SAT Inequalities & Absolute Value: Flip Rule, Compounds & Graphs

Inequalities show up about 2–4 times on a typical Digital SAT Math section. You will solve single-variable inequalities, compound chains (a < x < b), absolute value inequalities, and word problems with constraint language (“at least,” “no more than,” “fewer than”).

You will also see graph-based items: which half-plane satisfies a linear inequality, or which inequality matches a shaded region. Some items never ask you to “solve for x” — they ask which values are allowed, or which graph is correct.

If linear equations still feel shaky, fix isolation habits first (see Linear Equations). Inequalities reuse the same algebra with one extra rule that decides most misses.

Format A — Pure solve: Isolate the variable. Remember the flip rule when you multiply or divide by a negative.

Format B — Compound: Two bounds on the same expression (or two inequalities joined by “and”). Solve each bound, then combine into one range.

Format C — Absolute value: |expression| compared to a positive number. “Less than” becomes a bounded AND range; “greater than” becomes an OR of two rays.

Format D — Word / graph: Translate English into ≤/≥/< />, or match a shaded region to an inequality. Direction and boundary (solid vs dashed) matter as much as algebra.

Two-second habit: name the format before writing steps. Absolute value items are not “just like equations with bars.”

Solve: −3x + 6 > 15.

Step 1: Subtract 6: −3x > 9.

Step 2: Divide by −3. Because you divided by a negative, flip the sign: x < −3.

Step 3: Check a test point. Try x = −4 (less than −3): −3(−4) + 6 = 12 + 6 = 18 > 15. True. Try x = 0: 6 > 15 is false. Good.

The flip applies only when you multiply or divide both sides by a negative number. Adding or subtracting a negative does not flip. Multiplying by a positive leaves the sign alone.

Trap: flipping when you only moved terms by addition. −3x > 9 is not yet flipped until the division step.

Solve: −1 < 2x + 3 ≤ 9.

Treat the chain as two inequalities sharing the middle expression: −1 < 2x + 3 and 2x + 3 ≤ 9.

Left: −1 < 2x + 3 → −4 < 2x → −2 < x (or x > −2).

Right: 2x + 3 ≤ 9 → 2x ≤ 6 → x ≤ 3.

Combine: −2 < x ≤ 3. On a number line, open circle at −2, closed at 3, shade between.

You can also subtract 3 from all three parts, then divide all three by 2: −4 < 2x ≤ 6 → −2 < x ≤ 3. Same result — just keep the same operation on every part of the chain.

Trap: reversing only one bound when dividing a three-part chain by a negative. Flip both inequality symbols if you multiply/divide the whole chain by a negative.

Case AND (distance less than): |x − 4| < 3 means the distance from x to 4 is less than 3 → −3 < x − 4 < 3 → 1 < x < 7.

Case OR (distance greater than): |x − 4| > 3 means x is more than 3 units from 4 → x − 4 < −3 or x − 4 > 3 → x < 1 or x > 7.

Memory hook: “less than” = inside the interval (AND). “Greater than” = outside (OR). Equal versions use ≤/≥ and include the endpoints.

Worked solve for |2x + 1| ≥ 5: 2x + 1 ≤ −5 or 2x + 1 ≥ 5 → 2x ≤ −6 or 2x ≥ 4 → x ≤ −3 or x ≥ 2.

If the right side is negative and the comparison is |expression| < negative, no solution (absolute value cannot be negative). If |expression| > negative, every real x works. The SAT loves that one-line trap.

Word problem: A shipping box can weigh at most 50 lb. The empty box is 8 lb; each item is 3 lb. How many items n can you pack?

“At most 50” → total weight ≤ 50. Equation form: 8 + 3n ≤ 50 → 3n ≤ 42 → n ≤ 14. Since n is a count, n ≤ 14 and n ≥ 0 (and usually integer). Answer often “14 or fewer.”

Translation cheat sheet: at least → ≥; at most / no more than → ≤; fewer than / less than → <; more than → >; between a and b inclusive → a ≤ x ≤ b.

Graph region: For y ≥ 2x − 1, boundary is the line y = 2x − 1 (solid because ≥ includes equals). Shade the side that satisfies a test point. (0,0): 0 ≥ −1 is true, so shade that side. For strict > or <, the boundary is dashed and not included.

Trap: matching the correct line but the wrong half-plane. Always plug one easy point after you draw the boundary.

Solve like equations, then enforce the flip rule only on multiply/divide by negatives. Write “FLIP?” as a mental checkpoint before you leave the problem.

For absolute value, sketch a number line for distance from a center. Visual AND/OR beats memorizing formulas under stress.

Word problems: underline the constraint phrase and write the symbol before any algebra.

Graph items: (1) solid vs dashed, (2) test a point, (3) only then look at choices.

If choices are solution sets, test a boundary value and one interior point instead of fully solving when algebra is messy — legitimate and fast.

After solving, re-read whether the question wants the inequality, a specific integer count, or a graph match.

Forgetting to flip when dividing by a negative — answer is often the exact reverse of correct. Test one number from your solution set in the original inequality.

Mixing AND and OR for absolute value: |x| < a is not two separate “or” rays.

Wrong direction on “at least / at most.” Say the English sentence with a number: “at least 5” includes 5, 6, 7…

Treating |expression| = negative as solvable — absolute value is never negative.

On graphs: open vs closed endpoints (or dashed vs solid) ignored while the algebra was right.

Answering x when the question asked for 2x or for the maximum integer n.

Day 1: Single-variable inequalities with mandatory flip checks and substitution tests.

Day 2: Compound chains only (including one chain divided by a negative).

Day 3: Absolute value AND cases only; number-line every problem.

Day 4: Absolute value OR cases + “no solution / all real” edge items.

Day 5: Word constraints only — translate to symbols before solving.

Day 6: Graph region matching (half-planes).

Day 7: Mixed 12–15; tag misses as flip, AND/OR, direction, or careless. Timed mini-set of 6.

Ready bar: you flip only when required, convert |x − c| < d and |x − c| > d without notes, and translate “at most / at least” in one glance.

Algebra topic sessions mix inequalities with equations so you practice the flip rule under mixed order — closer to a real module than a worksheet that only does inequalities for ten pages.

After a miss, ask the AI tutor (Pro) “did I need to flip?” or “show the number-line version of this absolute value.” Narrow prompts fix the exact leak.

Pair with Linear Equations if isolation is slow, and with Systems when you meet systems of inequalities or overlapping constraints later.