Solving and Understanding Inequalities in College Algebra

Inequalities

Definition and Basic Concepts

An inequality is a mathematical sentence that uses the symbols <, >, ≤, or ≥ to compare two expressions.

• Example: $3x - 5 < 6 - 2x$

• To solve an inequality is to find all values of the variable that make the inequality true.

• Each of these values is a solution of the inequality, and the set of all such solutions is its solution set.

• Inequalities that have the same solution set are called equivalent inequalities.

Principles for Solving Inequalities

Addition and Multiplication Principles

• For any real numbers a, b, and c:

• Addition Principle for Inequalities: If $a < b$ is true, then $a + c < b + c$ is true.

• Multiplication Principle for Inequalities:

  • If $a < b$ and $c > 0$, then $ac < bc$.

  • If $a < b$ and $c < 0$, then $ac > bc$.

• Similar statements hold for ≤, ≥, and >.

• Important: When both sides of an inequality are multiplied or divided by a negative number, the inequality sign must be reversed.

Solving Linear Inequalities

Examples and Applications

• Example 1: Solve $3x - 5 < 6 - 2x$

• Example 2: Solve $13 - 7x rac{1}{2} > 10 - 4$

• Example 3: Solve $-3 < 2x + 5 ext{ and } 2x + 5 ext{ ≤ } 7$ (compound inequality)

Compound Inequalities

Conjunctions and Disjunctions

When two inequalities are joined by the word "and" or "or," a compound inequality is formed.

• Conjunction contains the word "and." Example: $-3 < 2x + 5 ext{ and } 2x + 5 ext{ ≤ } 7$ This can be written as $-3 < 2x + 5 ext{ ≤ } 7$.

• Disjunction contains the word "or." Example: $2x - 5 ext{ ≤ } -7 ext{ or } 2x - 5 > 1$

Graphing Solution Sets and Interval Notation

Representing Solutions

• After solving an inequality, the solution set can be represented on a number line (graphically) and in interval notation.

• Interval Notation: Uses parentheses ( ) for open intervals and brackets [ ] for closed intervals. Example: $x > 2$ is written as $(2, \, \infty)$.

Applications of Inequalities

Word Problems and Real-World Contexts

• Example 1: Natalia can be paid in one of two ways for her interior decorating job: Plan A: $1200 + $10 per hour Plan B: $120 per hour Suppose a job takes n hours. For what values of n is Plan B better for Natalia?

• Example 2: Curt can be paid in one of two ways for the furniture he sells: Plan A: Salary of $900 per month plus a commission of 10% of sales. Plan B: Salary of $1150 per month, plus a commission of 15% of sales in excess of $9000 per month. For what amount of monthly sales is Plan B better than Plan A, assuming Curt's sales are always more than $9000?

Finding the Domain of Functions

Domain Restrictions Involving Inequalities

• The domain of a function is the set of all real numbers for which the function is defined.

• For functions involving square roots or denominators, inequalities are used to determine the domain.

• Example 1: $f(x) = \sqrt{5 - 2x} + 1$ Set $5 - 2x \geq 0$ and solve for $x$.

• Example 2: $f(x) = \frac{9}{x^2 - 7x}$ Set $x^2 - 7x \neq 0$ and solve for $x$.