Download Unit 4: Solving Inequalities Inequalities are used to one amount to

Unit 4: Solving Inequalities
Inequalities are used to ___________________ one amount to another. There are _____ symbols that can
show inequality and we read them from ________________ to ____________
>
<
>
<
x > 4 means that x is ________________ than 4. We don’t know what number it is, but it must be something
bigger than 4.
Use your inequality symbols to translate these problems from words into math.
X is less than 4: ______________
5 is greater than or equal to y: _______________
3 is greater than w: ______________ b is less than or equal to 8: ______________
Graphing Inequalities on the Number Line:
Step 1: On your number line, find the number in your inequality and put a _______ where it is.
Step 2: Leave the circle empty if your inequality is ____ or _____. Fill in the circle if your inequality
is_____ or________.
Step 3: Shade your number line based on what the inequality says… if your variable is greater or
greater than or equal to a number, you shade to the __________, where the numbers are ___________. If
your variable is less than or less than or equal to a number, you shade to the ____________, where the
numbers are ________________.
Example 1: x > 4
Step 1: Put a circle on the number line at the _________.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Step 2: Leave the circle empty if your inequality is > or <. Fill in the circle if your inequality is > or <.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Step 3: Shade your number line based on what the inequality says… where are the numbers greater than 4?
-5
-4
-3
-2
-1
0
1
2
3
4
5
Example 2: 3 < x
Step 1: Put a circle on the number line at the ______.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Step 2: Leave your circle empty if your inequality is > or <. Fill in the circle if your inequality is > or <.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Step 3: Shade your number line based on what the inequality says… where are the numbers greater than or
equal to 3?
-5
-4
-3
-2
-1
0
1
2
3
4
5
You try! Graph these on the number line.
1. x > 2
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
2. -1 < x
3. 4 > x
4. x < -3
-5
-4
-3
-2
-1
0
1
2
3
4
5
When you have a negative x, you need to turn it into a positive! How do we get rid of a negative in
front of an x? _________________________________________________
When we divide by -1, our inequality sign ______________. If it was >, it becomes < and if it was <, it
becomes >. So if it was >, it becomes < and if it was _____ it becomes _____.
Example 3: -x > 2
You try!
1. 4 < -x
2. –x > 3
3. –x < -1
4. 5 > -x
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
Tuesday - Solving One Step Inequalities
For one step equations, we only need to use ___________________ _________________ one time to
________________ the variable. Use what you learned yesterday to try these!
Solve:
1. x + 3 > 7
2. 3x ≤ 18
3. -6n < - 12
4. 4 + x > 2
5. Z – 8 ≥ -7
Now, solve the inequality and graph it on the number line!
1. -8x > 32
2. x + 9 ≤ 15
Solving Two Step Inequalities
Solving two step inequalities is almost exactly the same as solving two step __________________________.
REMEMBER: The goal is ALWAYS TO _____________________ the variable OR get it by _______________
Step 1: Do the inverse operation of ___________________________ or __________________________ first.
Step 2: Do the inverse operation of ___________________________ or ______________________________ next.
If you are multiplying or dividing by a _____________________ number, you must _________________ the
inequality sign.
Step 3: __________________ for the variable.
Step 4: Plug your answer back in.
Examples:
x
1. 2x – 9 < 5
2. + 7 > 3
3. -3x – 10 < -16
2

More Practice 
1. 7x – 12 ≤ 23
2.
x
+7≥3
6
4. -3x – 10 < -16
5. 30 > -x + 18

7. 30 > 4 - x
8. 9x + 5 ≤ 43
Now solve and graph!
3. 5x + 6 < -21
6.
x
+ 6 > 23
4
9. -7x + 3 ≥ 14

1. 4 – 9c ≥ 13
2. -18 < 5m – 3
3. x + 4 < 6
2
4. 4d + 7 ≥ 23
5. -10 ≤ 5m – 3
6. -5x + 9 > -14
Challenge : Solve each (do not graph)
1. -2(h+2) < -14
2. -3t + 7 ≤ -5t + 9
3. -4(x -5) > 6(x + 3)
Wednesday – Absolute Value Inequalities
There are two types of inequalities with absolute values.
Type 1: “Less Than” Inequalities
Remember that absolute value is the distance from a number to ________ on the number line. Let’s
think about x < 3. The problem is saying that any answer to x needs to be less than ____ units from 0
on the number line. Where on the number line could x fall?

Generally speaking, if an absolute value equation is _______________ another value, the answers will all
fall BETWEEN two end points. It will follow the pattern below:
a b
-b < a < b
Example: 6x  12


Step 1. Rewrite the equation:
< 6x <
Step 2: Split up the two inequalities
< 6x
and
6x <
Step 3: Solve each inequality as if it were an equation.
________________________________________________
You try!
5p  20
2m  40




4g 1 18
6  3y  31
Type 2: “Greater Than” Inequalities
Let’s think about x  2 . The problem is saying that any answer to x needs to be more than ____ units
from 0 on the number line. Where on the number line could x fall?

When an absolute value equation is ____ _________________ another value, it can go on infinitely in both
directions. All of the answers will be ___ ____________ one number and __ ______________ its opposite. It
will follow the pattern below:
a b
a > b or a < -b
Example: 9x  36
Step 1. Rewrite the equation:

9x >
or
 2: Solve each inequality as if it were an equation.
Step
________________________________________________
9x <
You try!
6p  30
8w  40

4  2y 12

Thursday - Multi-Step Inequalities

The process for solving inequalities is the __________ as solving equations with ONE exception: when
you divide by a ___________________ you must __________ the inequality symbol!
Examples:
1.) 4(x+ 3) ≤ 24
2.) -2(x – 8) > 32
3.) 4x – 5 ≤ -6x + 15
4.) 8x + 6 < 2x - 12
You Try!
1.) 5(x+ 2) ≥ 30
2.) -3(x + 3) < 9
3.) 8x – 4 ≥ -10x + 32
4.) 4x + 2 > 10x - 22
Challenge Problems:
5.) 4  x  2  4


7.) 6  x  4  6

6.) 5  x 10 15
Friday - Compound Inequalities
Vocabulary
Compound Inequality:
Solving a Compound Inequality with AND
-
What will it look like? _______________ < _______________ < ____________________
Step
1
Write the _________________
inequality
2
Isolate the ____________
between the two inequality
symbols by first __________
or __________
3
__________ or __________ the
number with x to both
numbers on the outside of
the inequalities.
Example # 1
Example # 2
Example # 3
-2 < 3x – 8 < 10
-3 < 4x – 9 < 11
4x - 24 < 30 < 9x - 15
**** Remember to switch both
inequality symbols if you
multiplied or divided by a negative
number!!!
4
Write your solution as one
inequality _____ <x<_____
5
Plot both inequalities on
the ___________ number line
X>2
0
AND
X<6
2
6
Guided Practice Problems – Solve the following Compound Inequalities and Plot on a Number Line
1. -5 < 3x + 4 < 19
2. – 4 < 2 + x < 1
3. -25 < 5x < -20
Solving a Compound Inequality with OR
-
What will it look like? _______________ OR ____________________
Step
1
Write the _________
inequality
2
Solve each inequality
_______________ for x
Example # 1
Example # 2
2x + 7 < 3 or 5x + 5 > 10
3x + 8 > 17 or 2x+5< 7
**** Remember to switch the
inequality symbol if you
multiplied or divided by a
negative number!!!
3
Write your solution as
_________ inequalities with
OR in the middle
x < -2
6
Plot both inequalities on
the __________ number line
-2
x>1
1
Example # 3
-3x -7>8 or -2x -11< -31
Guided Practice Problems – Solve the following Compound Inequalities and Plot on a Number Line
6+2x > 20 or 8 + x < 0
3x + 1 < 4 or 2x – 5 > 7
-2x> 6 or 2x + 1 > 5
Monday - Inequality Word Problems
Why do we have inequalities? Think about the statements below:
1) John has 5 dollars. ________ 2) John has at least 5 dollars. __________ 3) John has less than 5 dollars._______
Why are they different? Why are they the same? How could you write each one?
When we know an exact amount, we use an ___________________ sign to show that the two sides are the
___________________. When we don’t know an exact amount, we use ________________________ to compare
sides.
Inequality Symbol
>
≥
<
≤
Words Associated with Symbol
Example Problems:
1) Yolando needs more than $5,000 to start her own business.
2) The speed limit is 80 MPH. a) Write an inequality and b) draw it on the number line.
3) Yellow Cab charges a $1.25 flat rate and an additional $0.30 per mile traveled. Jared has no more than
$10.00 to spend on his cab ride.
a) Write an inequality to represent Jared’s situation.
b) How many miles can Jared travel without going over his budget?
4) Domino’s sells medium 2-topping pizzas (p) for $8.00 and 2-liter sodas for $2.00 (s). The company
wants to make at least $80.00 per night. Write an inequality that expresses this situation.
5) The minimum age for the concert was 14 years old.
You Try!
1) The school bus fits no more than 40 students. a) Write an inequality to describe this and b) graph it.
2) Sarah needs more than 6 hours of sleep, but at most 9 hours.
a) Write an inequality to describe this and b) graph it.
3) Jacob has a lawn-mowing business. He makes $100 per week and $35 per lawn he mows. He needs to
make at least $240 per week to save up for his new car.
a) Write an inequality to represent Jacob’s situation.
b) How many lawns must he mow to make $240 per week?
4) To ride a rollercoaster, a person must be a minimum of 48” tall, but cannot exceed 80”.
5) The maximum weight limit for the bridge is 6000 pounds