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Solving Linear and Quadratic Equations and Absolute Value Equations

Slide 31 Solving Absolute Value Equations
Subtract 5
Mult. by -1
Subtract 5
Mult. by
Slide 32 Solving Absolute Value Equations

Highly recommended: Check your solutions by plugging them back
into the original equation:



Slide 33 Solving Absolute Value Equations

Note that simplifying absolute values should come early in the
order of operations, similar to brackets and parentheses.

|3x + 5| = 2 is not the same as |3x| + 5 = 2
Slide 34 Example: Using Absolute Values in Statistics

You are studying the relationship between head injury and helmet
use among people involved in bicycle accidents (Pagano and
Gauvreau, page 311).

You have information for 793 people involved in accidents. Of
those, 147 people were wearing helmets and 235 people experienced
a head injury. You can make a table that illustrates what you
know, and add what you would expect to see if there were no
association between helmet use and head injury:
  Wearing Helmet  
Yes No
Head Yes   235
Injury No  


147   793
Slide 35 Example: Using Absolute Values in Statistics

Here is the table of expected values we just created:
Wearing Helmet Yes No Total
Head Yes 43.6 191.4 235.0
Injury No 103.4 454.6 558.0
  Total 147.0 646.0 793.0

When you add the information you have about helmet use among
people who experienced a head injury, here is what you observed:

Wearing Helmet Yes No Total
Head Yes 17 218 235
Injury No 130 428 558
  Total 147 646 793
Slide 36 Example: Using Absolute Values in Statistics

We can use a “test statistic” to see whether the difference between
observed and expected values is big enough to signify that wearing
helmets is associated with fewer head injuries. Here’s what the test
statistic looks like:

means “add the first through the 4th” of what comes next

is the observed value for position i, where i is either 1, 2, 3, or 4

is the expected value in the corresponding position

χ2 (Chi-squared) is the name of this particular test statistic

Slide 37 Example: Using Absolute Values in Statistics

What does this mean? Is there an association between helmet use
and head injury? Stay tuned!

Slide 38 Inequalities

An inequality is like an equation, but it says that two expressions
are not equal.

a ≠b a is not equal to b

a < b a is less than b

a > b a is greater than b

a ≥b a is greater than or equal to b

a ≤b a is less than or equal to b

Note that “less than” means “to the left on the number line” and
“greater than” means “to the right on the number line”.

Slide 39 Example: Using Inequalities

The value of the test statistic, χ2, that we obtained in the last
example was 27.27. We can look this up in a table and see how
“statistically significant” this is. The smaller the p-value, the more
statistically significant.

As we can see from the table, the p-value gets smaller as the χ2
values get bigger. Since 27.27 > 10.83, p < 0.001.

We conclude that wearing helmets is associated with lower rates of
head injury.

Slide 40 Solving Linear Inequalities

As with equalities, we can use the additive and multiplicative
properties of inequality.

One wrinkle: If you multiply both sides of an inequality by a
negative number, the direction of the inequality changes. So
multiplying or dividing both sides of an equation by a negative
number means you should reverse the inequality.

multiply by -1
Not true!
reverse direction of inequality
Slide 41 Solving Linear Inequalities: Example
add -4x to both sides
combine like terms
add -3 to both sides
multiply by
& change direction of

Check: Does x = −2 work? Does x = −3 work?