Key Concepts of Inequalities

Why This Matters

Inequalities are the backbone of comparison thinking in algebra—they let you describe relationships where things aren't perfectly equal, which is actually how most real-world situations work. You're being tested on your ability to manipulate inequalities using properties and operations, interpret their solutions, and apply them to problems involving constraints like budgets, speed limits, or resource allocation. Understanding inequalities also sets you up for success in graphing systems and optimization problems later on.

Don't just memorize the symbols and steps—know why the inequality sign flips when you multiply by a negative, how compound inequalities combine conditions, and what different solution representations actually mean. The concepts here connect to number sense, algebraic manipulation, and mathematical modeling, all of which show up repeatedly on exams.

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Understanding Inequality Symbols and Definitions

Before you can solve inequalities, you need to speak their language fluently. Inequality symbols express the relative size or position of two quantities on a number line.

Definition of Inequalities

• An inequality compares two expressions—it states that one value is greater than, less than, or possibly equal to another
• Variables, constants, and operations can all appear in inequalities, just like in equations
• Unlike equations, inequalities describe a range of possible values rather than a single solution

Inequality Symbols

• The symbol $<$ means "less than" and $>$ means "greater than"—the arrow always points toward the smaller value
• The symbols $\leq$ and $\geq$ include equality—read as "less than or equal to" and "greater than or equal to"
• Strict inequalities ($<$, $>$) exclude the boundary value, while non-strict inequalities ($\leq$, $\geq$) include it

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Properties That Govern Inequality Operations

These properties are the rules of the game—they tell you what you can do to both sides of an inequality without breaking it. The key insight is that inequalities behave like equations, except when negative numbers enter the picture.

Addition and Subtraction Property

• Adding or subtracting the same number from both sides preserves the inequality—this works exactly like equations
• No sign change required regardless of whether you add positive or negative values
• This property lets you isolate variable terms by moving constants to the other side

Multiplication and Division Property

• Multiplying or dividing by a positive number keeps the inequality direction unchanged—straightforward and predictable
• Multiplying or dividing by a negative number REVERSES the inequality sign—this is the most common error on exams
• Why the flip? Multiplying by a negative reflects values across zero on the number line, reversing their order

Transitive Property

• If $a < b$ and $b < c$, then $a < c$—values maintain their relative order through a chain
• Useful for combining information from multiple inequalities into one conclusion
• This property underlies how we solve compound inequalities and compare multiple quantities

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Solving Techniques for Different Inequality Types

Different inequality structures require different approaches, but all rely on the same core properties. The goal is always to isolate the variable while tracking what happens to the inequality sign.

Solving Linear Inequalities

• Isolate the variable using inverse operations—treat it like an equation until you multiply or divide by a negative
• Express solutions in multiple forms: inequality notation ($x > 3$), interval notation ($(3, \infty)$), or graphically
• Check your answer by substituting a value from your solution set back into the original inequality

Inequalities with Variables on Both Sides

• Collect variable terms on one side first—subtract the smaller variable term from both sides to keep coefficients positive when possible
• Then isolate the variable using the same techniques as simple linear inequalities
• Watch for special cases: some inequalities are always true (all real numbers) or never true (no solution)

Absolute Value Inequalities

• Absolute value measures distance from zero—so $|x| < 3$ means $x$ is within 3 units of zero
• "Less than" creates an AND compound inequality: $|x| < 3$ becomes $-3 < x < 3$
• "Greater than" creates an OR compound inequality: $|x| > 3$ becomes $x < -3$ or $x > 3$

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Compound Inequalities and Combined Conditions

Compound inequalities let you express more complex constraints by combining two conditions. The connecting word—AND or OR—determines whether you need the overlap or the union of solutions.

AND Compound Inequalities

• Both conditions must be satisfied simultaneously—the solution is the intersection (overlap) of the two individual solutions
• Can be written in compact form: $-2 < x < 5$ means $x > -2$ AND $x < 5$
• Graphically, shade only the region where both conditions overlap on the number line

OR Compound Inequalities

• At least one condition must be true—the solution is the union of both individual solutions
• Cannot be written in compact form—you must state both parts: $x < -2$ or $x > 5$
• Graphically, shade all regions that satisfy either condition—often two separate rays

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Representing and Communicating Solutions

Being able to express your answer in different formats is essential—exams may require a specific notation, and each representation highlights different aspects of the solution.

Graphing Inequalities on a Number Line

• Open circles (○) mark excluded endpoints—use for strict inequalities ($<$ or $>$)
• Closed circles (●) mark included endpoints—use for non-strict inequalities ($\leq$ or $\geq$)
• Shade the region containing all solutions—the direction of shading shows which values satisfy the inequality

Interval Notation

• Parentheses ( ) indicate excluded endpoints—corresponds to open circles and strict inequalities
• Brackets [ ] indicate included endpoints—corresponds to closed circles and non-strict inequalities
• Use $\infty$ or $-\infty$ for unbounded intervals—always with parentheses, since infinity isn't a reachable value

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Real-World Applications

Inequalities aren't just abstract math—they model situations where you have limits, constraints, or ranges of acceptable values. Translating word problems into inequalities is a key skill.

Applications in Real-World Problems

• Budgeting and resource constraints use inequalities to express limits: "spend no more than $100" becomes $x \leq 100$
• Optimization problems often involve finding the maximum or minimum value within inequality constraints
• Interpreting solutions in context matters—a negative number of items or fractional people may not make sense even if mathematically valid

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Quick Reference Table

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Self-Check Questions

1. What operation requires you to reverse the inequality sign, and why does this happen?

2. How do the solutions to $|x| < 4$ and $|x| > 4$ differ in terms of their graphical representation?

3. Compare and contrast AND vs. OR compound inequalities—when does each type produce a bounded solution region?

4. If you solve an inequality and get $x > 5$, write this solution in interval notation and describe how you would graph it.

5. A student claims that $-2x > 6$ simplifies to $x > -3$. Identify and correct the error, explaining which property was misapplied.