An absolute value inequality is a type of algebraic inequality that involves absolute value symbols. Specifically, these are mathematical statements where an algebraic expression within absolute value bars is compared to a constant or another expression using inequality symbols like <, >, ≤, or ≥. The key difference from standard inequalities is the presence of the absolute value, which always results in a non-negative value.
Understanding Absolute Value Inequalities
According to provided references, absolute value inequalities:
- Are inequalities in algebra.
- Involve algebraic expressions with absolute value symbols and inequality symbols.
- Use absolute value symbols which always give a non-negative value for any number.
- Replace the equals (=) sign with inequality symbols such as greater than (>) or less than (<).
Essentially, an absolute value inequality asks you to find all the values of a variable that make the absolute value of a given expression greater than or less than a certain value.
Key Concepts
- Absolute Value: The absolute value of a number is its distance from zero on the number line, denoted by vertical bars (e.g., |x|). It always results in a non-negative value. For example, |-3| = 3 and |3| = 3.
- Inequality Symbols: These symbols indicate the relationship between two values:
- < (less than)
-
(greater than)
- ≤ (less than or equal to)
- ≥ (greater than or equal to)
Examples of Absolute Value Inequalities
Here are some examples to illustrate absolute value inequalities:
- |x| < 5
- This means the absolute value of 'x' is less than 5. The solutions would include all numbers between -5 and 5, not inclusive (e.g. -4, 0, 2, 4, but not -5 or 5)
- |x - 2| ≥ 3
- This means the absolute value of the expression 'x - 2' is greater than or equal to 3. The solutions could be values of x that satisfy this condition such as 5 or -1.
- |2x + 1| > 7
- This inequality specifies that the absolute value of '2x+1' is greater than 7. The solutions would be values of x where '2x+1' is either greater than 7 or less than -7.
Solving Absolute Value Inequalities
Solving absolute value inequalities involves splitting them into two separate inequalities and solving each individually. This is because the absolute value makes both positive and negative values inside the absolute value bars evaluate to positive numbers.
For example, to solve |x| < 5, you split it as follows:
- -5 < x < 5
Similarly, to solve |x - 2| ≥ 3, you would split it into:
- x - 2 ≥ 3 OR x - 2 ≤ -3
- x ≥ 5 OR x ≤ -1
Summary
Absolute value inequalities are inequalities that use absolute value symbols. They require special treatment when being solved since they involve considering both positive and negative scenarios. The references explicitly state that they are inequalities involving algebraic expressions with absolute value symbols. Understanding absolute value inequalities is crucial for solving various problems in algebra and higher math courses.
