An absolute value inequality is an inequality that contains an absolute value expression (like ∣x∣) and uses inequality signs such as <, >, ≤, and ≥.
The absolute value of a number represents its distance from zero on the number line, regardless of direction. It is always non-negative.
Graphing Absolute Value Inequalities
A number line is used to show solutions of inequalities using points and shading. Points (dots) show boundary values, and shading/arrow shows the range of solutions. Open dot "◌" is used to represent an open interval, and a closed dot "⚈" is used to represent a closed interval in the graph.
1. Open Intervals (◌)
- Used for <or >
- The endpoint is not included
Example: a<x<b → x is between a and b, but not equal to them
2. Closed Intervals (⚈)
- Used for ≤ or ≥
- The endpoint is included
Example: a ≤ x ≤ b→ x is between a and b, including both
3. With Infinity (Open)
- x<a: values less than a
- x>b: values greater than b
(Here, we use open dot ◌)
4. With Infinity (Closed)
- x≤a : less than or equal to a
- x≥b : greater than or equal to b
(Here, we use a closed dot ⚈)
Solving Absolute Value Inequalities
We use the number line approach to solve an inequality and follow the steps added below:
Step 1: Write down the inequality and assume it to be an equality making it an equation instead of inequation.
Step 2: Draw a number line depending on the intervals and represent the equation on the number line.
Step 3: From each interval, select a number and check if it satisfies the inequality.
Step 4: Perform step 3 for every interval and the intervals for which a random number satisfies the inequality will be included in your final answer.
Step 5: Take the union of all the intervals to get the answer.
Types of Absolute Value Inequalities
Depending on the type of sign in the inequality, there are different types of inequalities:
1. Greater Than Inequalities: These inequalities use a greater than sign i.,e. the number is greater than the value of some other mathematical value.
Example : x > 5
2. Less Than Inequalities: These inequalities use a less-than sign ,i.e. the number is less than the value of some other mathematical value.
Example : x < 57
3. Absolute Value Inequalities:These inequalities involve both greater than and less than cases, i.e., the number is less than and greater than the value of some other mathematical value. These inequalities use the absolute value.
Example: |x - 5| < 7
Intersection and Union in Absolute Value Inequalities
Union of Inequalities: For a given set of values, if the inequality is x>=a or x<b then we need to find the union of the values of x which can be given by
Case 1: x >= a or x < b
{x: x < b U x ≥ a}
Case 2: x < a or x >= b
{x: x <a U x ≥ b} = {x: x < a} U {x: x ≥ b}.
The solution i.e. the union can be calculated using a graph. Consider the example x <= 3 || x >= -4, then the union of the inequalities will give an overlapping interval which will include all real numbers as shown below.
Intersection of Inequalities: For a given set of values, if the inequality is x >= a and x < b then we need to find the intersection of the values of x which can be given by
Case 1: a <= x < b
{x: a≤x < b}
Case 2: a <= x U b > x
{x: a ≤ x U x < b}
The solution i.e. the intersection can be calculated using a graph. Consider the example x <= 4 U x >= -5, then the intersection of the inequalities will give an interval that will include all real numbers from -5 and 4 as shown below.
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Solved Examples
Example 1: Solve for inequality |x + 24| > -5 using the formula-based approach.
Given Inequality,
- -∞ < x + a < -b
- b < x + a < +∞
Solving both of them individually
Case 1:
-∞ < x < -a - b
Case 2:
b - a < x < ∞
x ϵ (-∞,-a-b) ⋃ (b-a, ∞)
Example 2: Solve this less than equal to absolute inequality |y + 5| <= 3y
Given Inequality: |y + 5| <= 3y
Case 1:
y + 5 <= 3y
5 <= 2y
5/2 <= yy ϵ [5/2, ∞)...(i)
Case 2:
-3y <= y + 5
-4y <= 5
y >= -1.25y ϵ (-∞, -5/4]...(ii)
From eq. (i) and eq. (ii)
y ϵ [5/2, -5/4]
Practice Problems
Question 1: Use the union and intersection method to find the solution for x given |x+7|<1001 and |x+2|>24.
Question 2: Use the graphical representation method to find the solution of |2x+5|+y>7x.
Question 3: Solve the inequality ∣2x + 3∣ < 5.
Question 4: Find all values such that ∣x − 4∣ ≥ 2.