Absolute Value Inequalities

The absolute value inequalities combine the concepts of absolute values with inequality operators >, <, ≤ ,≥. These inequalities describe sets or ranges of possible values that pertain to a certain condition. 

There are two main types of absolute value inequalities based on the inequality symbols.

1. The less than type (<, ≤)

These inequalities represent values within a certain distance from a number. The solution is generally a bounded interval.

For example, |x - 2| <3 means that x is less than 3 units away from 2 or -1 < x < 5.

2. The greater than type (>, ≥)

These inequalities represent values of x that are further away than a certain distance from a point. The solution is generally in two separate parts, forming a union of intervals.

For example, |x + 1| >4 means that x lies farther than 4 units from -1, giving x < -5 or x > 3.

To solve these inequalities, we convert the absolute value expression into one or two standard inequalities, depending on the inequality sign:
For "<" or "≤", we use a compound inequality.
For ">" or "≥", we split into two separate inequalities joined by "or"
Follow these steps to solve an absolute value inequality:
Step 1: Rewrite the inequality as an equation. This helps identify boundary points where the inequality may change.
Step 2: Use a number line to plot these boundary points. The spaces between each point are the intervals.
Step 3: Choose a value from each interval and substitute it into the inequality to see if the given condition is satisfied.
Step 4: Repeat this for all intervals and note the intervals where the values satisfy the condition and make the inequality true.
Step 5: Combine these intervals and write the solution using union notation.
Let’s apply these steps to an example.
Question: Solve the given inequality. 
∣x − 3∣ > 2 
Solution: 
Step 1: Convert into an equation and solve
x-3=2
x-3=2 or x-3=-2
x=5 or x=1
Step 2: Draw a number line and plot points
Mark the points x = 1 and x = 5 on the number line. These points will divide the line into 3 intervals:
-1    0    1    2    3    4    5    6    7
----|----|----|----|----|----|----|----|----
              ↑                   ↑
             x=1                x=5
Interval 1, where x < 1
Interval 2, where 1 < x < 5, and
Interval 3, where x > 5
Step 3: Choose a test value in each interval, 1 x = 0
 ∣0 – 3∣ = 3 > 2 → satisfies inequality
From Interval 2: (1 < x < 5), choose x = 3
∣3 − 3∣ = 0 < 2→  does not satisfy inequality
From Interval 3: (x > 5), choose x = 6 
∣6 − 3∣ = 3 > 2 → satisfies inequality
Step 4: Identify valid intervals
Intervals 1 and 3 are valid, and 2 isn't.
Step 5: Final answer with all valid intervals
x(-,1)(5,)

There are three different types of inequalities based on the type of sign, these are:
Inequalities with the ‘greater than’ condition
These inequalities use the greater than sign (|x| > a), meaning that x lies at a distance farther than a unit from x on the number line.
Some examples are,
 |x| > 6
 |2x - 1| > 4 
 |x + 3|  7
Inequalities with ‘less than’ condition 
These inequalities use the less than sign (|x| < a), which means that the distance from a point to x is less than a.
Some examples include,
|x| < 5
|x - 2|  3
|3x + 1| < 8
Compound inequalities involving absolute values
This refers to inequalities that include absolute value expressions along with other operations that require isolating the absolute value.
For example, 
∣2x − 4∣ < 10
∣x + 5∣ > 12
∣3x − 2∣ ≤ 9

An intersection means the solution includes only values satisfying both conditions at the same time. They are denoted by A  B. In absolute value inequalities, intersection happens when you have inequalities like |x| < a  -a < x < a.
The union means that the solution set includes values that satisfy any one of the two conditions. It is denoted by A  B. In absolute value inequalities, union occurs when solving |x| > a  x > -a or x > a.
Let us see how we can find the union and intersection in absolute value inequalities.

For a given set of values, if the inequality is xa or x<b then, the union of values is given by x:x<bx:xa
Note that all values taken are less than b and greater than or equal to a, combining them into one set.
Here, there are 2 cases to be considered.
Case 1: b > a
The two intervals touch or overlap. 
For instance, x  2 or x < 5
x  2 (from 2 to ) 
x < 5 (from - to 5)
Since “or” includes both ranges, all real numbers are covered, and there is an overlap between 2 and 5. So the solution is xR.
Case 2: b  a
There is a gap between the intervals, and they do not cover everything. 
For instance, x  5 or x < 2 are two separate regions that do not overlap. 
x5 (from 5 to )
x<2(from - to 2)
So, the solution is x(-,2)(5, ).

For a given set of values, if the inequality is xa and x < b then the intersection of inequalities is given by {x:ax<b}.
This indicates that all values of x comply with both conditions at the same time. This means that x starts from a and goes up to but doesn't include b.
Case : ax<b
This is the standard form where both conditions need to be true, and the result is an interval x(a,b) , which is the intersection of two inequalities.

Absolute value inequalities are commonly used in real-life situations to define limits and ranges. Some real-life applications of absolute value inequalities are listed below:

• Noting temperature fluctuations in weather forecasts
  
  Absolute value inequalities are used to describe how much temperature may vary on average.

• Checking quality standards in manufacturing products
  
  During the manufacturing stages of a product, companies apply quality checks using absolute value inequalities. For example, the size of a part should be within ±0.02 cm of 5 cm is represented as ∣x − 5∣ ≤ 0.02.

• Defining zones in GPS and location tracking
  
  Absolute value inequalities help define zones of uncertainty around a specific location. This is helpful in navigation systems.

• Analyzing stock price movements
  
  Investors analyze stock price movements using absolute value inequalities to track how far a stock deviates from a target price.

• Managing medical dosages
  
  Doctors use absolute value inequalities to make sure that medication dosages remain in specific ranges that are safe for consumption. In some cases, a dosage must not vary more than 2 mg from the ideal dose; this is written as ∣d − 50∣ ≤ 2.