Solving Absolute Value Less-Than Inequalities
Solving absolute value less-than inequalities involves rewriting |x| < c as a compound AND inequality: -c < x < c. This rule applies when the absolute value is isolated and compared to a positive number, and is taught in Openstax Intermediate Algebra 2E, Chapter 2: Solving Linear Equations. The solution is a bounded interval on the number line.
Key Concepts
Property When an isolated absolute value inequality uses a "less than" ($<$) or "less than or equal to" ($\leq$) symbol with a positive number, it translates into a compound "AND" inequality. If $|u| < a$, the equivalent compound inequality is $ a < u < a$. This means the expression inside must be trapped between the negative and positive boundaries of that distance.
Examples Simple Less Than: Solve $|x| < 9$. You are looking for all numbers whose distance from zero is less than 9. Rewrite as a compound inequality: $ 9 < x < 9$. The solution is $( 9, 9)$. Multi Step Less Than: Solve $|2x 5| \leq 3$. Rewrite as a compound "and" inequality: $ 3 \leq 2x 5 \leq 3$. Add 5 to all three parts: $2 \leq 2x \leq 8$. Divide all three parts by 2: $1 \leq x \leq 4$. The solution is $[1, 4]$.
Explanation When an absolute value is "less than" a number, it means the expression is constrained. It is kept close to the center point. To solve it algebraically, you drop the absolute value bars and sandwich the inner expression between the negative and positive limits. You then solve all three parts of the chain at the same time to find the exact overlapping region.
Common Questions
How do you solve an absolute value less-than inequality?
An inequality |expression| < c translates to -c < expression < c (an AND compound inequality). Solve both parts to find the solution interval.
What if the right side is negative in an absolute value less-than inequality?
If |expression| < negative number, there is no solution since absolute value is always non-negative.
Where is solving absolute value less-than inequalities taught?
This is in Openstax Intermediate Algebra 2E, Chapter 2: Solving Linear Equations.
What type of interval is the solution to |x| < c?
The solution is a bounded interval (-c, c) or [-c, c] depending on whether strict or non-strict inequality, always centered at 0.
How is |x| < c different from |x| > c?
|x| < c gives a bounded AND interval between -c and c, while |x| > c gives an unbounded OR union of two intervals.