4.3 Solving and Graphing Linear Inequalities

Inequality Symbols, Number Line Graphing, and Interval Notation

A linear inequality expresses a relationship of inequality between algebraic expressions. Unlike equations, which typically have a single discrete solution, linear inequalities possess continuous solution sets containing infinitely many real numbers.

Inequality Symbols and Verbal Translations

Graphing on a One-Dimensional Number Line

To graph a single-variable inequality along the real number line:

1. Plot the Endpoint (Boundary): Draw an open circle if the inequality is strict ($<, >$) or a solid filled circle if the inequality is inclusive ($\le, \ge$).
2. Shade the Solution Region:
   • If the variable is on the left and the symbol points left ($x < k$ or $x \le k$), shade to the left (toward $-\infty$).
   • If the variable is on the left and the symbol points right ($x > k$ or $x \ge k$), shade to the right (toward $+\infty$).

                    Number Line Graphing Conventions
                    
   Strict Inequality:  x > 2 (Open Circle at 2, shaded right)
   <───┼───┼───┼───┼───┼───○════════════════════════════════════════════════►
      -2  -1   0   1   2   3   4   5
                          (2, ∞)
                          
   Inclusive Inequality: x ≤ -1 (Closed Circle at -1, shaded left)
   ◄═══════════════════●───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───>
      -4  -3  -2  -1   0   1   2   3   4   5
       (-∞, -1]

Interval Notation Conventions

Interval notation provides a standardized format for writing solution sets using boundary pairs:

• Parentheses $(, )$: Indicate open (excluded) endpoints, strict inequalities ($<, >$), and infinity ($-\infty, +\infty$).
• Brackets $[, ]$: Indicate closed (included) endpoints and inclusive inequalities ($\le, \ge$).

Solving Linear Inequalities and the Sign Reversal Rule

Solving linear inequalities follows the exact same procedures as solving linear equations (distribute, combine terms, isolate variables), with one critical exception.

The Negative Multiplication/Division Rule

The Cardinal Rule of Inequalities: Whenever you multiply or divide both sides of an inequality by a negative number, you MUST reverse (flip) the inequality symbol.

$\text{If } a < b \text{ and } c < 0, \quad \text{then } a \cdot c > b \cdot c \quad \text{and} \quad \frac{a}{c} > \frac{b}{c}$

Why the Sign Flips: Geometric Demonstration

Consider the true numerical inequality $2 < 5$:

• On the positive side of the number line, $2$ is to the left of $5$ ($2 < 5$).
• Multiply both sides by $-1$: $2(-1) \quad \text{and} \quad 5(-1) \implies -2 \quad \text{and} \quad -5$
• On the negative side of the number line, $-2$ is to the right of $-5$, meaning $-2 > -5$.
• If you did not reverse the sign, you would get the false assertion $-2 < -5$.

   Original:               2 < 5  (2 is to the left of 5)
   <───┼───┼───┼───┼───┼───┼───┼───●───┼───┼───●───┼───┼───┼───>
      -5  -4  -3  -2  -1   0   1   2   3   4   5
                                   ▲           ▲
                                   │           │
   Multiply by -1:                 ▼           ▼
   <───●───┼───┼───●───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───>
      -5  -4  -3  -2  -1   0   1   2   3   4   5
      
   Negated:               -5 < -2   or   -2 > -5  (Order Inverted!)

Common Inequality Pitfalls to Avoid

1. Do NOT flip the sign when adding or subtracting a negative number: In $x - 5 > 2$, adding $+5$ gives $x > 7$. The sign does not flip.
2. Do NOT flip the sign when dividing a negative numerator by a positive denominator: In $4x < -20$, dividing by $+4$ gives $x < -5$. The sign remains unchanged because the divisor ($+4$) was positive.
3. DO flip the sign when dividing by a negative coefficient: In $-4x < 20$, dividing by $-4$ yields $x > -5$.

Compound Inequalities: Conjunctions (AND) vs. Disjunctions (OR)

A compound inequality combines two distinct inequality statements using either the logical connector AND or OR.

1. Conjunctions ("AND" / Double Inequalities)

A conjunction requires that the variable satisfies both conditions simultaneously. It represents the intersection ($\cap$) of two solution sets. Conjunctions are frequently written in condensed three-part form:

$a < x < b \iff x > a \quad \text{and} \quad x < b$

• Graph: A bounded line segment connecting two endpoints.
• Solving 3-Part Inequalities: Perform identical inverse operations on all three parts simultaneously.

Worked Example: Solving a 3-Part Conjunction

$\text{Solve: } -8 \le 3x + 4 < 19$

1. Subtract 4 from all three regions: $-8 - 4 \le 3x < 19 - 4$ $-12 \le 3x < 15$
2. Divide all three regions by $+3$: $\frac{-12}{3} \le \frac{3x}{3} < \frac{15}{3}$ $-4 \le x < 5$
3. Interval Notation: $[-4, 5)$

2. Disjunctions ("OR")

A disjunction requires that the variable satisfies at least one of the conditions. It represents the union ($\cup$) of two solution sets:

$x < a \quad \text{or} \quad x \ge b$

• Graph: Two separate rays pointing outward in opposite directions away from each other.
• Solving Disjunctions: Solve each inequality branch completely independently.

Worked Example: Solving a Disjunction

$\text{Solve: } 5 - 2x \ge 13 \quad \text{or} \quad 4x - 7 > 9$

1. Solve Left Branch ($5 - 2x \ge 13$): $-2x \ge 13 - 5 \implies -2x \ge 8$ $x \le \frac{8}{-2} \implies x \le -4 \quad (\text{Sign reversed!})$
2. Solve Right Branch ($4x - 7 > 9$): $4x > 9 + 7 \implies 4x > 16$ $x > \frac{16}{4} \implies x > 4$
3. Combined Solution: $x \le -4 \text{ or } x > 4$
4. Interval Notation: $(-\infty, -4] \cup (4, \infty)$

Step-by-Step Worked Inequality Examples

Example 1: Multi-Step Inequality with Distribution and Negative Division

Problem: Solve the inequality and state the solution set in interval notation:

$-3(2x - 5) + 8 \le 4(x + 7) - 5$

Solution:

1. Distribute across parentheses: $-6x + 15 + 8 \le 4x + 28 - 5$
2. Combine like terms per side: $-6x + 23 \le 4x + 23$
3. Collect variable terms on the left: $-6x - 4x + 23 \le 23 \implies -10x + 23 \le 23$
4. Subtract 23 from both sides: $-10x \le 23 - 23 \implies -10x \le 0$
5. Divide by $-10$ and reverse the inequality sign: $x \ge \frac{0}{-10} \implies x \ge 0$
6. Interval Notation: $[0, \infty)$

Example 2: Negative Multiplier in a 3-Part Inequality

Problem: Solve the compound inequality:

$-14 < 6 - 4x \le 18$

Solution:

1. Subtract 6 from all three parts: $-14 - 6 < -4x \le 18 - 6$ $-20 < -4x \le 12$
2. Divide all three parts by $-4$ and REVERSE BOTH inequality signs: $\frac{-20}{-4} > \frac{-4x}{-4} \ge \frac{12}{-4}$ $5 > x \ge -3$
3. Rewrite in standard ascending order (smallest value on left): $-3 \le x < 5$
4. Interval Notation: $[-3, 5)$