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Solving Inequalities with One Absolute Value (I)

 

Step 1: Isolate the Absolute value

 

Treat the absolute value signs like a set of parenthesis that you can’t break, or like the whole thing was one variable, then solve for that on the left side.  Remember to flip the inequality if you multiply by a negative number.

 

1-2|x-3| > -5  è -2|x-3| > -6 è |x-3| < 3

 

If the right side is a constant, solve using the steps below.  If the right side includes a variable, this problem requires special handling (learn how in part II of this lesson)

 

Step 2: Is the right side negative or zero?

 

If the right side is or could be negative, this problem needs special treatment.  If the right side is a negative constant, then the answer will either be “no solution” or “all real numbers” depending on whether the equality is “greater than” or “less than”

 

|x| < -1             è no solution

|x| <= -1           è no solution

|x| > -1             è all real numbers

|x| >= -1           è all real numbers

 

The same goes for a right side of zero, UNLESS the inequality is “less than or equal.” In this very special case the contents of the absolute value equal zero.

 

|x| <= 0            è x = 0

 

Step 3: Get rid of the absolute value marker

 

Now we split the equation into two parts.  We will compare the left side without the absolute value markers to the right side and it’s opposite.  If the inequality uses “less than” then we bound the contents of the absolute value marker between the right side of the equation and it’s opposite.  If the inequality uses “greater than” then we split the equation into the union of two inequalities.

 

|x+1| < 3          è -3 < x +1 < 3

|x| <= 5            è -5 <= x <= 5

|x-3| > 7           è x -3 > 7 OR x -3 < -7

|x| >= 13          è x >= 13 OR x <= -13

 

Step 4: Simplify (if needed)

Use basic algebra to simplify as needed.