When you come across signs like < or ,> , and even ≤ or ≥ ,you will know it is an inequality.
Inequalities define ranges of possible values rather than single solutions, whether one value is less than, greater than, or equal to another, helping to describe situations where quantities can vary within certain limits. Therefore, The result of the inequality will be a certain range of values that you will have to find.
An important rule to keep in mind: when you double or divide both sides of the operation, the sign of the inequality is reversed!
Absolute value inequality is an inequality that involves the absolute value of a variable or expression. The absolute value represents the distance of a number from zero on the number line, always as a non-negative quantity. In inequalities, this means solving for a range of values that satisfy the given condition, either greater than or less than a certain value, regardless of sign. Absolute value inequalities often split into two cases, one considering the positive side and the other the negative, providing two sets of possible solutions.
We can solve absolute value inequalities in 2 ways:
The geometric method
- We will draw a number line, and mark the point from which we are interested in the distance to X.
- Then we find the points whose distance from the relevant point is exactly the distance mentioned in the condition.
- Now, let's pay attention to the condition: > or < And we will find the results.
The algebraic method
- Step 1: Divide into two cases:
- Case 1: Assume the expression inside the absolute value is positive.
- Case 2: Assume the expression inside the absolute value is negative.
- Step 2: Return to the original inequality:
- Case 1: Remove the absolute value and solve.
- Case 2: Remove the negative sign, solve the inequality.
- Step 3: Find the common domain for both cases.
- Step 4: Use a number line to mark valid ranges.
- Step 5: Determine all values of XXX that satisfy the inequality.