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Introduction to Inequalities

1. What are Inequalities?

An inequality is like an equation, but instead of showing that two values are equal, it shows that one value is greater than, less than, or equal to another.

Example:

For example, if you have more than $10 in your bank account, you could write: x > 10.

Here, x represents the amount of money in your account and the symbol > means "greater than."

2. Understanding Inequality Symbols

  • ✔ Greater than (>) → Example: x > 5
  • ✔ Less than (<) → Example: y < 10
  • ✔ Greater than or equal to (≥) → Example: a ≥ 3
  • ✔ Less than or equal to (≤) → Example: b ≤ 7

3. Real-Life Examples of Inequalities

  • ✔ Shopping: A sale on items costing $20 or less → x ≤ 20
  • ✔ Speed Limits: A car should not exceed 60 mph → s ≤ 60
  • ✔ Age Restrictions: You must be at least 18 years old to vote → x ≥ 18
  • ✔ Weather: Temperature expected above freezing (0°C) → t > 0

4. Solving One-Step Inequalities

Solving inequalities is similar to solving equations. The goal is to isolate the variable by performing the same operation on both sides.

Addition & Subtraction:

Example: Solve x + 4 < 12:
Subtract 4 from both sides: x < 8

Multiplication & Division:

Example: Solve 3x ≥ 15:
Divide both sides by 3: x ≥ 5

5. The Golden Rule: Multiplying or Dividing by a Negative Number

When you multiply or divide an inequality by a negative number, you must flip the inequality sign.

Example: Solve -2x > 10:
Divide both sides by -2 (and flip the sign): x < -5

6. Graphing Inequalities on a Number Line (Rewritten & Clearer Explanation)

Graphing inequalities on a number line helps us visualize the range of possible values that satisfy the inequality. Let’s break it down step by step.

Step 1: Identify the Boundary Number
Look at the inequality and find the number that separates the possible values. For example, in x > 4, the boundary number is 4.

Step 2: Determine the Type of Circle
The type of circle you place on the number line depends on the inequality sign:

  • Open Circle (○): Use for strict inequalities (< or >).
    Example: x > 4 or x < -2 (the boundary value is not included).
  • Closed Circle (●): Use for inequalities that include "equal to" (≥ or ≤).
    Example: x ≥ 3 or x ≤ -1 (the boundary value is included).

Step 3: Shade in the Correct Direction
Once the circle is placed, shade the number line in the correct direction:

  • For x > a or x ≥ a: Shade to the right (values increasing).
  • For x < a or x ≤ a: Shade to the left (values decreasing).

Examples with Graphs:

Example 1: Graph x > 3
Boundary number: 3
Circle type: Open circle (○) because x is not equal to 3.
Shading: Right (values increasing).
Graph:
3 4 5 6 7 →
◯-----→

Example 2: Graph x ≤ -2
Boundary number: -2
Circle type: Closed circle (●) because x can equal -2.
Shading: Left (values decreasing).
Graph:
←-----●
-2

Example 3: Graph -4 < x ≤ 2 (Compound Inequality)
At -4: Use an open circle (since x is greater than -4 but not equal).
At 2: Use a closed circle (since x can equal 2).
Shading: Shade between -4 and 2.
Graph:
◯=====●
-4 2

Quick Summary for Graphing Inequalities

Inequality Circle Type Shading Direction
x > a Open (○) Right →
x ≥ a Closed (●) Right →
x < a Open (○) Left ←
x ≤ a Closed (●) Left ←

7. Solving One-Step Inequalities

Solving inequalities is similar to solving equations. The goal is to isolate the variable by performing the same operation on both sides.

Addition and Subtraction Inequalities:

Example: Solve x + 4 < 12:
Subtract 4 from both sides to get x < 8.

Multiplication and Division Inequalities:

Example: Solve 3x ≥ 15:
Divide both sides by 3 to get x ≥ 5.

8. Writing Inequalities from Word Problems

Example 1:
👉 A roller coaster requires riders to be at least 48 inches tall.
Write an inequality for the height h: h ≥ 48

Example 2:
👉 A basketball team can have a maximum of 12 players.
Write an inequality for the number of players p: p ≤ 12

Practice Questions (Basic Level) 🎯

  1. Solve for x: x - 4 < 10
  2. Solve for y: 3y ≥ 12
  3. Solve for z: z/2 ≤ 6
  4. Graph x > 5 on a number line.
  5. Write an inequality: A number n is greater than 20.
  6. The temperature must stay below 30°C. Write an inequality.
  7. Solve 4x + 2 ≤ 14.
  8. Solve -5x ≥ 20 and graph it.
  9. John wants to buy a laptop that costs at most $800. Write an inequality for the price p.
  10. Solve (x/3) - 2 > 5.

Want a Bigger Challenge? 🤔🔥

  1. Solve for x: 5(x - 2) > 10
  2. Solve for y: y/3 + 4 = 10
  3. Jack spends $20 on movie tickets and buys p popcorns at $4 each. If he spends $40 total, write and solve an equation for p.
  4. The sum of three consecutive numbers is 36. Write and solve an equation to find them.
  5. A cell phone plan costs $15 per month plus $0.10 per text. If your total bill was $35, how many texts (t) did you send?

11. Summary

  • ✅ Inequalities show relationships where one value is greater or smaller than another.
  • ✅ Symbols: >, <, ≥, ≤.
  • ✅ Solving inequalities follows similar rules as equations (remember to flip the sign when multiplying/dividing by a negative).
  • ✅ Graphing uses open circles for > and <, and closed circles for ≥ and ≤.
  • ✅ Real-world applications include shopping, speed limits, age restrictions, and more.

🎉 Awesome work! Now that you understand Inequalities, you're ready for the next topic: Working with Integers and Rational Numbers! 🚀

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