Ch. 8: Inequalities and Absolute Value – Review of Linear Inequalities in One Variable

Ch. 8: Inequalities and Absolute Value

8.1 Review of Linear Inequalities in One Variable

This section reviews the process of solving linear inequalities in one variable, including the rules for manipulating inequalities, the use of interval notation, and the graphical representation of solution sets. It also introduces compound inequalities and the concept of set intersection.

Definition and Properties of Inequalities

• Inequality: A mathematical statement that compares two expressions using symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).

• Key Property: When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.

• Example: To make $-5 > -9$ a true statement, multiply both sides by $-1$ and reverse the sign: $5 < 9$.

Solving Linear Inequalities

• Solve inequalities using the same steps as equations, with the additional rule for multiplying/dividing by negatives.

• Example: Solve $\frac{3x+1}{4} > -5$.

• Step 1: Multiply both sides by 4: $3x+1 > -20$

• Step 2: Subtract 1: $3x > -21$

• Step 3: Divide by 3: $x > -7$

• Solution in interval notation: $(-7, \infty)$

Interval Notation and Number Lines

• Interval Notation: Uses parentheses ( ) for strict inequalities (<, >) and brackets [ ] for inclusive inequalities (≤, ≥).

• Infinity Symbols: $\infty$ for positive infinity, $-\infty$ for negative infinity. Always use parentheses with infinity.

• Example: $x > 3$ is written as $(3, \infty)$.

• Example: $x \leq 6$ is written as $(-\infty, 6]$.

Solving and Graphing Compound Inequalities

• Compound Inequality: An inequality that combines two simple inequalities, often using the word "and" or "or".

• Example: $11 < x \leq 15$ is written as $(11, 15]$ in interval notation.

• Graph the solution on a number line and rewrite in interval notation.

Special Cases in Inequalities

• If the variable is eliminated and a true statement remains (e.g., $3 < 5$), the solution is all real numbers ($\mathbb{R}$).

• If the variable is eliminated and a false statement remains (e.g., $20 < 9$), there is no solution ($\emptyset$).

Solving Chains of Inequalities

• A chain of inequalities (e.g., $5 < 3x-4 \leq 15$) can be solved as two separate inequalities and then combined.

• Express the solution set in both graph and interval form.

• Example: $9 < 3x \leq 19$

• Step 1: Divide all parts by 3: $3 < x \leq \frac{19}{3}$

• Interval notation: $(3, \frac{19}{3}]$

Intersection of Sets and Compound Inequalities

• The intersection of two sets $A$ and $B$, written $A \cap B$, is the set of elements common to both $A$ and $B$.

• For compound inequalities with "and", the solution is the intersection of the solution sets.

• Example: $A = \{3, 4, 5, 6, 7\}$, $B = \{5, 6\}$, then $A \cap B = \{5, 6\}$

• Use "AND" for intersections.

Application Example: Students with Multiple Criteria

• Let set $A$ be students taking 12 or more credit hours.

• Let set $B$ be students working 20 or more hours per week.

• The intersection $A \cap B$ is students who meet both criteria ("super busy").

Summary Table: Interval Notation Symbols

Inequality	Interval Notation	Graphical Representation
$x > a$	$(a, \infty)$	Open circle at $a$, arrow right
$x \geq a$	$[a, \infty)$	Closed circle at $a$, arrow right
$x < b$	$(-\infty, b)$	Open circle at $b$, arrow left
$x \leq b$	$(-\infty, b]$	Closed circle at $b$, arrow left
$a < x \leq b$	$(a, b]$	Open at $a$, closed at $b$

Additional info: This table summarizes the correspondence between inequality symbols, interval notation, and their graphical representation on a number line.

Key Points to Remember

• Always reverse the inequality sign when multiplying or dividing by a negative number.

• Use interval notation to express solution sets.

• Compound inequalities with "and" represent intersections of sets.

• Check for special cases: all real numbers or no solution.