6.3 Linear Inequalities: Solving, Graphing on Number Lines, & Word Problems
Key Takeaways
- A linear inequality describes a relationship of relative magnitude ($<, \le, >, \ge$) whose solution set is an infinite continuous range of real numbers rather than a single point.
- On a 1D number line, strict inequalities ($<, >$) are plotted with open boundary circles (indicating non-inclusion), while inclusive inequalities ($\le, \ge$) are plotted with solid closed circles (indicating inclusion).
- The Golden Rule of Inequalities dictates that whenever both sides are multiplied or divided by a negative number, the inequality sign must be reversed ($< \leftrightarrow >$, $\le \leftrightarrow \ge$).
- Compound inequalities connect two conditions: 'AND' inequalities represent the intersection ($a \le x \le b$) where both conditions hold simultaneously, while 'OR' inequalities represent the union ($x < a \text{ or } x > b$) of two separate solution rays.
6.3 Linear Inequalities: Solving, Graphing on Number Lines, & Word Problems
While linear equations determine exact, specific numerical values where expressions are equal, real-world constraints—such as staying within an operating budget, maintaining a minimum speed, or observing maximum load limits—are governed by inequalities. On the TABE assessment, you must be able to solve linear inequalities, reverse direction signs appropriately, graph continuous solutions on number lines, and evaluate compound relationships.
1. Inequality Symbols & The Concept of Continuous Solution Sets
An inequality compares two algebraic expressions using one of five relational symbols:
| Symbol | Mathematical Meaning | Key Word Phrases | Boundary Circle Type |
|---|---|---|---|
| $<$ | Strictly less than | "fewer than", "under", "below", "less than" | Open Circle ($\circ$) |
| $\le$ | Less than or equal to | "at most", "maximum of", "no more than", "does not exceed" | Closed Circle ($\bullet$) |
| $>$ | Strictly greater than | "more than", "exceeds", "above", "greater than" | Open Circle ($\circ$) |
| $\ge$ | Greater than or equal to | "at least", "minimum of", "no less than", "a bottom limit of" | Closed Circle ($\bullet$) |
| $\ne$ | Not equal to | "is not equal to", "is different from" | Open Circle at point |
Discrete vs. Continuous Solution Sets
Unlike the equation $x = 3$ (which contains exactly one discrete point), the inequality $x > 3$ contains an infinite continuum of real numbers ($3.0001, 3.5, 4, 100, 10^{12}$). Because an infinite set of numbers cannot be listed individually, solutions are represented graphically on a number line or via algebraic interval notation.
2. Graphing Inequalities on a 1D Number Line
Graphing an inequality on a number line requires two decisions:
- Boundary Point (Circle Type):
- Use an Open Circle ($\circ$) for strict inequalities ($<$ or $>$) to show the endpoint is excluded.
- Use a Closed / Solid Circle ($\bullet$) for inclusive inequalities ($\le$ or $\ge$) to show the endpoint is included.
- Shading Direction (Ray):
- Shade to the Right (toward $+\infty$) when the variable is greater than the boundary value ($x > c$ or $x \ge c$).
- Shade to the Left (toward $-\infty$) when the variable is less than the boundary value ($x < c$ or $x \le c$).
Graph of x > 2 (Open Circle at 2, Shaded Right):
<---+---+---+---(o)=======>
-1 0 1 2 3 4
Graph of x <= -1 (Closed Circle at -1, Shaded Left):
<=======*---+---+---+---+--->
-3 -2 -1 0 1 2 3
[!TIP] The Variable-First Alignment Rule: The inequality symbol points in the direction of the arrow only when the variable is on the left side! If you obtain $5 < x$, rewrite it with the variable on the left as $x > 5$ before shading to the right.
3. The Golden Rule of Inequalities: Multiplying or Dividing by a Negative
Solving a linear inequality follows the exact same inverse operations as solving a linear equation, with one vital, fundamental exception:
[!CAUTION] THE GOLDEN RULE OF INEQUALITIES: Whenever you multiply or divide both sides of an inequality by a negative number, you MUST REVERSE (FLIP) THE INEQUALITY SYMBOL:
Why Does the Inequality Symbol Reverse?
Consider the true numerical inequality: $2 < 6$.
- If we multiply both sides by positive $2$: $2(2) < 6(2) \implies 4 < 12$ (Still True).
- If we multiply both sides by negative $1$: $2(-1) \text{ vs } 6(-1) \implies -2 \text{ vs } -6$. On a number line, $-2$ is to the right of $-6$, which means $-2 > -6$! Multiplying by a negative reflects quantities across zero, completely inverting their relative order.
Step-by-Step Worked Demonstration
Problem: Solve $-4x + 15 \le 39$.
\text{Step 1 (Undo Addition):} &\quad -4x + 15 - 15 \le 39 - 15 \\ &\quad -4x \le 24 \\ \text{Step 2 (Divide by } -4 \text{ and FLIP sign):} &\quad \frac{-4x}{-4} \ge \frac{24}{-4} \quad (\le \text{ flips to } \ge) \\ &\quad x \ge -6 \end{aligned}$$ *Graphical Representation:* A solid closed circle at $-6$ with shading extending infinitely to the right. --- ## 4. Multi-Step Linear Inequalities **Example:** Solve $3(2x - 5) - 4x < 7x + 9$. $$\begin{aligned} \text{Step 1 (Distribute):} &\quad 6x - 15 - 4x < 7x + 9 \\ \text{Step 2 (Combine Like Terms):} &\quad 2x - 15 < 7x + 9 \\ \text{Step 3 (Collect Variables on Left):} &\quad 2x - 7x - 15 < 9 \\ &\quad -5x - 15 < 9 \\ \text{Step 4 (Undo Subtraction):} &\quad -5x < 24 \\ \text{Step 5 (Divide by } -5 \text{ and FLIP):} &\quad x > -\frac{24}{5} \quad (x > -4.8) \end{aligned}$$ --- ## 5. Compound Inequalities: "AND" vs. "OR" A **compound inequality** connects two individual inequalities with either the conjunction "AND" or the disjunction "OR". ```text Compound Inequalities: ├── 1. "AND" Conjunction (Intersection ∩) ──> a <= x <= b (Bounded Overlapping Segment) └── 2. "OR" Disjunction (Union ∪) ──────────> x < a OR x > b (Two Diverging Rays) ``` ### 1. "AND" Compound Inequalities (Intersection $\cap$) In an "AND" inequality, both conditions must be true simultaneously. It is typically written as a continuous double-ended inequality: $a < x < b$. **Worked Example:** Solve $-8 \le 2x + 4 < 14$. - To solve, perform inverse operations to **all three parts simultaneously**: $$\begin{aligned} \text{Subtract 4 from all 3 parts:} &\quad -8 - 4 \le 2x + 4 - 4 < 14 - 4 \\ &\quad -12 \le 2x < 10 \\ \text{Divide all 3 parts by 2:} &\quad \frac{-12}{2} \le \frac{2x}{2} < \frac{10}{2} \\ &\quad -6 \le x < 5 \end{aligned}$$ *Graph:* A closed circle at $-6$, an open circle at $5$, and the segment between them shaded. ### 2. "OR" Compound Inequalities (Union $\cup$) In an "OR" inequality, a value of $x$ is a solution if it satisfies **at least one** of the individual inequalities. The two parts are solved separately. **Worked Example:** Solve $4x - 3 < -15 \quad \text{or} \quad 2x + 7 \ge 17$. - Left inequality: $4x < -12 \implies x < -3$. - Right inequality: $2x \ge 10 \implies x \ge 5$. - **Combined Solution:** $x < -3 \quad \text{or} \quad x \ge 5$. *Graph:* An open circle at $-3$ shaded to the left, and a solid circle at $5$ shaded to the right. --- ## 6. Real-World Constraints & Word Problems ### Workplace Problem 1: Freight Elevator Maximum Load **Problem:** A warehouse cargo elevator has a safe working load limit of $2,500\text{ lbs}$. An operator weighing $190\text{ lbs}$ must transport pallets of engine parts weighing $140\text{ lbs}$ each. What is the maximum number of pallets $p$ that can be safely loaded? 1. **Set Up Inequality:** $140p + 190 \le 2500$ 2. **Solve:** $$\begin{aligned} 140p &\le 2500 - 190 \\ 140p &\le 2310 \\ p &\le \frac{2310}{140} = 16.5 \end{aligned}$$ 3. **Practical Interpretation:** Because you cannot load a partial pallet without exceeding safe capacity, round down to the nearest integer: **$16$ pallets maximum**. ### Academic Target Problem 2: Average Score Target **Problem:** A nursing student needs an overall average of at least $84$ across four pharmacology exams to pass. On the first three exams, the student scored $79, 88,$ and $81$. What score $s$ must be earned on the fourth exam? $$\begin{aligned} \frac{79 + 88 + 81 + s}{4} &\ge 84 \\ \frac{248 + s}{4} &\ge 84 \\ 248 + s &\ge 84 \times 4 \\ 248 + s &\ge 336 \\ s &\ge 336 - 248 \\ s &\ge 88 \end{aligned}$$ *Conclusion:* The student must score at least **$88$** on the final exam.Solve the linear inequality: -5(x - 2) + 8 <= 38.
Which graph correctly represents the solution set for the compound inequality: -14 <= 3x + 1 < 16?
A catering banquet hall charges a flat setup fee of $380 plus $24 per guest. A corporate event coordinator has a strict maximum budget of $1,700 for the banquet. What is the maximum number of guests g that can be invited without exceeding the budget?