Linear Inequalities: Understanding the Boundaries
From Equations to Inequalities: The Core Idea
You are already familiar with linear equations. An equation, like $x + 3 = 7$, states that two expressions are equal. The solution is a single number: $x = 4$.
A linear inequality is similar but uses symbols to show that one expression is greater than, less than, or possibly equal to another. Instead of a single answer, inequalities usually have a whole range of answers, called a solution set.
$<$: Less than
$>$: Greater than
$\le$: Less than or equal to
$\ge$: Greater than or equal to
Consider this simple example: "You must be at least 16 years old to get a driver's license." If $a$ represents age, this situation is written as $a \ge 16$. The solution isn't just 16; it's 16, 17, 18, and so on—infinitely many numbers! This is the primary shift in thinking from equations to inequalities.
Solving Linear Inequalities Step-by-Step
The process of solving a linear inequality is nearly identical to solving a linear equation. You use inverse operations (addition/subtraction, multiplication/division) to isolate the variable on one side. Let's solve $2x - 5 < 9$.
Step 1: Add 5 to both sides to undo the subtraction.
$2x - 5 + 5 < 9 + 5$
$2x < 14$
Step 2: Divide both sides by 2 to isolate $x$.
$\frac{2x}{2} < \frac{14}{2}$
$x < 7$
The solution is all numbers less than 7. We can check by picking a number less than 7, like 5: $2(5) - 5 = 10 - 5 = 5$, and $5 < 9$ is true.
Example: Solve $-3x \le 12$.
Divide by $-3$: $\frac{-3x}{-3} \color{#ffb600}{\ge} \frac{12}{-3}$.
Result: $x \ge -4$.
Graphing Solutions on a Number Line
Visualizing the solution set makes it much clearer. We use a number line with specific markers:
- An open circle (○) for $<$ or $>$: The number is not included.
- A closed circle (●) for $\le$ or $\ge$: The number is included.
- A shaded arrow or line extending in the direction of all solutions.
Let's graph the solutions from our previous examples:
| Inequality | Solution | Number Line Description |
|---|---|---|
| $x < 7$ | All numbers less than 7 | Open circle at 7, shaded line extending to the left. |
| $x \ge -4$ | All numbers greater than or equal to -4 | Closed circle at -4, shaded line extending to the right. |
| $1 < x \le 5$ | All numbers between 1 and 5, excluding 1 but including 5 | Open circle at 1, closed circle at 5, shaded line segment between them. |
Graphing Inequalities in Two Variables
Just as $y = 2x + 1$ graphs as a line, the inequality $y > 2x + 1$ graphs as a half-plane—all the points on one side of that boundary line.
Steps to Graph a Two-Variable Linear Inequality:
- Graph the boundary line. First, graph the related equation (e.g., $y = 2x + 1$).
- Use a solid line for $\le$ or $\ge$ (points on the line are included in the solution).
- Use a dashed line for $<$ or $>$ (points on the line are not included).
- Test a point. Pick a test point not on the line, usually the origin $(0,0)$ is easiest if it's not on the line. Substitute its coordinates into the inequality.
- Shade the correct region. If the test point makes the inequality true, shade the side of the line containing that point. If false, shade the opposite side.
Example: Graph $y \le -x + 3$.
1. Graph $y = -x + 3$ with a solid line (because of $\le$).
2. Test $(0,0)$: $0 \le -(0) + 3$ simplifies to $0 \le 3$, which is true.
3. Therefore, shade the region that contains $(0,0)$, which is the area below and to the left of the line.
Applying Inequalities to Real-World Scenarios
Inequalities are perfect for modeling situations with minimums, maximums, or ranges. Let's explore a detailed budgeting example.
Scenario: Maria is shopping for notebooks and pens. Notebooks cost $4 each, and pens cost $2 each. She needs at least 3 notebooks. She has $30 to spend and wants to know her possible combinations.
Step 1: Define variables.
Let $n$ = number of notebooks.
Let $p$ = number of pens.
Step 2: Write inequalities.
1. Cost constraint: Total cost must be $30$ or less. $4n + 2p \le 30$.
2. Minimum notebooks: $n \ge 3$.
3. Non-negativity: You can't buy negative items. $n \ge 0, p \ge 0$ (though $n \ge 3$ is stricter for n).
Step 3: Graph and interpret. We graph the system of inequalities. The feasible solutions are all the points with integer coordinates (you can't buy half a pen) in the shaded region that satisfies all conditions. For instance, (3, 9) costs $4(3)+2(9)=30$, using all her money. (4, 5) costs $4(4)+2(5)=26$, leaving $4 unspent. This visual model helps Maria see all her options at once.
Important Questions
Think about a true statement: $3 < 5$. If we multiply both sides by $-1$ without flipping, we get $-3 < -5$, which is false. On a number line, multiplying by -1 reflects numbers to the opposite side of zero, reversing their order. To keep the statement true, we must reverse the inequality: $-3 > -5$.
"And" means both conditions must be true at the same time, like $x > 1$ and $x \le 5$ (often written as $1 < x \le 5$). The solution is the overlap of the two sets.
"Or" means either condition can be true, like $x < -2$ or $x \ge 3$. The solution combines both sets, often leaving a gap in between.
You substitute the coordinates of the point into each inequality in the system. If the point makes all of the inequalities true, then it is a solution. If it fails even one, it is not a solution. For example, for the point (2, 10) in Maria's problem: Check $4(2)+2(10)=28 \le 30$ (true) and $2 \ge 3$ (false). So (2,10) is not a solution because it fails the notebook requirement.
Footnote
[1] LTR (Left-to-Right): The standard text direction for English and many other languages, as opposed to right-to-left (RTL) languages. It ensures formulas and numbers display correctly.
[2] Solution Set: The collection of all values that satisfy a given equation or inequality.
[3] Half-plane: The region on one side of a line in a coordinate plane. When graphing a two-variable inequality, the solution is always a half-plane.
[4] Compound Inequality: Two or more inequalities joined by the word "and" or "or."