Solving Compound Inequalities

TL;DR

Compound inequalities combine two simple inequalities with "and" or "or." To solve them, you solve each part separately and then combine the solutions based on the connector. The solution usually looks like a range of numbers on a number line.

1. The Mental Model

Think of compound inequalities as two separate rules that numbers must follow. For "and," a number must follow both rules. For "or," it just needs to follow at least one of the rules.

2. The Core Material

Compound inequalities link two simple inequalities using either "and" or "or." How you combine their solutions depends entirely on which word is used.

"And" Compound Inequalities

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When you see "and," it means the solution must satisfy both inequalities at the same time. Think of it as looking for the overlap between the two individual solution sets.

Often, "and" inequalities are written in a compact form, like $2 < x + 3 < 7$. This is just a shorthand for $2 < x + 3$ and $x + 3 < 7$. When solving these, you can either split them into two separate inequalities or solve all three parts simultaneously.

Here's how you usually solve them:
1. Isolate the variable in the middle part (if written compactly) or in each individual inequality.
2. Graph each solution on a number line.
3. Find the intersection (the overlapping region) of the two graphs. That's your final solution.

"Or" Compound Inequalities

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When you see "or," it means the solution must satisfy at least one of the inequalities. A number is in the solution if it works for the first inequality, or the second, or both. Think of it as combining both solution sets.

Here's how you solve them:
1. Solve each inequality separately to find its solution set.
2. Graph each solution on a number line.
3. Combine the two graphs. The solution is all numbers covered by either graph (the union of the two sets).

Sometimes, the solutions for "or" inequalities might overlap, or they might be completely separate. Both are valid.

graph TD
    A["Start with Compound Inequality"] --> B{Is it "AND" or "OR"?};
    B -- "AND" --> C["Solve Inequality 1"];
    B -- "AND" --> D["Solve Inequality 2"];
    C --> E["Graph Solution 1"];
    D --> F["Graph Solution 2"];
    E --> G["Find Overlap (Intersection)"];
    F --> G;
    G --> H["Write Final Solution"];
    B -- "OR" --> I["Solve Inequality 1"];
    B -- "OR" --> J["Solve Inequality 2"];
    I --> K["Graph Solution 1"];
    J --> L["Graph Solution 2"];
    K --> M["Combine Both Graphs (Union)"];
    L --> M;
    M --> H;

3. Worked Example

Let's solve the compound inequality: $4x - 2 < 10$ and $-3x \le 9$.

1. Solve the first inequality:
   $4x - 2 < 10$
   $4x < 12$ (Add 2 to both sides)
   $x < 3$ (Divide by 4)

2. Solve the second inequality:
   $-3x \le 9$
   $x \ge -3$ (Divide by -3 and flip the inequality sign!)

3. Combine the solutions with "and":
   We need numbers that are both less than 3 and greater than or equal to -3.
   This means $-3 \le x < 3$.

On a number line, you'd draw a closed circle at -3, an open circle at 3, and shade everything in between.

4. Key Takeaways

• "And" means the solution must satisfy all parts of the inequality simultaneously.
• "Or" means the solution must satisfy at least one part of the inequality.
• When solving "and" inequalities written as $a < x < b$, you can perform operations on all three parts at once.
• Remember to flip the inequality sign whenever you multiply or divide by a negative number.
• Graphing solutions on a number line helps visualize the "overlap" for "and" and the "combination" for "or."
• The solution to an "and" inequality is often a single interval, while "or" solutions can be two separate intervals or one combined one.

Common Mistakes to Avoid

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• Not flipping the sign: Forgetting to reverse the inequality direction when multiplying or dividing by a negative number.
• Confusing "and" and "or": Incorrectly finding the union instead of the intersection, or vice-versa.
• Incorrectly writing compact form: Trying to write an "or" inequality in the compact $a < x < b$ form; this form only applies to "and" inequalities.
• Graphing errors: Using open circles instead of closed circles (or vice-versa) for strict vs. inclusive inequalities.

5. Now Try It

Solve the following compound inequality and graph its solution on a number line:

$2x + 5 \le 1$ or $3x - 1 > 8$

What to do:
1. Solve the first inequality for $x$.
2. Solve the second inequality for $x$.
3. Combine these two solutions using the "or" condition.
4. Draw a number line and mark the solutions, using open/closed circles and shading appropriately.

What success looks like: You'll have two separate inequalities solved for $x$, and your number line will show two distinct shaded regions, or one continuous region if they overlap or meet.