Skip to content

Linear Inequalities

Definition

A linear inequality is a mathematical statement that compares two linear expressions using an inequality symbol (<, >, ≤, ≥). It describes a region on the coordinate plane rather than just a line.

The general form often looks like: ax + by < c, ax + by > c, ax + by ≤ c, or ax + by ≥ c

Prerequisites

To understand and solve linear inequalities, you should have mastered:

  1. Linear Equations
  2. Basic Equation Solving
  3. Coordinate Plane (for graphing)

Learning Objectives

After mastering linear inequalities, you should be able to:

  1. Identify linear inequalities.
  2. Solve linear inequalities for a specific variable (remembering to flip the sign when multiplying/dividing by a negative number).
  3. Graph the solution set of a linear inequality on a number line (for one variable) or coordinate plane (for two variables).
  4. Distinguish between strict (<, >) and non-strict (≤, ≥) inequalities in graphing (dashed vs. solid lines).
  5. Test points to determine the correct region to shade on a graph.
  6. Model real-world situations using linear inequalities.

Examples

Example 1: Solving a One-Variable Inequality (Early Middle School)

Problem: Solve for x: 3x - 4 > 11

Solution: 1. Add 4 to both sides: 3x - 4 + 4 > 11 + 4 3x > 15 2. Divide both sides by 3: 3x ÷ 3 > 15 ÷ 3 x > 5

Graph: An open circle at 5 on the number line, with an arrow pointing to the right.

Example 2: Graphing a Two-Variable Inequality (Late Middle School/High School)

Problem: Graph the inequality y ≤ -2x + 1

Solution: 1. Graph the boundary line y = -2x + 1. Since it's ≤, use a solid line. * y-intercept is (0, 1). * Slope is -2 (down 2, right 1). Plot another point, e.g., (1, -1). * Draw the solid line through (0, 1) and (1, -1). 2. Test a point not on the line, e.g., (0, 0). * Substitute into the inequality: 0 ≤ -2(0) + 1 * 0 ≤ 1. This is true. 3. Shade the region containing the test point (0, 0), which is the area below the line.

Example 3: Real-World Application (High School)

Problem: You have at most $50 to spend on snacks. Chips cost $3 per bag, and sodas cost $2 per bottle. Write an inequality representing the possible combinations of chips (c) and sodas (s) you can buy.

Solution: * Cost of chips: 3c * Cost of sodas: 2s * Total cost must be less than or equal to $50. * Inequality: 3c + 2s ≤ 50 * (Also, c ≥ 0 and s ≥ 0, since you can't buy negative snacks).

Common Misconceptions

  1. Forgetting to flip the inequality sign: When multiplying or dividing both sides by a negative number, the inequality symbol must be reversed.
  2. Confusing dashed and solid lines: Use a dashed line for strict inequalities (<, >) and a solid line for non-strict inequalities (≤, ≥).
  3. Shading the wrong region: Always test a point to confirm which side of the boundary line represents the solution set.

Applications in SAT

Linear inequalities appear in:

  1. Heart of Algebra questions: Solving inequalities, interpreting their graphs.
  2. Word problems: Modeling constraints or limitations.
  3. Systems of inequalities: Finding the feasible region defined by multiple inequalities.

Advanced Connections

Linear inequalities are foundational for:

  1. Systems of Equations (understanding regions helps visualize solutions)
  2. Linear Programming (Optimization problems with constraints)
  3. Calculus (defining domains and analyzing functions)

Practice Problems

  1. Basic: Solve for y: 10 - 2y ≤ 4
  2. Intermediate: Graph the inequality x - 3y > 6.
  3. Advanced: A student needs to score at least 80% on a test with 20 multiple-choice questions (worth 3 points each) and 10 short-answer questions (worth 4 points each). Write an inequality representing the possible combinations of correct multiple-choice (m) and short-answer (s) questions to achieve this score. Total possible points = 203 + 104 = 60 + 40 = 100. Required score = 80% of 100 = 80. Inequality: 3m + 4s ≥ 80.
  4. SAT-Level: If 2x + 5 > 9 and 3x - 1 < 11, which of the following integer values could x be? (Solve both: x > 2 and x < 4. Integer value must be 3).