Solving 'Less Than' Absolute-Value Inequalities
Learn how to solve Solving 'Less Than' Absolute-Value Inequalities with clear steps and practice problems for Grade 9 algebra. Build confidence solving equations and checking your solutions.
Key Concepts
Property For an inequality in the form $|K| < a$, where $K$ represents a variable expression and $a 0$, the solution is $ a < K < a$. This is an "AND" compound inequality, meaning $K a$ AND $K < a$. Explanation Think of this as a 'distance limit.' Your value must be so close to zero that its distance is less than a certain number. This traps your answer between two boundaries, creating a single, cozy interval. You must be greater than the negative boundary AND less than the positive one simultaneously. Examples $|x 5| \le 3$ is solved as $ 3 \le x 5 \le 3$, which simplifies to the solution $2 \le x \le 8$. $|x| + 7.4 \le 9.8$ first simplifies to $|x| \le 2.4$, which means the solution is $ 2.4 \le x \le 2.4$.
Common Questions
How do you solve an absolute-value inequality?
Rewrite |x| < c as -c < x < c (conjunction) or |x| > c as x < -c or x > c (disjunction). Solve each part separately and graph the solution on a number line.
When do you write an absolute-value inequality as 'and' vs 'or'?
'Less than' absolute-value inequalities become compound 'and' statements (conjunctions). 'Greater than' inequalities become 'or' statements (disjunctions).
What does the solution set of an absolute-value inequality look like on a number line?
A 'less than' solution is a bounded interval between two values. A 'greater than' solution consists of two separate rays extending in opposite directions.