Absolute Value Equations
I can solve absolute value equations, including when there is no solution.
The problem this sheet solves
Because a quantity and its opposite sit the same distance from zero, |x − 3| = 5 splits into two cases, and the second one is the one that goes missing. The companion error runs the other way: when the right side is negative students grind out two answers anyway, even though a distance can never be negative and there is nothing to find. This sheet puts both solutions on a number line so the symmetry is visible rather than asserted, which is what makes the second case feel necessary instead of arbitrary. Isolating the absolute value before splitting is treated as a required first move, and the sheet covers all three outcomes, including the negative case with no solution and the zero case with exactly one.
The key rule
Isolate the absolute value, then split into two cases: the inside equals c or the inside equals negative c. Solve each. If c is negative there is no solution, and if c is zero there is exactly one.
Two answers, one each side of center. |x − 3| = 5 gives x = 8 and x = −2.
Watch out
The mistake: Solving only x − 3 = 5 and stopping at x = 8.
The fix: Also solve x − 3 = −5 for x = −2. Two opposite insides give the same absolute value.
What is included
- Reference page — student-friendly definitions, a worked example, a Watch Out error pair, and the key rule
- Practice page — six problems, building from foundational to multi-step
- Apply page — a word problem, a reasoning prompt, and an error-analysis task
- Assess — a half-page exit ticket, two to a sheet, with a score box
- Full answer key — every item worked, plus a grading guide that says what each wrong answer tells you
- Ink-saver edition of all 7 pages
- 14-slide teacher deck (PowerPoint) with presenter notes on every slide
Standard codes refer to the Common Core State Standards for Mathematics. Math Class 678 is an independent publisher, not affiliated with, sponsored by, or endorsed by NGA Center or CCSSO. Every product is sold on Teachers Pay Teachers.