2.3Two-Step Inequalities
Two operations stand between you and the variable — undo them in reverse order, watch for the negative flip, and you have an entire solution set.
Why This Matters
Two-step inequalities model everyday constraints like budgets and time limits. You'll use this skill in linear programming, economics, and any situation where you need to find all values that satisfy multiple conditions.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do you solve a two-step inequality, and when must you reverse the inequality symbol during the process?
Lesson Overview
A two-step inequality requires exactly two inverse operations to isolate the variable — the same order of operations logic as two-step equations, but with the inequality symbol carried through every step. The strategy is identical to solving two-step equations: undo addition or subtraction first, then undo multiplication or division. The one rule that makes inequalities different still applies here: if the second step requires multiplying or dividing by a negative number, flip the inequality symbol at that step only. Adding or subtracting in the first step never flips the symbol, regardless of whether the number is positive or negative. After solving, graph the solution set on a number line and verify by substituting a test value from the shaded region into the original inequality.
When Does the Symbol Flip?
Symbol stays — these NEVER flip
- Adding any number (+ or −)
- Subtracting any number (+ or −)
- Multiplying by a positive number
- Dividing by a positive number
Symbol flips — ONLY these cases
- Multiplying by a negative number
- Dividing by a negative number
In a two-step problem, the flip happens at Step 3 only — never at Step 2.
Need a deeper explanation? Read our article on why the inequality sign reverses.
Worked Examples
Solve and graph: 2x + 3 > 11
Step 1 — Identify operations on x: multiply by 2, then add 3.
Step 2 — Undo addition: subtract 3 from both sides.
2x + 3 − 3 > 11 − 3
2x > 8
Step 3 — Undo multiplication: divide both sides by 2 (positive → no flip).
x > 4
Graph: open circle at 4, shade right. Interval: (4, ∞)
Check: Test x = 6: 2(6) + 3 = 15 > 11 ✓. Test x = 2: 2(2) + 3 = 7 > 11 ✗ (correct).
Graph: x > 4
Open circle at 4 · shaded right → (4, ∞)
Solve and graph: 3x − 5 ≤ 10
Step 2 — Undo subtraction: add 5 to both sides.
3x − 5 + 5 ≤ 10 + 5
3x ≤ 15
Step 3 — Undo multiplication: divide both sides by 3 (positive → no flip).
x ≤ 5
Graph: closed circle at 5, shade left. Interval: (−∞, 5]
Check: Test x = 0: 3(0) − 5 = −5 ≤ 10 ✓. Test x = 8: 3(8) − 5 = 19 ≤ 10 ✗ (correct).
Graph: x ≤ 5
Closed circle at 5 · shaded left → (−∞, 5]
Solve and graph: −4x + 2 < 18
Step 2 — Undo addition: subtract 2 from both sides (no flip — this is addition/subtraction).
−4x + 2 − 2 < 18 − 2
−4x < 16
Step 3 — Undo multiplication: divide both sides by −4 (NEGATIVE → FLIP the symbol).
−4x ÷ (−4) > 16 ÷ (−4) ← < becomes >
x > −4
Graph: open circle at −4, shade right. Interval: (−4, ∞)
Check: Test x = 0: −4(0) + 2 = 2 < 18 ✓. Test x = −6: −4(−6) + 2 = 26 < 18 ✗ (correct).
Graph: x > −4
Open circle at −4 · shaded right → (−4, ∞)
Solve and graph: −2x − 7 ≥ 1
Step 2 — Undo subtraction: add 7 to both sides (no flip).
−2x − 7 + 7 ≥ 1 + 7
−2x ≥ 8
Step 3 — Undo multiplication: divide both sides by −2 (NEGATIVE → FLIP the symbol).
−2x ÷ (−2) ≤ 8 ÷ (−2) ← ≥ becomes ≤
x ≤ −4
Graph: closed circle at −4, shade left. Interval: (−∞, −4]
Check: Test x = −6: −2(−6) − 7 = 5 ≥ 1 ✓. Test x = 0: −2(0) − 7 = −7 ≥ 1 ✗ (correct).
Graph: x ≤ −4
Closed circle at −4 · shaded left → (−∞, −4]
Solve and graph: x/3 + 4 > 6
Step 2 — Undo addition: subtract 4 from both sides (no flip).
x/3 + 4 − 4 > 6 − 4
x/3 > 2
Step 3 — Undo division: multiply both sides by 3 (positive → no flip).
x > 6
Graph: open circle at 6, shade right. Interval: (6, ∞)
Check: Test x = 9: 9/3 + 4 = 7 > 6 ✓. Test x = 3: 3/3 + 4 = 5 > 6 ✗ (correct).
Graph: x > 6
Open circle at 6 · shaded right → (6, ∞)
Guided Practice
Answers are in the Answer Key section.
Guided Practice Video: Two-Step Inequalities
Watch the guided practice walkthrough for two-step inequalities, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Solve and graph: 2x + 1 < 9
Hint: Step 2: subtract 1 from both sides → 2x < 8. Step 3: divide by 2 (positive, no flip) → x < 4. Open circle at 4, shade left.
Graph your answer (2x + 1 < 9):
Solve and graph: 4x − 3 ≥ 9
Hint: Step 2: add 3 to both sides → 4x ≥ 12. Step 3: divide by 4 (positive, no flip) → x ≥ 3. Closed circle at 3, shade right.
Graph your answer (4x − 3 ≥ 9):
Solve and graph: −3x + 6 > 0
Hint: Step 2: subtract 6 from both sides → −3x > −6. Step 3: divide by −3 (NEGATIVE → flip) → x < 2. Open circle at 2, shade left.
Graph your answer (−3x + 6 > 0):
Solve and graph: −5x − 4 ≤ 11
Hint: Step 2: add 4 to both sides → −5x ≤ 15. Step 3: divide by −5 (NEGATIVE → flip) → x ≥ −3. Closed circle at −3, shade right.
Graph your answer (−5x − 4 ≤ 11):
Solve and graph: x/2 − 3 < 1
Hint: Step 2: add 3 to both sides → x/2 < 4. Step 3: multiply by 2 (positive, no flip) → x < 8. Open circle at 8, shade left.
Graph your answer (x/2 − 3 < 1):
Key Vocabulary
Two-Step Inequality
An inequality that requires two inverse operations to isolate the variable.
Example: 2x + 3 > 11: subtract 3, then divide by 2.
Reverse Order of Operations
To solve, undo operations in the opposite order they were applied: undo + or − first, then × or ÷.
Example: In 3x − 5 ≤ 10: add 5 first, then divide by 3.
Flip the Symbol
Reverse the inequality direction only when multiplying or dividing both sides by a negative number.
÷ positive → stays
x > 4
÷ negative → flips
x < −4
Solution Set
All values of the variable that satisfy the inequality — graphed as a ray on a number line.
Example: x > 4: all numbers greater than 4.
Test Value
A number substituted into the original inequality to verify the solution set is correct.
Example: For x > 4, test x = 6: 2(6) + 3 = 15 > 11 ✓
Interval Notation
A compact way to express a solution set: (2, ∞) means x > 2; (−∞, 5] means x ≤ 5.
Example: x ≤ 5 → (−∞, 5]; x > 4 → (4, ∞)
Interval Notation — Visual Reference
Interactive Practice — 5 Questions
When solving a two-step inequality, at which step can the inequality symbol flip?
Solve: −3x + 1 > 7
Which inequality does NOT require flipping the symbol?
Solve: 2x + 6 ≤ 2
Solve: −2x − 4 < 6
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Solve and graph: 2x + 5 > 13
Solve and graph: −2x + 4 > 10
Solve and graph: x/2 + 3 > 7
Solve and graph: −3x − 5 ≤ 4
Solve and graph: 4x − 7 ≥ 1
Problem 1 graph:
Problem 2 graph:
Problem 3 graph:
Problem 4 graph:
Problem 5 graph:
Common Mistakes
Flipping the inequality after subtracting or adding — the sign only flips when multiplying or dividing by a negative.
Only flip the inequality symbol when you multiply or divide both sides by a negative number.
Undoing multiplication before undoing addition/subtraction — e.g., dividing by 3 before subtracting 6 in 3x + 6 > 15.
Undo addition/subtraction first, then undo multiplication/division — same order as two-step equations.
Forgetting to flip the sign when the coefficient is negative after combining like terms.
Check the coefficient of x after simplifying. If it's negative, flip the sign when you divide.
Dropping the inequality symbol mid-solution and treating the problem like an equation.
Write the inequality symbol in every line of your work — it's easy to drop it and forget to flip.
Visual: Forgetting to Flip — −4x + 2 < 18 → −4x < 16
✗ Common Error
−4x < 16
→ x < −4 (symbol NOT flipped — wrong!)
Divided by −4 but forgot to flip < to >.
✓ Correct
−4x < 16
→ x > −4 (symbol flipped — correct)
Divided by negative → symbol flipped.
Math Tips
Same order as two-step equations: undo + / − first, then × / ÷. The only difference is the flip rule at Step 3.
Circle the coefficient before you start. If it's negative, write "FLIP" next to it as a reminder before you divide.
Test from the shaded region: pick a "nice" number clearly inside the solution (e.g., if x > 3, test x = 5). Avoid the boundary value.
SAT/ACT tip: rewrite if x is on the right. If you end up with 4 < x, rewrite as x > 4 so the variable is on the left.