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1310, Section 2.6
Linear Inequalities
Linear – means that the variable is to the first power
Inequality – means that the comparative symbol is one of the following:




So we’re comparing and there’s a boundary involved. For example if I say x > 5, I mean
for you to think of numbers strictly bigger than 5.0….5.55 is an example of this type of x.
I can use interval notation to summarize all these x’s: (5, )
Or I can use a graph with a numberline featuring a hollow dot at 5:
(Notice, this can become an x axis if I put in the y axis)
5
Inequalities have the feature that you may add and subtract a number to both sides and
you will get an equivalent inequality. You may multiply and divide by a positive number
and you will get an equivalent inequality.
If you multiply or divide by a negative number you induce a flip about the centerpoint
zero and we symbolize this by flipping the inequality symbol.
So: Solve  5x  3  8
Subtract 3 from both sides…
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Now the claim is that negative 5 times x is greater than positive 5. What kind of numbers
are our x’s if the product of x and  5 is a positive?
Indeed, the x’s are negatives. Divide both sides by 5
What happens if we forget to flip the inequality?
Put your answer in interval notation and sketch a graph of your answer:
How about:
6x + 3 < 9x + 15
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Now we can have compound inequalities, too. You must read these carefully.
Here is a simple one: the x’s I care about are between and including 3 and 5:
3  x  5 . Saying [3,5] and
both mean the same x’s
3
5
Now it’s possible to get compound inequality problems, too.
1
3
 2x  3 
2
5
Let’s solve this one.
3  5  x  12
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Now compound inequalities can be “outies” as well as “innies”
Let’s look at the inequality expression for
-3
5
How would you write the manglish for this picture using inequality symbols? The key
word is “or”
How would you write this in interval notation? “Or” is written “”, a union of two rays.
Let’s review CTS
x 2  26x  7  0
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Now let’s move into Absolute Value, Section 2.8
Most everybody can tell me what the absolute value of 3 is: ____________
But – I want you to start thinking of absolute value as a distance. Indeed as a distance
from zero.
If I tell you to read out loud and draw a picture of the following equation:
x 3
I’ll be expecting to hear: “The distance from x to zero is 3”
and I’ll expect a picture that looks like this:
-3
3
There are two places on the number line that are 3 away from zero. The answer is x = 3
or x = 3.
When you get absolute value equations, then the “problem” is then two problems:
the negative problem and the positive problem.
Look at 3x  5  11 .
From order of operations, you know that you need to remove the absolute value problem
delimiters before continuing. We’ll do that by reading the problem and translating our
words to set of problems.
This says: The distance from 3x  5 to zero is 11. Well, now there are two places that
are 11 away from zero on the number line.
So we’ll solve both problems:
5
3x  5 =  11
Let’s do another:
And another:
0
3x  5 = 11
3  x  15
2 2x  3  12
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