Download Lesson 1-6: Solving Linear Inequalities What are inequalities? Can

Lesson 1-6: Solving Linear Inequalities
What are inequalities?
You know what an equality is. It is a math statement that two expressions are equal; it
has an equal sign in it. Ok, fine…so what if we have two equations that are not equal.
What are the options? Well, seems pretty obvious: one expression will be greater than
the other right? If they aren’t equal, one’s bigger, the other is smaller. That would be
what we call an inequality…a math statement that says two expressions are not equal
and tells you which of the two is greater.
Can an inequality be both equal and greater (or less)?
Great question! Answering that question will help us get our arms around the real
meaning of inequality: two expressions that are not strictly equal. You can think of this
as an at least or at most statement. “I know I have at least $1.00 in my wallet, so I
have greater than or equal to $1.00. So, yes it can be both equal and greater.
Translating an inequality to math
How to we write an inequality in math? I’m sure you’ve seen this before, but here you go:
English
Math
Greater than
>
The symbols are easy to remember if you
remember that the open part (wider part) is on
the bigger value (greater) side.
Greater than or equal ≥
Less than
<
The point (or smaller end) points to the smaller
value.
Less than or equal
≤
The bar underneath means equal.
Practice translating
Translate the following into math:
1. x is less than seven
x<7
2. Seven is less than x
7<x
3. State #2 using greater than
x is greater than seven … x > 7
Can you swap sides with inequalities?
If you look at #2 and #3, you can see they really are saying the same thing, just reversed.
With equalities we learned we can swap sides…does swapping work with inequalities?
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Lesson 1-6: Solving Linear Inequalities
The answer is no. If you swap sides without also flipping the inequality sign, you are
saying almost the opposite! If you take #2 (seven is less than x) and swap sides, you
get x is less than seven. If you are having a hard time seeing that they aren’t the same,
pick a number that works for the 1st: 8 works, seven is less than eight. Now try 8 in the
swapped version: is eight less than seven? Nope; can’t swap sides with inequalities. If
you do swap sides, you will also need to flip the inequality sign.
Solutions of linear inequalities
When you solve a linear equality you get a single number. When you solve a linear
inequality, you get all the numbers that satisfy the inequality. Take for instance the
equality x – 1 = 5. There is only one number that satisfies that equality: 6. Now if you
take the inequality x > 5, you get lots of numbers: 5.5, 6, 7, 8.1, 9.3242, etc, etc, etc.
How many numbers satisfy x > 5? Actually there is an infinite number: every single
number that is larger than 5!
Practice determining if a number is a solution of an inequality
Just replace the variable with the number and simplify. If the result makes sense (is a
true statement) then the number is a solution for that inequality. Remember that it isn’t
the only solution, but it is a solution.
1. 4t – 9 ≥ 7; 3
No it isn’t. 4(3) – 9 = 3 and 3  7
2. 4t – 9 ≥ 7; 4
Yes it is. 4(4) – 9 = 7 and 7 = 7 so 7 ≥ 7
3. 4t – 9 ≥ 7; 5
Yes it is. 4(5) – 9 = 11 and 11 > 7 so 11 ≥ 7
4. 6y + 2 < 4; 0
Yes it is. 6(0) + 2 = 2 and 2 < 4
5. 6y + 2 < 4;
1
3
1
No it isn’t. 6( ) + 2 = 4 and 4 = 4 so 4  4
3
Graphing inequalities
A great way to understand an inequality is to graph it on a number line. To do so, you
simply draw a heavy line over the numbers that satisfy the inequality. One very
important thing to note is whether or not the inequality is an “or equal” statement. If it is
(≥ or ≤) that means it includes the base number and we’ll put a solid dot on the graph
at that number. If it is not (> or <) that means it does not include the base number and
we’ll put an open dot on the graph at that number. Examples:
x≥5
x<5
3 4 5 6 7
3 4 5 6 7
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Lesson 1-6: Solving Linear Inequalities
Solving simple linear inequalities
All the inequalities we’ve talked about so far are simple linear inequalities. They only
have one inequality sign in the statement. Here are the rules for solving simple linear
inequalities:
1. Treat it as a regular equality statement
2. Except you need to flip the inequality sign if you multiply or divide by a negative.
Examples
1. 3x  7  13
7 7
Subtract 7 fromboth sides
3x  6
3x 6

3 3
x2
2.
Divideby 3   by positive so don ' t flip inequality sign !
63  15 y  45
63
 63
 15 y  18
15 y 18

15 15
6
y
5
3. 8 x  3  23  5 x
3 3
Subtract 63 fromboth sides
Divideby  15   by negative so must flip inequality sign !
Notethe flipped inequality sign !!!
Add 3 to both sides
8 x  26  5 x
 5x
 5x
Add 5 x to both side
13x  26
13 x 26

13 13
x2
Divideboth sides by 13( by positive so don ' t flip inequality )
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Lesson 1-6: Solving Linear Inequalities
Compound linear inequalities
A compound linear inequality is two simple inequalities joined by “or” or “and.”
An “and” compound inequality looks like this: 4 ≤ x < 8. You would read this as
“four is less than or equal to x and x is less than eight.” The two inequalities joined
together are 4 ≤ x and x < 8. Think of the x’s overlapping…
Another way of reading it could be “x is a number between 4 and 8 including 4.” An
“and” compound inequality is basically a “between” statement.
The graph of “4 ≤ x < 8” is:
3 4 5 6 7 8 9
An “or” compound inequality looks like this: x > -1 or x < -5. You would read this
as “x is greater than negative one or x is less than negative five.” The two inequalities
joined together are easier to see here: x > -1 is joined with x < -5.
The “or” compound inequality can be thought of as an “outside of” statement.
The graph of “x > -1 or x < -5” is:
-7
-5
-1 0 1
Please note that we always state compound inequalities from least to greatest:
instead of 8 > x ≥ 3, we would say 3 ≤ x < 8.
Solving compound linear inequalities
A compound inequality is two inequalities joined together. To solve one, you need to
solve each of the joined inequalities.
For an “and” compound inequality, what you do to one part, do to all parts.
For an “or” compound inequality, do each side by itself.
Also, if you multiply or divide by a negative, you need to flip all inequality signs.
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Lesson 1-6: Solving Linear Inequalities
1.
4  4 x  12  16
12
 12  12
Add 12 to all parts
16  4 x  28
16 4 x 28


4
4
4
4 x7
2.
 8  7  3 x  10
7  7
7
15  3x  3
15 3x 3


3
3 3
5  x  1
1  x  5
Divide all parts by 4 ( positive so don ' t flip inequality )
Subtract 7 from all parts
Divideby  3( by negative so flip inequalites )
Noteboth inequalities flipped
Express the statement fromleast to greatest  reverseit
3. 5  9 x  14 or  4 x  17  3
1) 5  9 x  14
5
Solve each part individually
5
9 x 9

9 9
x  1
2)  4 x  17  3
 17  17
4 x 20

4 4
x  5
Answer : x  1 or x  5
Dividing by negative so flip the inequality
Dividing by negative so flip the inequality
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