Math · Algebra & word problems
Inequalities and Solution Sets
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Start here: key ideas
- An inequality describes a set of values.
- Open endpoints exclude; closed endpoints include.
- Multiplying or dividing by a negative reverses order.
- Context determines whether whole-number answers make sense.
What you’ll be able to do
- Solve and graph one-variable inequalities.
- Reverse the sign after negative multiplication or division.
- Interpret bounds in context.
Solve like an equation while protecting the order relationship.
Complete Solving Linear Equations first.
A set rather than one answer
An inequality compares quantities with <, >, ≤, or ≥. Its solution set includes every value making it true. The boundary separates solutions from nonsolutions. An open circle excludes the boundary; a closed circle includes it.
OpenStax, via Mathematics LibreTexts · CC BY 4.0 · https://math.libretexts.org/Bookshelves/Algebra/Elementary_Algebra_1e_(OpenStax)/02%3A_Solving_Linear_Equations_and_Inequalities/2.07%3A_Solve_Linear_Inequalities · https://creativecommons.org/licenses/by/4.0 · Source image reproduced unchanged.
OpenStax, via Mathematics LibreTexts · CC BY 4.0 · https://math.libretexts.org/Bookshelves/Algebra/Elementary_Algebra_1e_(OpenStax)/02%3A_Solving_Linear_Equations_and_Inequalities/2.07%3A_Solve_Linear_Inequalities · https://creativecommons.org/licenses/by/4.0 · Source image reproduced unchanged.
Solve and protect order
Use balanced inverse operations as with equations. One special rule preserves order: multiplying or dividing both sides by a negative reverses the sign. For −3x>12, division by −3 gives x<−4. A compound inequality such as 2≤x<5 requires both comparisons: close 2, open 5, and shade between.
Interpret a bound
“At most 40” means x≤40; “at least 6” means x≥6; “more than 10” means x>10. Context can restrict answers to whole counts. If $7 tickets must fit a $30 budget, 7t≤30 means no more than 4 whole tickets.
Worked examples
Worked example 1: 5x−7≤18. Add 7: 5x≤25. Divide by 5: x≤5. Close 5 and shade left.
Worked example 2: 8−2x>14. Subtract 8: −2x>6. Divide by −2 and reverse: x<−3. Testing −4 gives 16>14.
A reliable way to reason
The reversal rule follows from number-line order. Since 2<5, multiplying both values by −1 gives −2>−5: reflection across zero reverses left and right. Adding or subtracting does not reflect the line, so it does not reverse the symbol. Keep this cause in mind instead of flipping whenever a negative merely appears.
Graphing is also a verification method. For x>3, test 4; it works, so shade toward 4 and larger values. Test the boundary separately: 3>3 is false, so use an open circle. For x≤−2, −2 itself works, requiring a closed circle, and −3 confirms shading left. Context may convert a numerical boundary into a usable count. If 6 notebooks must cost less than $25, 6n<25 gives n<25/6 dollars, but if n is a count instead of price, only whole-number values would be allowed. Always define the variable before deciding whether rounding or discrete interpretation is appropriate.
Guided reasoning and error checks
Worked context: A van can carry at most 1,200 pounds. Existing cargo weighs 375 pounds, and each crate weighs 55 pounds. If c is the number of additional crates, write 375+55c≤1,200. Subtract 375 to get 55c≤825; divide by positive 55 to get c≤15. Because c counts crates, the feasible solutions are whole numbers from 0 through 15. Substitution confirms the boundary: 375+55(15)=1,200, so 15 is included.
Optional extension: solving a compound bound
For a compound bound, solve all parts together. From −4<2x+2≤10, subtract 2 throughout: −6<2x≤8. Divide every part by positive 2: −3<x≤4. The graph opens at −3, closes at 4, and shades between. Testing 0 confirms the interior.
Final self-check
When writing an answer from a graph, translate the endpoint and arrow separately. An open point at 2 with shading right means x>2; a closed point at 2 with shading left means x≤2. Neither the circle nor the arrow alone is enough. Read both features, then test one shaded value and one unshaded value against the original inequality.
Key terms
- inequality: A comparison using <, >, ≤, or ≥.
- solution set: All values that make an inequality true.
- boundary: The endpoint separating solutions from nonsolutions.
- open circle: A graph endpoint that is excluded.
- closed circle: A graph endpoint that is included.
- compound inequality: Two comparisons describing one set.
Summary
- Isolate the variable with balanced operations.
- Reverse only after multiplying or dividing by a negative.
- Graph the endpoint and test the shaded side.
- Translate words such as “at most” and “at least” into inclusive bounds that fit the context.
Negative multiply or divide? Flip the sign. Open excludes; closed includes.
Sources
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