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.

Parent Tip: You can explain absolute value inequalities using distance as an example. 

• |x| > 2, means your child can move more than 2 steps in left or right direction, (assuming 2D plane)
   
• Similarly, |x| < 2, means your child can move only within 2 steps.

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"

Do You Know?
Compound inequality is simply a way of writing two separate inequalities using "and" or "or".
For example, if you can only play outside if it is before 6 pm or after 6 am. This can be written as time < 6 pm or time > 6 am

To solve absolute value inequalities, follow the steps mentioned below:

• 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 solve absolute value inequalities step-by-step.

Question: Solve the given inequality. 
∣x − 3∣ > 2 

Solution: 

1. Convert into an equation and solve
   
   \(x-3=2\\ x-3=2 \ or \ x-3=-2\\ x=5 \ or \ x=1\)
    
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:
   
   
   
   Interval 1, where x < 1
   Interval 2, where 1 < x < 5, and
   Interval 3, where x > 5
    
3. Choose a test value in each interval,
   
   From Internal 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
    
4. Identify valid intervals
   Intervals 1 and 3 are valid, and 2 isn't.
    
5. Final answer with all valid intervals
   \(x \in (- \infty,1) \cup (5, \infty)\)

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,

1.  |x| > 6
2.  |2x - 1| > 4 
3.  |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,

1. |x| < 5
2. |x - 2|  3
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, 

1. ∣2x − 4∣ < 10
2. ∣x + 5∣ > 12
3. ∣3x − 2∣ ≤ 9

Intersection
 

• An intersection means the solution includes only values satisfying both conditions at the same time.
• They are denoted by \(A \cap B\).
• In absolute value inequalities, intersection happens when you have inequalities like |x| < a ⇒ -a < x < a.

Union
 

• The union means that the solution set includes values that satisfy any one of the two conditions.
• It is denoted by \(A \cup B\).
• 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 \(x \ge a \ or \ x<b\) then, the union of values is given by \(\{x:x<b \} \cup \{ x:x \ge a \}\)

Pro Tip: 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, \(x \ge 2 \ or\ x < 5\)
  \(x \ge  2 \ (from \ 2 \ to\ \infty ) \)
  \(x < 5 \ (from \ - \infty \ to \ 5)\)
  
  Since “or” includes both ranges, all real numbers are covered, and there is an overlap between 2 and 5.
  
  So the solution is \(x \in R\).

• Case 2: \(b \le a\)
  There is a gap between the intervals, and they do not cover everything. 
  
  For instance, \(x \ge  5 \ or \ x < 2\) are two separate regions that do not overlap. 
  \(x \ge 5\ (from \ 5 \ to \ \infty )\)
  \(x < 2 \ (from \ -\infty \ to \ 2)\)
  
  So, the solution is \(x \in (- \infty ,2) \cup (5, \infty)\).

For a given set of values, if the inequality is \(x \ge a \ and \ x < b\) then the intersection of inequalities is given by \(\{x:a \le x<b\}\). 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:\( a \le x<b\)

This is the standard form where both conditions need to be true, and the result is an interval \(x \in (a,b) \), which is the intersection of two inequalities.

For young students, understanding and solving absolute value inequalities can be difficult and confusing. To make it easy, here are s few tips and tricks:

1. You can use a number line to visualize an inequality.
    
2. Remember, for |x| < a, it is less than inequality. And |x| > a is greater than inequality.
    
3. When the inequality has \(\le \ or \ \ge\), use a solid circle to represent the endpoints on the number line.
    
4. Simplify the inequalities if needed.
    
5. When doing multiplication or division by a negative number, reverse the inequality sign.

Parent Tip: You can use real life scenarios like time limit, temperature or distance to explain absolute value inequalities. Encourage your child to practice problems from absolute value inequalities worksheet.

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

1. Noting temperature fluctuations in weather forecasts
   
   Absolute value inequalities are used to describe how much temperature may vary on average.
   
    
2. 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.
   
    
3. 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.
   
    
4. Analyzing stock price movements
   
   Investors analyze stock price movements using absolute value inequalities to track how far a stock deviates from a target price.
   
    
5. 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.