 > 2x – 4
4. 4y – 3 < -y + 12
Word Problems
 You plan to make and sell candles.You pay $12 for
instructions. The materials for each candle cost $0.50. You
plan to sell each candle for $2. Let x be the numbers of
candles you sell. How many candles must be sold to make at
least $300 profit.
Sec 6.4; Solving compound inequalities
 How do you combine two thoughts in English class?
 What are they called?
 What words do we use?
How to Solve:
-2 < x + 2 < 4
2. -1 < x + 3 < 7
3. -6 < -3x < 12
4. 0 < x – 4 < 12
1.
These are “AND”
problems. These
problems must be
re-written into two
problems.
Word Problems
1.
In the summer it took a Pony Express rider about 10 days
to ride from St. Joseph, Missouri to Sacramento,
California. In winter it took as many as 16 days. Write an
inequality to describe the number of days that the trip
might have taken.
Word Problems
 Frequency is used to describe the pitch of a sound.
Frequencies are measured in hertz. Write an inequality for
the following
 Sound of a human voice: 85 hertz to 1100 hertz
 Sound of a bats signal: 10,000 hertz to 120,000 hertz
 Sound heard by a dog: 15 hertz to 50,000 hertz
 Sound heard by a dolphin: 150 hertz to 150,000 hertz
Pg 346 #’s 30-46 even
Sec 6.5; Solving Compound
Inequalities
 These are called “OR” problems
 These problems are already written and ready to be solved
Examples
X-4<3 or 2x> 18
2. 3x + 1 < 4 or 2x – 5 > 7
3. X + 5 < -6 or 3x > 12
4. 6x – 5 < 7 or 8x + 1 > 25
1.
Word Problem
 A baseball is hit straight up in the air. Its initial velocity is 64
ft per second. Its formula is v = -32t + 64. Find the values
of “t” for which the velocity of the baseball is greater than 32
or less than -32 feet per second.
Sec 6.6; Absolute Value Equations
 These problems will create some re-writing
 1st – make sure the absolute value bars are on a side by
themselves
 2nd – drop the bars and write two problems
 3rd – in the second problem you will need the opposite of the
symbol (equals or inequality) and the opposite of the number.
 Now solve the equations
 If its an inequality then you will also graph the solutions.
For examples we will use p. 356
Pg. 358 #”s 16-26 even
pg 359 #’s 32-40 even
Sec 6.7; Solving absolute value
inequalities
 We will use the same rules as the previous section (6.6)
 Make sure to remember to make the changes to the second
problem that you re-write.
We will use pg 364 and 365 for
examples
Sec 6.8; Graphing Linear Inequalities
in Two Variables
 We use our previous knowledge from Chapter’s 4 and 5.
To graph we will use y= mx + b
 Slope intercept form: y = mx + b
 “b” is the y-intercept. I must always use this
number first when graphing. It is always
located on the y-axis, either above or below
the origin.
 “m” is the slope. Its always a fraction and
remember to use rise over run
To graph given when given an
equation; turn it into y=mx+b
 If the symbol is < or > we
will draw a dashed line
 If the symbol is < or > we
will draw a normal line
 To shade will have to use a
test point and the slope
intercept form
 If the test result is false shade
away from the TP, if the test is
true shade to the TP.
 Flow Chart
 Let x jump the tracks
 If there is a number on
y, then we will set up
three fractions and
divide or reduce
 Collect “b” and find it
on the y-axis
 Collect “m” and use
rise over run to get to
the next point on my
line.
 -2x + y < 3
MORE EXAMPLES:
1.
2.
3.
4.
5.
X < -2
Y<1
X +Y > 3
2X –Y > -2
3X –Y < 4
 P 371 #’S 26-30 EVEN
 P 371 #’S 36-50 EVEN