Absolute value measures distance on a number line, so it is always zero or positive. This idea is useful when a problem asks how far a quantity is from a target value, such as an error tolerance, temperature change, or position from the origin. Absolute value equations and inequalities often have two possible directions because the same distance can lie to the left or right of a center point.
Learning the number line meaning makes the algebra rules easier to remember and use correctly.
To solve an absolute value equation, isolate the absolute value expression first, then split it into two linear cases. Inequalities also split into cases, but the direction depends on whether the statement says the distance is less than a number or greater than a number. Statements like |x - a| < b describe values within b units of a, while |x - a| > b describes values more than b units away from a.
Graphing the solution helps confirm whether the answer is a bounded interval, two separate rays, one point, all real numbers, or no solution.
Key Facts
- |x| is the distance from x to 0 on the number line, so |x| >= 0.
- |A| = b with b > 0 means A = b or A = -b.
- |A| = 0 means A = 0, and |A| = b with b < 0 has no solution.
- |A| < b with b > 0 means -b < A < b.
- |A| > b with b > 0 means A < -b or A > b.
- |x - a| < b means a - b < x < a + b, while |x - a| > b means x < a - b or x > a + b.
Vocabulary
- Absolute value
- The distance of a number or expression from zero on a number line.
- Compound inequality
- A statement that combines two inequalities using and or or.
- Interval notation
- A compact way to describe a set of numbers using parentheses, brackets, and endpoints.
- Extraneous solution
- A value found during solving that does not satisfy the original equation or inequality.
- Boundary point
- A value where an inequality changes from true to false or false to true.
Common Mistakes to Avoid
- Splitting |A| = b into only A = b is wrong because a distance b can occur in the positive or negative direction, so you must also solve A = -b when b > 0.
- Forgetting to isolate the absolute value first is wrong because the two-case rules apply only to an expression of the form |A| compared directly to a number.
- Using and for |A| > b is wrong because greater than means outside the interval, so the solution is A < -b or A > b.
- Keeping solutions when |A| = -3 or |A| < -3 is wrong because an absolute value cannot be negative, so these cases have no solution.
Practice Questions
- 1 Solve |2x - 5| = 9 and check both solutions in the original equation.
- 2 Solve and graph the solution set for |3x + 1| <= 10.
- 3 Without solving by algebra first, explain why |x - 4| < 2 represents all points within 2 units of 4 and describe the solution interval.