# Linear Inequalities ## What is an Inequality? An **inequality** is a mathematical statement that compares two expressions using inequality symbols instead of an equals sign. **Inequality Symbols:** - **<** less than - **>** greater than - **≤** less than or equal to - **≥** greater than or equal to Examples: - x < 5 (x is less than 5) - x ≥ 3 (x is greater than or equal to 3) - 2x + 1 > 7 (2x + 1 is greater than 7) ## Difference from Equations **Equations** have ONE solution (or specific number of solutions) Example: x + 2 = 5 has solution x = 3 **Inequalities** have INFINITE solutions (a range of values) Example: x + 2 < 5 has solutions x < 3 (all numbers less than 3) ## Solutions of Inequalities A **solution** to an inequality is any value that makes the inequality true. Example: Is x = 2 a solution to x < 5? Check: 2 < 5? Yes ✓ So x = 2 is a solution. Is x = 7 a solution to x < 5? Check: 7 < 5? No ✗ So x = 7 is NOT a solution. ## Graphing Solutions on a Number Line We represent solutions visually on a number line: **Open Circle (○):** Use for < or > The number is NOT included in the solution **Closed Circle (●):** Use for ≤ or ≥ The number IS included in the solution **Arrow:** Shows direction of other solutions Examples: x < 3 ○ at 3, arrow pointing left Numbers less than 3 x ≥ -2 ● at -2, arrow pointing right -2 and numbers greater than -2 ## Solving One-Step Inequalities Solve inequalities just like equations, with ONE IMPORTANT EXCEPTION: flip the inequality sign when multiplying or dividing by a NEGATIVE number. **Addition/Subtraction (no sign flip):** Example 1: x + 4 < 9 Subtract 4 from both sides: x < 5 Example 2: x - 3 ≥ 7 Add 3 to both sides: x ≥ 10 **Multiplication/Division by POSITIVE (no sign flip):** Example 3: 3x > 12 Divide both sides by 3: x > 4 Example 4: x/2 ≤ 5 Multiply both sides by 2: x ≤ 10 **Multiplication/Division by NEGATIVE (FLIP the sign!):** Example 5: -2x < 8 Divide both sides by -2 AND flip sign: x > -4 Example 6: -x ≥ 5 Multiply both sides by -1 AND flip sign: x ≤ -5 ## Why Flip the Sign for Negatives? When you multiply or divide by a negative, the order of numbers reverses. Example: Start with true statement 3 < 5 Multiply both sides by -1: -3 ? -5 On a number line, -3 is to the RIGHT of -5, so: -3 > -5 The inequality flipped! ## Solving Two-Step Inequalities Use the same process as equations: undo addition/subtraction first, then multiplication/division. Example 1: 2x + 5 < 13 Step 1: Subtract 5 from both sides 2x < 8 Step 2: Divide both sides by 2 x < 4 Example 2: -3x + 7 ≥ 16 Step 1: Subtract 7 from both sides -3x ≥ 9 Step 2: Divide both sides by -3 (FLIP sign!) x ≤ -3 Example 3: x/4 - 3 > 2 Step 1: Add 3 to both sides x/4 > 5 Step 2: Multiply both sides by 4 x > 20 ## Solving Multi-Step Inequalities Follow the same steps as multi-step equations: Example 1: 3(x - 2) ≤ 15 Step 1: Distribute 3x - 6 ≤ 15 Step 2: Add 6 3x ≤ 21 Step 3: Divide by 3 x ≤ 7 Example 2: 5x - 3 < 2x + 9 Step 1: Subtract 2x from both sides 3x - 3 < 9 Step 2: Add 3 3x < 12 Step 3: Divide by 3 x < 4 Example 3: -2(x + 4) > 6 Step 1: Distribute -2 -2x - 8 > 6 Step 2: Add 8 -2x > 14 Step 3: Divide by -2 (FLIP!) x < -7 ## Compound Inequalities **Compound inequalities** combine two inequalities. **"And" Compound Inequalities:** Written as a < x < b or using AND The solution satisfies BOTH inequalities (overlap) Example: -2 < x < 5 Read as: "x is greater than -2 AND less than 5" Solution: numbers between -2 and 5 Graph: ○ at -2, ○ at 5, line between them Solving: -3 < 2x + 1 < 7 Solve as three parts: -3 < 2x + 1 AND 2x + 1 < 7 -4 < 2x AND 2x < 6 -2 < x AND x < 3 Solution: -2 < x < 3 **"Or" Compound Inequalities:** The solution satisfies EITHER inequality (union) Example: x < -1 OR x > 3 Solution: numbers less than -1 or greater than 3 Graph: ○ at -1 with arrow left, ○ at 3 with arrow right ## Graphing Linear Inequalities in Two Variables When graphing inequalities like y > 2x + 1: **Step 1:** Graph the boundary line (as if it were an equation) - Use solid line for ≤ or ≥ - Use dashed line for < or > **Step 2:** Shade the appropriate region - For y > or y ≥: shade ABOVE the line - For y < or y ≤: shade BELOW the line **Test Point Method:** Pick a test point (often (0,0) if not on the line) Substitute into inequality If true, shade region containing that point If false, shade the other region Example: Graph y < 2x + 1 Step 1: Graph y = 2x + 1 with DASHED line (slope 2, y-intercept 1) Step 2: Test point (0, 0) 0 < 2(0) + 1 0 < 1 ✓ true Step 3: Shade region containing (0, 0) (below line) ## Checking Solutions Always check by substituting a value from your solution back into the original inequality. Example: If solving 3x - 4 > 8 gives x > 4 Check with x = 5: 3(5) - 4 > 8 15 - 4 > 8 11 > 8 ✓ ## Special Cases **All Real Numbers:** When you get a statement that's always true Example: x + 3 > x + 1 After simplifying: 3 > 1 (always true) Solution: all real numbers **No Solution:** When you get a false statement Example: x + 3 < x + 1 After simplifying: 3 < 1 (never true) Solution: no solution (empty set) ## Common Mistakes to Avoid 1. **Forgetting to flip the inequality sign** When dividing by -2, must flip! -2x < 8 becomes x > -4, NOT x < -4 2. **Using wrong circle type** < or > use open circle ○ ≤ or ≥ use closed circle ● 3. **Wrong direction on number line** x < 3 means numbers to the LEFT of 3 x > 3 means numbers to the RIGHT of 3 4. **Not distributing negative correctly** -2(x - 3) = -2x + 6, not -2x - 6 5. **Treating inequality like an equation** Remember: infinity solutions, not just one! ## Real-World Applications Inequalities model many real situations: **Example 1: Budget** "You have $50. Each movie ticket costs $12. How many tickets can you buy?" Let x = number of tickets 12x ≤ 50 x ≤ 4.17 You can buy at most 4 tickets (can't buy partial ticket) **Example 2: Speed Limit** "The speed limit is 65 mph" s ≤ 65 (your speed must be at most 65) **Example 3: Minimum Age** "You must be at least 16 to drive" a ≥ 16 (your age must be 16 or greater) **Example 4: Temperature** "Water is liquid between 32°F and 212°F" 32 < T < 212 (temperature is between 32 and 212) ## Problem-Solving Strategy 1. Read problem and identify what you're solving for 2. Define a variable 3. Identify the inequality symbol from words: - "at least" means ≥ - "at most" means ≤ - "more than" means > - "less than" means < - "no more than" means ≤ - "no less than" means ≥ 4. Write the inequality 5. Solve the inequality 6. Interpret answer in context ## Key Words and Phrases | Phrase | Symbol | |--------|--------| | Greater than | > | | Less than | < | | At least | ≥ | | At most | ≤ | | No more than | ≤ | | No less than | ≥ | | More than | > | | Fewer than | < | | Minimum | ≥ | | Maximum | ≤ | ## Quick Reference | Inequality | Read As | Number Line | |------------|---------|-------------| | x < 3 | x less than 3 | ○ at 3, arrow left | | x > 3 | x greater than 3 | ○ at 3, arrow right | | x ≤ 3 | x at most 3 | ● at 3, arrow left | | x ≥ 3 | x at least 3 | ● at 3, arrow right | ## Practice Tips - Always check if you need to flip the sign - Test a value to verify your solution - Draw number lines clearly with correct circles - Remember: ≤ and ≥ include the endpoint - Word problems: translate carefully to inequality symbols - Keep track of negative signs when distributing