SAT Absolute Value Equations and Inequalities: Solving and Graphing

The absolute value of a number is its distance from zero on the number line. |3|=3 and |-3|=3. When solving absolute value equations like |x-2|=5, recognize that the expression inside the absolute value can be either 5 or -5. This leads to two cases: x-2=5 (giving x=7) or x-2=-5 (giving x=-3). Both solutions are valid. The fundamental strategy for solving absolute value equations is to split the equation into two cases, one positive and one negative, then solve each case separately and verify both solutions in the original equation. This case-by-case approach catches both solutions and prevents the common mistake of finding only one.

Absolute value equations sometimes have no solution. If you set up |expression|=-7, there is no solution because absolute value is always non-negative. Recognizing when an absolute value equation or inequality has no solution (when the equation says |something|=negative number or |something|

Absolute value inequalities follow different rules than equations. For |x-2|<5, you solve by creating a compound inequality: -55, you solve by splitting into two cases: x-2>5 (giving x>7) or x-2<-5 (giving x<-3), which you write as x<-3 or x>7 (a union of two intervals, not an intersection). The key rule: less-than absolute value inequalities create AND (compound), while greater-than absolute value inequalities create OR (two separate regions). Confusing these rules is a frequent error; remembering this AND vs OR distinction prevents most mistakes.

A practical checklist for solving |expression| inequality: (1) Identify the operation: < or >. (2) If <, set up compound inequality: -k, split into two cases: expression>k or expression<-k. (3) Solve to get your solution interval or union of intervals. (4) Graph the solution on a number line to visualize. This checklist, applied to five daily problems, builds automaticity so you do not pause to remember the rule during a test.

The graph of y=|x| is a V-shape with its vertex at the origin. Transformations shift and shape this graph: y=|x-h|+k shifts the vertex to (h,k). y=a|x-h|+k also scales the V by factor a. Solving |x-2|=5 geometrically means finding where the graph of y=|x-2| intersects the horizontal line y=5. The graph hits at two points (x=-3 and x=7), confirming there are two solutions. Solving |x-2|<5 means finding where the graph of y=|x-2| is below the line y=5, which is the interval between the two intersection points. Connecting algebraic solutions to their geometric meaning—where graphs intersect or where one graph is above/below another—solidifies understanding and provides a way to verify algebraic answers visually.

On test day, when you solve an absolute value inequality and get a compound solution like -37, quickly sketch a number line and mark the solution regions to verify your answer makes sense. This visual check catches sign errors and mistakes in your case-splitting logic. For more complex absolute value expressions, graphing provides an independent way to verify your algebraic work.

Common errors include: forgetting to check both solutions in original equation (especially for equations), confusing AND vs OR in inequalities (the biggest pitfall), and making sign errors when negating expressions for the negative case. A 3-check error-prevention routine: (1) After solving an equation, substitute both solutions back into the original |expression|=k and verify both produce the right value. (2) For inequalities, trace your AND/OR choice: less-than gives AND (a single interval), greater-than gives OR (two separate regions). (3) Graph your final solution on a number line to check that it passes the visual smell test. A focused 2-week drill of 5-10 absolute value problems daily—mixing equations and inequalities—builds the skill to automaticity, making these problems feel routine rather than tricky on test day. Focus your drills on problems where the expression inside the absolute value is more complex (like |2x-3| or |x/2+1|), since these expose the case-splitting logic more clearly.

Review your errors from practice tests specifically for absolute value. Did you forget a second solution? Did you confuse the AND/OR rule? Use your error pattern to refine your checking routine. Some students find they always make the same mistake (like forgetting to verify solutions); building an automated check for that specific mistake prevents it from recurring.