Introduction to Inequalities and Two-Step Inequalities

TL;DR

Inequalities are like equations but use symbols like < (less than) or > (greater than) instead of an equals sign, meaning there's a range of possible solutions. Solving two-step inequalities uses the same inverse operations as two-step equations, but with one crucial difference: you flip the inequality sign if you multiply or divide by a negative number. This helps you find the set of values that make the inequality true.

1. The Mental Model

Think of inequalities as setting a boundary or a limit, rather than finding one exact answer. For example, if you need to be at least 18 to vote, your age can be 18, 19, 20, and so on – a whole range of numbers.

2. The Core Material

When we talk about inequalities, we're comparing two expressions that aren't necessarily equal. Instead, one might be greater than, less than, or greater/less than or equal to the other.

Here are the symbols you'll see:
* >: Greater than (e.g., x > 5 means x can be 6, 7, 8...)
* <: Less than (e.g., x < 5 means x can be 4, 3, 2...)
* ≥: Greater than or equal to (e.g., x ≥ 5 means x can be 5, 6, 7...)
* ≤: Less than or equal to (e.g., x ≤ 5 means x can be 5, 4, 3...)

Solving Two-Step Inequalities

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Solving two-step inequalities is very similar to solving two-step equations. You use inverse operations to isolate the variable. The main difference, and it's a big one, comes when you multiply or divide both sides by a negative number.

Here's the general process:

graph TD
    A["Start with the inequality"] --> B{"Isolate the term with the variable"};
    B --> C["Add or subtract the constant from both sides"];
    C --> D{"Isolate the variable"};
    D --> E["Multiply or divide both sides"];
    E --> F{Was the multiplier/divisor negative?};
    F -- Yes --> G["FLIP the inequality sign!"];
    F -- No --> H["Keep the inequality sign the same"];
    G --> I["Solution set"];
    H --> I;

Let's break down the "flipping" rule:

• Rule: If you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.
• Why? Think about it: 2 < 5 is true. If you multiply both sides by -1, you get -2 and -5. Now, -2 is actually greater than -5. So, 2 < 5 becomes -2 > -5. The sign flipped!

Example of Flipping the Sign

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Let's solve -2x + 5 > 15:

1. Subtract 5 from both sides:
   -2x + 5 - 5 > 15 - 5
-2x > 10

2. Divide by -2 (and remember to flip the sign!):
   -2x / -2 < 10 / -2 (Notice the > became <)
   x < -5

So, any number less than -5 (like -6, -7, etc.) will make the original inequality true.

3. Worked Example

Let's solve 3x - 7 ≤ 8

1. Identify the constant term you need to move first.
   Here, it's -7. The inverse operation is addition.

2. Add 7 to both sides of the inequality.
   3x - 7 + 7 ≤ 8 + 7
3x ≤ 15

3. Identify the coefficient of the variable.
   Here, it's 3. The inverse operation is division.

4. Divide both sides by 3.
   3x / 3 ≤ 15 / 3
x ≤ 5

   Did we divide by a negative number? No, we divided by 3 (a positive number), so we do not flip the inequality sign.

The solution is x ≤ 5. This means any number less than or equal to 5 will make the inequality true.

4. Key Takeaways

• Inequalities represent a range of solutions, not just one specific answer.
• The four main inequality symbols are >, <, ≥, and ≤.
• To solve two-step inequalities, you use inverse operations to isolate the variable, just like with equations.
• The critical rule is to flip the inequality sign only when you multiply or divide both sides by a negative number.
• Always perform addition/subtraction before multiplication/division to isolate the variable term.
• The solution to an inequality is a set of numbers that satisfy the condition.

Common Mistakes to Avoid

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• Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
• Flipping the sign when adding or subtracting a negative number (you only flip for multiplication/division).
• Performing multiplication/division before addition/subtraction when trying to isolate the variable.
• Confusing the meanings of > and < or ≥ and ≤.

5. Now Try It

Solve the following inequality: -4x + 10 < 2.

What to do:
1. Subtract 10 from both sides.
2. Divide both sides by -4. Remember to consider if you need to flip the inequality sign!

What success looks like:
You should arrive at a solution in the format x [inequality symbol] [number]. If you do it correctly, your final inequality symbol should be >.