Linear Inequalities

While equations ask "what equals what?", inequalities ask "what is greater or less than what?" This chapter extends the algebraic toolkit from equations to inequalities, exploring what happens when relationships aren't exact but bounded. Linear inequalities model real constraints: budgets limit spending, capacity limits storage, regulations set minimum and maximum standards. Understanding inequalities is essential for optimization problems, feasibility analysis, and linear programming—powerful techniques used in business, engineering, and economics to make optimal decisions within real-world constraints.

From Equations to Inequalities

An equation like 2x + 3 = 7 has one solution: x = 2. An inequality like 2x + 3 > 7 has infinitely many solutions: x > 2. Instead of a single point, the solution is a region—a ray or interval on the number line.

The inequality symbols are:

• < means "less than"
• ≤ means "less than or equal to"
• > means "greater than"
• ≥ means "greater than or equal to"

These simple symbols unlock a different way of thinking mathematically.

Properties of Inequalities

Inequalities behave mostly like equations, with one crucial exception:

Adding or subtracting the same value preserves the inequality: If a > b, then a + c > b + c

Multiplying or dividing by a positive number preserves the inequality: If a > b and c > 0, then ac > bc

BUT: Multiplying or dividing by a negative number reverses the inequality: If a > b and c < 0, then ac < bc

This reversal is crucial. If you have 5 > 3 and multiply both sides by -2, you get -10 < -6. The inequality flipped. This happens because negative multiplication reverses the order on the number line.

Linear Inequalities in One Variable

A linear inequality in one variable has the form ax + b > c (or <, ≤, ≥).

Solving is like solving an equation, but watch the reversal rule:

3x - 5 < 10 3x < 15 x < 5

The solution is all numbers less than 5: the interval (-∞, 5).

Graph this on a number line with an open circle at 5 (since x = 5 is not included) and an arrow pointing left toward -∞.

For x ≤ 5, use a closed circle at 5 (it is included) and an arrow pointing left.

Linear Inequalities in Two Variables

A linear inequality in two variables looks like 2x + 3y > 6. The solution is not a line but a region on the coordinate plane.

Start by graphing the boundary line: 2x + 3y = 6. Then test a point not on the line, say (0, 0): 2(0) + 3(0) = 0, which is not > 6. So (0, 0) is not in the solution region.

Shade the opposite side of the line. If the inequality includes equality (≥ or ≤), use a solid line; if strict (> or <), use a dashed line.

Systems of Linear Inequalities

Often we have multiple constraints. A system of inequalities represents multiple conditions that must all be satisfied simultaneously.

For example, a bakery problem:

• Let x = loaves of bread, y = cakes made
• Flour constraint: 2x + 3y ≤ 120
• Time constraint: x + 2y ≤ 50
• Non-negativity: x ≥ 0, y ≥ 0

The solution is the region where all four inequalities are satisfied—the intersection of all shaded regions. Any point in this region represents a feasible production plan.

Feasibility Region

In constrained optimization problems, the feasible region is the set of all points satisfying every constraint. The vertices (corner points) of this region are where the boundaries intersect. In linear programming, optimal solutions always occur at vertices.

If you want to maximize profit as 5x + 4y (where each loaf of bread gives 5 units of profit, each cake gives 4), you evaluate this expression at each vertex. The vertex yielding the highest value is optimal.

Graphical Representation

Visual representation is powerful. On a 2D coordinate plane:

• Each inequality creates a half-plane (everything on one side of a line)
• Multiple inequalities intersect to form the feasible region
• Constraints define boundaries; the region inside is possible

Real-World Applications

Business uses inequalities for resource allocation and budgeting. "We have at most 40 hours of labor and at least 100 units of output." Manufacturing uses them for production constraints. Medicine uses inequalities to determine safe dosage ranges. Agriculture uses them to optimize crop allocation given limited water and land.

Key Concepts

• Solution set: All values satisfying the inequality
• Boundary: The equality case, marking where solution region changes
• Feasible region: Area satisfying all constraints simultaneously
• Linear programming: Optimizing an objective within constraints

Socratic Questions

1. Why does multiplying or dividing an inequality by a negative number reverse the direction? What does this reveal about how multiplication by negative numbers transforms the number line?

1. When graphing a linear inequality in two variables like 2x + y > 5, why is the solution a region rather than a line? What do all points in that region have in common?

1. In a system of inequalities modeling a real business problem, what does each vertex of the feasible region represent? Why do optimal solutions occur at vertices rather than in the interior?

1. If a constraint is "x ≥ 0" (a non-negativity constraint), what problem does this represent in the real world? Why are such constraints essential in practical optimization?

1. Consider two scenarios: "budget of at most $100" versus "budget of at least $100." How would the corresponding inequalities and feasible regions differ? Which represents a more constrained situation?