Absolute value — Digital SAT Math practice
Distance from zero in equations and inequalities.
Key terms
- Absolute Value: The non-negative distance of a number from zero on the number line, denoted by vertical bars.
- Critical Point: The value that makes the expression inside absolute value bars equal to zero.
- Compound Inequality: A mathematical statement combining two inequalities with and or or to describe solution sets.
- Extraneous Solution: A false result produced during algebraic manipulation that fails in the original equation.
Common traps
- Forgetting to split an absolute value equation into two cases misses valid solutions.
- Using the "between" form for a greater-than problem; greater than needs "or".
- Forgetting that |x - a| = b has no solution when b is negative.
- Assuming |x + y| equals |x| + |y|; the signs of x and y change the result.
Key ideas to know
Absolute Value Definition and Equations
What does |x| represent, and how many solutions does |x - 4| = 7 have? List them.
|x| is the distance from zero, so it is never negative. |x - a| = b (b > 0) has two solutions: x = a + b and x = a - b. For |x - 4| = 7, x = 11 or x = -3.
Watch out: Assuming absolute value equations always have two solutions. If the right side is negative, there are zero solutions.
Isolating Before Splitting Absolute Value
Which step comes first when solving 3|x + 1| - 5 = 10?
Isolate the absolute value expression before splitting into cases. Add 5, then divide by 3 to get |x + 1| = 5. Then split into x + 1 = 5 and x + 1 = -5.
Watch out: Splitting before isolating creates messy expressions and arithmetic errors. Always isolate first, then check solutions in the original equation.
Absolute Value Inequality Directions
A student writes the solution to |x - 2| > 5 as -3 < x < 7. What is wrong?
The inequality sign was reversed. |x - a| > b means x < a - b or x > a + b, not between. Correct: x < -3 or x > 7.
Watch out: Confusing greater than with less than inequality rules. Greater than points outward; less than points inward toward the center.
Sample questions
Question 1
What is the sum of all solutions to the equation |x − 4| = 8 ?
- 4
- 16
- 8
- 0
Show answer and explanation
Answer: 8
x − 4 = 8 or x − 4 = −8, giving x = 12 and x = −4. Sum = 8.