3. Linear Inequalities & Interval Notation
Before you start
- Solve single-variable linear equations using inverse operations
- Recognize the four inequality symbols and read them aloud correctly
- Place positive and negative numbers on a number line in the correct order
- Distinguish open versus closed circles on a graph
By the end you'll be able to
- Solve a linear inequality and graph the solution set on a number line
- Apply the flip rule whenever multiplying or dividing both sides by a negative
- Translate among inequality, interval, and graph representations without ambiguity
- Solve compound inequalities and express the result as a union or intersection
- Choose between inclusive and exclusive endpoints based on the original symbol
Linear Inequalities & Interval Notation
A linear inequality is just a linear equation with <, >, ≤, or ≥ instead of =. The
solving steps are nearly identical, with one twist that catches everyone.
The procedure
Solve like an equation:
- Distribute and combine like terms.
- Move variable terms to one side, constants to the other.
- Divide by the variable’s coefficient.
The twist: if you multiply or divide both sides by a negative number, flip the
inequality direction. So −2x > 6 becomes x < −3, not x > −3.
Why the flip?
Multiplying by a negative reflects the number line through 0. A statement like “5 is greater than 2” reflects to “−5 is less than −2.” The relative ordering reverses. The flip rule preserves the truth of the statement under reflection.
Interval notation
| Notation | Meaning |
|---|---|
(a, b) |
a < x < b (both endpoints excluded) |
[a, b] |
a ≤ x ≤ b (both endpoints included) |
[a, b) |
a ≤ x < b (left included, right excluded) |
(−∞, a] |
x ≤ a (negative infinity always uses an open paren) |
(a, ∞) |
x > a |
Square brackets = “I include this endpoint.” Round parens = “I exclude this endpoint.” Infinity always uses a paren — it’s a direction, not a number you can include.
Compound inequalities
- AND (intersection):
−2 < x < 5means x must be greater than −2 and less than 5. The solution is one bounded interval. - OR (union):
x < −2 OR x > 5means x is in either ray. The solution is two unbounded intervals, written with the union symbol:(−∞, −2) ∪ (5, ∞).
When you have a three-section compound −2 < 2x − 3 ≤ 5, perform every operation on all three
parts simultaneously.
Connection to ML
- Algebraic structure learned. A linear inequality cuts the number line (or the plane) into a half-region; intersecting many of them produces a bounded convex feasible region.
- ML object with the same structure. Constrained optimization — non-negative matrix factorization, support-vector-machine margin constraints, linear programming relaxations — is a system of linear inequalities defining a polytope, with an objective optimized over that polytope.
- Computational payoff. Solvers exploit convexity to find the optimum in polynomial time. Recognizing your constraints as a linear-inequality system is what lets you reach for an LP solver instead of writing a slow custom search.
Common mistakes
These are the traps learners hit most often on this topic. Knowing them in advance is half the fix.
Forgetting to flip the inequality on a negative divide
becomes , not . Multiplying or dividing both sides by a negative number reverses the direction of the inequality. Mixing up open and closed brackets
is — closed bracket on the included endpoint. is — open paren on the excluded endpoint. Infinity always uses a paren. Treating compound 'OR' like a single inequality
OR is two disjoint regions. Don’t try to compress into a single chained inequality — it doesn’t exist for ‘OR’.
Practice problems
Try each on paper first. Click Show solution only after you've made a real attempt.
- Problem 1Solve and write in interval notation:
. Show solution
- Subtract 1:
. - Divide by 2:
.
Interval:
. - Subtract 1:
- Problem 2Solve:
. Show solution
- Subtract 2:
. - Divide by
— flip: .
Interval:
. - Subtract 2:
- Problem 3Solve the compound inequality:
. Show solution
- Subtract 5:
. - Divide by 3:
.
Interval:
. - Subtract 5:
- Problem 4Translate to interval notation:
OR . Show solution
→ . → .
Union:
. - Problem 5Solve:
. Show solution
- Subtract 3:
. - Multiply by
— flip: .
Interval:
. - Subtract 3:
- Problem 6Solve:
. Show solution
- Distribute:
. - Subtract
, add 5: . - Divide by 3:
.
Interval:
. - Distribute:
- Problem 7Express in interval notation: x is at least -1 and less than 5.
Show solution
‘At least’ is inclusive, ‘less than’ is strict.
Interval:
.
Practice quiz
- Question 1Solve: 2x + 1 < 9
- Question 2Solve: -3x ≥ 12
- Question 3Interval notation for x ≥ 2:
- Question 4Solve: 5 < 2x + 3 ≤ 11
- Question 5Solve: 4 - 2x > 10. Give the inequality.
- Question 6Write ‘x is between -2 and 5, inclusive’ in interval notation.
- Question 7Solve and write in interval notation: -3 ≤ 2x - 1 < 7
- Question 8Compound: x < 1 OR x > 4 in interval notation:
- Question 9ML connection: a ‘feasible region’ in linear programming is bounded by:
- Reflection 10Why does flipping the inequality when multiplying by a negative matter geometrically?
Week 3 recap
You solved single and compound linear inequalities, mastered interval notation, and internalized the one rule unique to inequalities: flip the direction when multiplying or dividing by a negative. Compound inequalities introduced the union and intersection language you will reuse in absolute-value problems next week and in probability later. Three trap families fell this week: the missed-flip trap (forgetting to reverse on a negative operation), the inclusive-vs-exclusive endpoint trap (wrong bracket on a boundary), and the AND-vs-OR confusion (writing a single chained inequality where two disjoint rays were needed). Each outcome compounds: half-plane intersections become feasible regions in linear programming; interval notation becomes the domain language for functions; the flip rule generalizes to any monotonic-decreasing transformation, including the negation that flips a maximization into a minimization in optimization.
Coming next: Week 4 — Absolute Value Equations & Inequalities
Next week wraps absolute value mechanics around equations and inequalities. You will learn the structural difference between |x| <= k (one bounded interval, an AND/intersection) and |x| >= k (two unbounded rays, an OR/union), the same pattern that will reappear when you study L1 regularization and confidence intervals. Lesson 4 also includes a cumulative review covering weeks 1-3.
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