2.2One-Step Inequalities
Solve inequalities in one move — just like one-step equations — but with one critical twist: multiplying or dividing by a negative number flips the inequality symbol.
Why This Matters
One-step inequalities are the simplest form of constraint problems. The key rule — flipping the sign when multiplying or dividing by a negative — is a concept you'll apply in Algebra 2, Precalculus, and optimization problems.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
Why does the inequality symbol reverse direction when you multiply or divide both sides by a negative number?
Lesson Overview
A one-step inequality requires exactly one inverse operation to isolate the variable — the same strategy as a one-step equation. You can add, subtract, multiply, or divide both sides by the same value to keep the inequality balanced. There is one crucial rule that does not apply to equations: when you multiply or divide both sides by a negative number, you must flip (reverse) the inequality symbol. For example, if you divide both sides of −3x > 12 by −3, the > becomes <, giving x < −4. This happens because multiplying by a negative reverses the order of numbers on the number line. Always graph your solution set and check by substituting a value from the shaded region.
The Flip Rule — Most Important Rule in This Chapter
When you multiply or divide both sides of an inequality by a negative number, you must reverse (flip) the inequality symbol.
Positive divisor — symbol stays
2x > 8
÷ 2 on both sides
x > 4 ✓ symbol unchanged
Negative divisor — symbol flips ↕
−2x > 8
÷ (−2) on both sides
x < −4 ← symbol flipped!
Worked Examples
Solve and graph: x + 5 > 12
Identify: x has +5 added to it.
Inverse: subtract 5 from both sides.
x + 5 − 5 > 12 − 5
x > 7
Symbol check: subtracted a positive → NO flip.
Graph: open circle at 7, shade right → (7, ∞)
Check: test x = 10: 10 + 5 = 15 > 12 ✓ | test x = 4: 4 + 5 = 9 > 12 ✗ (correct)
Graph: x > 7
Open circle at 7 · shaded right → (7, ∞)
Solve and graph: x − 8 ≤ 3
Identify: x has 8 subtracted from it.
Inverse: add 8 to both sides.
x − 8 + 8 ≤ 3 + 8
x ≤ 11
Symbol check: added a positive → NO flip.
Graph: closed circle at 11, shade left → (−∞, 11]
Check: test x = 0: 0 − 8 = −8 ≤ 3 ✓ | test x = 15: 15 − 8 = 7 ≤ 3 ✗ (correct)
Graph: x ≤ 11
Closed circle at 11 · shaded left → (−∞, 11]
Solve and graph: 4x < 20
Identify: x is multiplied by 4 (positive).
Inverse: divide both sides by 4.
4x ÷ 4 < 20 ÷ 4
x < 5
Symbol check: divided by a POSITIVE → NO flip.
Graph: open circle at 5, shade left → (−∞, 5)
Check: test x = 3: 4(3) = 12 < 20 ✓ | test x = 7: 4(7) = 28 < 20 ✗ (correct)
Graph: x < 5
Open circle at 5 · shaded left → (−∞, 5)
Solve and graph: −3x > 12
Identify: x is multiplied by −3 (NEGATIVE).
Inverse: divide both sides by −3.
−3x ÷ (−3) > 12 ÷ (−3)
*** Dividing by NEGATIVE → FLIP the symbol: > becomes < ***
x < −4
Graph: open circle at −4, shade left → (−∞, −4)
Check: test x = −6: −3(−6) = 18 > 12 ✓ | test x = 0: −3(0) = 0 > 12 ✗ (correct)
Graph: x < −4
Open circle at −4 · shaded left → (−∞, −4)
A student earns $15 per hour babysitting. She wants to earn more than $90. Write and solve an inequality for the number of hours h she must work.
Translate: earnings = 15h; must be more than 90.
15h > 90
Divide both sides by 15 (positive):
h > 6
Answer: She must work more than 6 hours.
Check: 15(7) = 105 > 90 ✓
Graph: h > 6 (hours)
Open circle at 6 · shaded right → (6, ∞)
Guided Practice
Answers are in the Answer Key section.
Guided Practice Video: One-Step Inequalities
Watch the guided practice walkthrough for one-step inequalities, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Solve and graph: x + 8 < 15
Hint: Subtract 8 from both sides. Adding/subtracting never flips the symbol.
Graph your answer (x + 8 < 15):
Solve and graph: x − 6 ≥ −2
Hint: Add 6 to both sides. Symbol stays the same.
Graph your answer (x − 6 ≥ −2):
Solve and graph: 5x > −25
Hint: Divide both sides by 5 (positive). Symbol stays.
Graph your answer (5x > −25):
Solve and graph: −6x < 18
Hint: Divide both sides by −6 (NEGATIVE). Flip the symbol: < becomes >.
Graph your answer (−6x < 18):
Solve and graph: x/3 ≤ 4
Hint: Multiply both sides by 3 (positive). Symbol stays.
Graph your answer (x/3 ≤ 4):
Key Vocabulary
One-Step Inequality
An inequality that requires exactly one inverse operation to isolate the variable.
Example: x + 5 > 12 → subtract 5 → x > 7
Inverse Operation
The opposite operation used to undo another. Addition ↔ Subtraction; Multiplication ↔ Division.
Example: To undo +8, subtract 8. To undo ×3, 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 make the inequality true — graphed as a ray on a number line.
Example: x > 4: all numbers greater than 4 (infinitely many solutions)
Interval Notation
A compact way to write a solution set. Parenthesis = excluded; bracket = included.
Example: x > 4 → (4, ∞); x ≤ 3 → (−∞, 3]
Test Value
A number substituted into the original inequality to verify the solution is correct.
Example: For x > 4, test x = 6: 6 > 4 ✓ (in solution set)
Interval Notation — Visual Reference
Interactive Practice — 5 Questions
When do you reverse the inequality symbol?
Solve: −4x > 12
Which of the following does NOT require flipping the inequality symbol?
Solve: x/3 ≥ −6
Solve: −x ≤ 7
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Solve and graph: x + 3 > 7
Solve and graph: −2x > 8
Solve and graph: x/5 < 3
Solve and graph: −4x ≤ 20
Solve and graph: x − 5 < 2
Problem 1 graph:
Problem 2 graph:
Problem 3 graph:
Problem 4 graph:
Problem 5 graph:
Common Mistakes
Forgetting to flip the inequality when dividing or multiplying by a negative number — e.g., −2x > 8 → x > −4.
Dividing by −2 flips the sign: −2x > 8 → x < −4.
Using a closed circle for strict inequalities (< or >) on the number line.
Open circle for < and >. Closed circle for ≤ and ≥.
Writing the solution set with the wrong bracket in interval notation — e.g., [−4, ∞) for x > −4.
x > −4 is (−4, ∞) with a parenthesis. Brackets are only for ≤ or ≥.
Flipping the symbol when subtracting a negative — e.g., x − (−3) > 5 → flipping.
Adding or subtracting — even a negative number — NEVER flips the symbol. Only multiplication/division by a negative does.
Visual: Forgetting to Flip — −2x > 8
✗ Common Error
−2x > 8
→ x > −4 (symbol NOT flipped — wrong!)
Divided by −2 but forgot to flip > to <.
✓ Correct
−2x > 8
→ x < −4 (symbol flipped — correct)
Divided by negative → symbol flipped.
Visual: Open vs. Closed Circle
Open Circle ○
Endpoint NOT included
Use for < or >
x > 3
Closed Circle ●
Endpoint IS included
Use for ≤ or ≥
x ≥ 3
Math Tips
The Flip Rule: multiply or divide by a negative → flip the symbol. This is the only time the symbol changes. Adding or subtracting never flips it.
Graph and check: after solving, graph the solution and test a value from the shaded region in the original inequality to confirm.
Rewrite if needed: if the variable ends up on the right (e.g., 3 < x), rewrite as x > 3 for clarity.
SAT/ACT tip: 0.5 is positive — do NOT flip when dividing by 0.5. Only negative values trigger the flip.