Linear inequalities and number lines (KCSE Mathematics Form 2)

TL;DR

Linear inequalities are like equations but use comparison symbols instead of an equals sign, showing a range of possible solutions. You solve them almost the same way as equations, but remember to flip the inequality sign if you multiply or divide by a negative number. Number lines are visual tools to represent these solution ranges clearly.

1. The Mental Model

Think of linear inequalities as setting boundaries for a value, not pinpointing an exact value. A number line is simply a picture that shows all the numbers that fit within those boundaries.

2. The Core Material

What are Linear Inequalities?

A linear inequality is a mathematical statement that compares two expressions using an inequality symbol. Unlike equations that use '=' (equals), inequalities use:
* < (less than)
* > (greater than)
* ≤ (less than or equal to)
* ≥ (greater than or equal to)

For example, x < 5 means 'x' can be any number smaller than 5 (like 4, 0, -10, 4.999), but not 5 itself. x ≥ 2 means 'x' can be 2 or any number larger than 2.

Solving Linear Inequalities

Solving linear inequalities is very similar to solving linear equations. You want to isolate the variable (like 'x').

Here's how it generally works:

graph TD
    A[Start: Inequality given] --> B{Are there brackets?};
    B -- Yes --> C[Expand brackets];
    B -- No --> D{Are there fractions?};
    C --> D;
    D -- Yes --> E[Multiply by LCM to clear denominators];
    D -- No --> F[Collect like terms on each side];
    F --> G[Move variable terms to one side, constants to the other];
    H{Did you multiply or divide by a negative number?};
    G --> H;
    H -- Yes --> I[Flip the inequality sign];
    H -- No --> J[Keep the inequality sign];
    I --> K[Isolate the variable];
    J --> K;
    K --> L[End: Solution in form x < a, x > a, etc.];

Key Rule to Remember: If you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign.

Example 1: Simple Inequality
Solve x + 3 > 7
1. Subtract 3 from both sides: x + 3 - 3 > 7 - 3
2. Simplify: x > 4

Example 2: With a Negative Coefficient
Solve -2x ≤ 6
1. Divide both sides by -2.
2. Remember to flip the sign! -2x / -2 ≥ 6 / -2
3. Simplify: x ≥ -3

Example 3: With Brackets and Fractions
Solve 2(x - 1) + 5 < x/3
1. Expand brackets: 2x - 2 + 5 < x/3
2. Simplify constants: 2x + 3 < x/3
3. Clear fraction (multiply by LCM, which is 3): 3 * (2x + 3) < 3 * (x/3)
6x + 9 < x
4. Collect like terms (move x to one side): 6x - x + 9 < x - x
5x + 9 < 0
5. Move constant: 5x < -9
6. Isolate x (divide by 5): x < -9/5 or x < -1.8

Representing Solutions on a Number Line

A number line is a visual way to show the range of numbers that satisfy an inequality.

• Open Circle (o): Used for < or > inequalities. It means the number itself is not included in the solution.
• Closed Circle (•): Used for ≤ or ≥ inequalities. It means the number itself is included in the solution.
• Arrow: Points in the direction of the solution.

How to draw:
1. Draw a straight line and mark a few key numbers, including the critical value from your solution.
2. Place an open or closed circle on the critical value.
3. Draw an arrow extending from the circle in the direction indicated by the inequality.

Examples:
* x > 4: Draw an open circle at 4, and an arrow pointing to the right.
* x ≤ -3: Draw a closed circle at -3, and an arrow pointing to the left.
* x < -1.8: Draw an open circle at -1.8, and an arrow pointing to the left.

3. Worked Example

Problem: Solve the inequality 3(x + 2) - 5 ≥ 7x - 1 and represent the solution on a number line.

Solution:

1. Expand the bracket:
   3x + 6 - 5 ≥ 7x - 1

2. Simplify constants on the left side:
   3x + 1 ≥ 7x - 1

3. Collect x terms on one side (let's move 3x to the right to keep x positive, if possible):
   1 ≥ 7x - 3x - 1
1 ≥ 4x - 1

4. Collect constant terms on the other side (move -1 to the left):
   1 + 1 ≥ 4x
2 ≥ 4x

5. Isolate x (divide by 4):
   2/4 ≥ x
1/2 ≥ x

6. Rewrite to have x on the left (optional, but often clearer):
   x ≤ 1/2 (or x ≤ 0.5)

Representing on a Number Line:
1. Draw a number line.
2. Mark 0.5 (or 1/2) on it.
3. Since the inequality is ≤ (less than or equal to), place a closed circle (•) at 0.5.
4. The inequality x ≤ 0.5 means x is less than or equal to 0.5, so draw an arrow pointing to the left from the closed circle.

<-----•--------------------->
      0.5

4. Key Takeaways

• Linear inequalities compare expressions using <, >, ≤, or ≥.
• Solve inequalities like equations by isolating the variable.
• Always flip the inequality sign if you multiply or divide both sides by a negative number.
• Use an open circle (o) on a number line for < or >.
• Use a closed circle (•) on a number line for ≤ or ≥.
• The arrow on the number line points in the direction of the solution.

Common Mistakes to Avoid:
- Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
- Confusing open and closed circles on the number line.
- Incorrectly simplifying expressions, especially with negative numbers.
- Drawing the arrow in the wrong direction on the number line.

5. Now Try It

Solve the inequality (x - 4)/2 + 3x > 5 - x and then draw a number line to represent your solution.
Success looks like: A correctly solved inequality in the form x > a or x < a (or x ≥ a, x ≤ a), and a number line with the correct type of circle and arrow direction.