Section 3-1:  Linear inequalities and Absolute Value
 
 
    Solving a linear inequality is very similar to solving a linear equation.  You can basically solve it the same way except for one situation.  When you multiply/divide both sides of an inequality by a negative number, change the sense (direction) of the inequality.
 
Sample problems
 
 
  Problem 1 
   3x - 5 < 9 + x 
   2x - 5 < 9 
        2x < 14 
          x < 7
Problem 2 
       8 - 2x >
           -2x > -2 
              x <
 
 
In problem 1, it was solved exactly like an equation.  The solution is all numbers less than 7 make the statement true.  The graph looks like:
The open dot means does not include!
 
In problem two, notice we changed the sense in the last step because we divided both sides by -2.  The graph looks like:
The closed dot means it includes this point!

 
Absolute Value
 
Recall that absolute value means the distance from any point to zero.  Study the following chart:
 
Sentence Meaning Graph Solution
|x| = a
The distance from  x to 0 is exactly a units
x = a or x = -a
|x| < a The distance from x to 0 is less than a units. -a < x < a
|x| > a The distance from x to 0 is greater than a units x < -a or x > a

 
Sample Problems
 
2|3x - 5| = 8
This is an equality, choice number 1 on the chart.  First, islolate the absolute value to use the chart above.  Divide both sides by 2.
|3x - 5| = 4
Now apply the definition in the last column.
3x - 5 = 4 or 3x - 5 = -4
Solve each one individually.  Add 5 and divide by 3.
3x = 9 or 3x = 1
x = 3 or x = 1/3
 
 
3|2x + 3| < 12
This is an inequality involving the less than symbol.  Use choice two from the chart.  Remember to isolate the absolute value first!  Divide by 3
|2x + 3| < 4
Now apply rule #2 from the above chart.
-4 < 2x + 3 < 4
Isolate for x in the middle by subtracting 3 and dividing by 2
-7 < 2x < 1
-7/2 < x < 1/2
The solution is all numbers between -7/2 and 1/2.

 
|x - 4|/3 > 2
This is the last choice involving a greater than symbol.  Again, isolate the absolute value before you apply the appropriate rule!  Multiply by 3
|x - 4| > 6
Now apply the last rule from the chart.
x - 4 > 6 or x - 4 < -6
Add 4 to both sides.
x > 10 or x < -2
All numbers greater than or equal to 10 work, and all numbers less than or equal to -2 work!

 A Slightly tougher problem!!
 
1 < |x| < 5
 
Method 1.
Split into two inequalities
1 < |x| and |x| < 5
Solve individually
|x| > 1 and |x| < 5
The first inequality is rule 3 and the second is rule 2.  Apply these rules:
x > 1 or x < -1 and -5 < x < 5
To find the solution, find the area which is common to both.
 
Look at these to graphs and what points do they have in common?  That's right,  the numbers between -1 and -5 and the numbers between 1 and 5!!
The solution looks like:

 
Let's head on to the next section: