뒤로Study Notes: Solving Inequalities in Precalculus
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Inequalities in Precalculus
Introduction to Inequalities
Inequalities are mathematical statements that compare two expressions using symbols such as <, >, ≤, or ≥. Unlike equations, inequalities often have infinitely many solutions, which are best described using interval notation.
Equation: An equality between two expressions, solved for specific values.
Inequality: A comparison between two expressions, solved for ranges of values.
Interval Notation: Used to describe solution sets for inequalities, e.g., or .
Solving Linear Inequalities
Linear inequalities are solved using similar techniques as linear equations, with special attention to the direction of the inequality sign.
Key Principle: If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
Example: Solve .
Subtract 7:
Divide by -2 (reverse sign):
Solution set:
No Solution Example:
Subtract : (which is false)
Solution set: (the empty set)
Compound Inequalities
Compound inequalities involve two or more inequalities joined by "and" or "or". They can be expressed as "three-sided" inequalities.
Intersection ("and"): Solution is the overlap of the individual solution sets.
Union ("or"): Solution is any value that satisfies at least one inequality.
Example: Solve .
Add 4:
Divide by 2:
Solution set:
Absolute Value Inequalities
Absolute value inequalities involve expressions like or . These represent distances from zero or another point.
Key Principle:
is equivalent to
is equivalent to or
Example: Solve
Solution set:
Example: Solve
or
Solution set:
Example: Solve
Subtract 3:
Divide by 3:
Solution set:
Example: Solve
or
or
Solution set:
"Sneaky" Absolute Value Inequalities:
is never true (absolute value is always non-negative).
is always true (absolute value is always greater than or equal to 0).
Solution set:
Solving Nonlinear Inequalities
Nonlinear inequalities involve polynomials or rational expressions. The solution process is more involved than for linear inequalities.
Step 1: Move everything to one side so that 0 is on the other side.
Step 2: Factor the expression as much as possible.
Step 3: Find the critical values (values that make each factor zero or undefined).
Step 4: Set up a number line and test intervals between critical values.
Step 5: Label each interval with the sign of the expression and select intervals that satisfy the inequality.
Example: Solve
Factor:
Critical values: ,
Test intervals: , ,
Solution set:
Example: Solve
Move all terms to one side:
Simplify:
Critical value:
Test intervals: and
Solution set:
Summary Table: Types of Inequalities and Solution Strategies
Type | Form | Solution Strategy | Example |
|---|---|---|---|
Linear | Isolate , reverse sign if dividing by negative | ||
Compound | Solve both inequalities, find intersection | ||
Absolute Value | Split into two inequalities | ||
Nonlinear | Factor, find critical values, test intervals |
Key Points to Remember
Always reverse the inequality sign when multiplying or dividing by a negative number.
Use interval notation to express solution sets.
For absolute value inequalities, interpret the meaning as distance from a point.
For nonlinear inequalities, always test intervals between critical values.
Some inequalities have no solution or are always true; check for these cases.
Additional info: These notes expand on the handwritten and printed content, providing full academic context, definitions, and examples for each type of inequality discussed in the source material.
