1 ) Sara studied a map that showed the routes people followed to explore and settle in Kentucky during the 1700s. The scale on t
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answer is -1 < x < 5
-|x - 2|+ 9 > 6
Rearrange the terms
-|x - 2| > 6 - 9
-|x - 2| > - 3
then divide both sides of the inequality by the co- efficient of variable
|x - 2| < 3
convert the absolute inequality to standard inequality
-3 <x - 2 < 3
separate compound inequalities into system of inequality
{x - 2}> -3
{x - 2 < 3}
Rearrange variable to the left side of the equation
x > -3 + 2
calculate the sum or difference
x > -1
x -2 < 3
Rearrange variable to the left side of the equation
x < 3 + 2
calculate the sum or difference
x < 5
x > -1 and x < 5
Find intersection
-1 < x < 5
Equations and inequalities are both mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are deemed equal which is shown by the symbol =. Where as in an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.
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We have:
Thus:
Adding
gives:
Factoring gives:
Thus, we have:
or
.
Note that because
, there will be no solutions 
of the form 
for positive integer 
and solution 
. So we find the basic values of 
giving 
.
For with positive , 
(this can be seen easily from the unit circle. But 
, a contradiction. So we examine the case in which .
In this case,
. So this is the only solution to this equation.
Substituting this back in, we have
, as desired. So this is a valid solution.
Thus,.
Thus:
Adding
Factoring gives:
Thus, we have:
or
Note that because
For
In this case,
Substituting this back in, we have
Thus,