Solving Absolute-Value Inequalities
Learn how to solve Solving Absolute-Value Inequalities with clear steps and practice problems for Grade 9 algebra. Build confidence solving equations and checking your solutions.
Key Concepts
Property First, isolate the absolute value expression. Then, rewrite it as a compound inequality. Use AND for "less than" ($|x| < c \rightarrow c < x < c$) and OR for "greater than" ($|x| c \rightarrow x < c$ or $x c$).
Explanation Think of it like unwrapping a gift! You have to get the absolute value expression all by itself on one side before you can see the two possible solutions hidden inside. It's the most important first step, so don't rush past it. After isolating, you split the problem into two separate paths.
Examples Solve $3|x| + 5 < 14$. First, isolate $|x|$ to get $|x| < 3$. Then, unwrap it: $ 3 < x < 3$. Solve $|x+2| 4 1$. First, isolate the absolute value to get $|x+2| 5$. Then, split it: $x+2 < 5$ or $x+2 5$, which simplifies to $x < 7$ or $x 3$. Solve $|\frac{x}{2} 1| + 6 \leq 10$. Isolate to get $|\frac{x}{2} 1| \leq 4$. This becomes $ 4 \leq \frac{x}{2} 1 \leq 4$. Solving gives $ 6 \leq x \leq 10$.
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.