Quick Answer
Absolute Value represents a number's distance from zero on a number line. On the Digital SAT, this concept typically appears in Algebra modules as equations or inequalities. It is a moderately frequent topic, often appearing as one or two questions per test, usually requiring students to solve for multiple possible solutions or interpret graphs.
The absolute value of a real number x, denoted by |x|, is its non-negative distance from zero regardless of its sign. Mathematically, |x| = x if x is greater than or equal to zero, and |x| = -x if x is less than zero.
Question: If |2x - 5| = 11, what is the sum of all possible values of x? Solution: Set up two equations: 2x - 5 = 11 and 2x - 5 = -11. 1) 2x = 16, so x = 8. 2) 2x = -6, so x = -3. Sum: 8 + (-3) = 5.
Forgetting the negative case: Students often solve only for the positive scenario (e.g., x - 3 = 5) and miss the second solution (x - 3 = -5).
Distributing signs incorrectly: Students may incorrectly change the signs inside the absolute value bars before removing them, rather than setting up two separate equations.
Ignoring the non-negative constraint: Students might try to solve an equation where the absolute value is set equal to a negative number (e.g., |x| = -2), which has no real solution.
Students targeting 750+ should know that absolute value inequalities like |x - m| <= k are the standard way to express a range where m is the midpoint and k is the tolerance or radius. Recognizing this structure allows you to bypass algebraic steps and identify the center and spread of a data set or interval instantly.
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