Absolute Value on the SAT

Why the SAT Emphasizes Absolute Value

Absolute value is about distance. If you are $5$ miles from home, you could be $5$ miles east or $5$ miles west, and both are equally valid. The SAT uses absolute value because it forces you to think in two directions at once.

Many students lose points by solving only one case or by accepting a negative value that cannot work. This lesson shows how to set up the two cases, solve carefully, and check the results so you are never surprised by an extraneous solution.

A Simple Definition Unlocks Absolute Value

The expression $|x|$ represents the distance from $0$, so it is always nonnegative. When you see $|x| = 7$, there are two possibilities, $x = 7$ and $x = -7$, because both are seven units away from zero.

For equations like $|2x - 3| = 9$, you create two linear equations, one for the positive case and one for the negative case. If the right side is negative, there are no solutions, because a distance can never be less than zero.

Work Through Absolute Value Step by Step

Work through both cases carefully and keep the equations separate so you do not mix the solutions.

Use Desmos to Check Absolute Value

Desmos can solve absolute value equations by graphing both sides and showing intersections. It is especially helpful when the numbers are messy or when you want to confirm both solutions.

Read the $x$ values at the intersection points. Algebra is usually faster for clean integers, but Desmos is fast when you want a visual check or the equation has decimals.

Desmos is faster for counting solutions or when the numbers are messy. Algebra is faster when the absolute value splits cleanly into two simple cases.

Expert move: Graph the left and right sides directly (no need to isolate $y$), then click the intersection(s) and read the $x$-value(s). If Desmos gives decimals, convert to fractions when the choices are exact and keep only values that fit the domain or context.

When to skip Desmos: If the algebra is one or two steps, solve by hand and use Desmos only to verify; Desmos is best for messy coefficients or checking setup.

• Desmos features used: absolute value graphing, intersections.
• Common mistake: reporting only one intersection when there are two.

Practice Absolute Value with SAT-Style Questions

Answer each one, then compare with the solution logic.

Key Takeaways to Remember for Absolute Value

• Absolute value represents distance, so two solutions are typical.
• Set up two cases and solve each one carefully.
• Desmos is a fast visual check for intersections and missed solutions.