9-17-13 Compound Inequalities

Welcome to Professor Baker's Math Class! Today, we're diving into the world of compound inequalities. A compound inequality is essentially two or more inequalities joined together. Understanding these is a key step in mastering algebra!

Objective

Our main goal for today is to understand the difference between simple inequalities and compound inequalities, and learn how to solve and graph them effectively.

Warm-up Questions:

  • What are the rules for how to place the dot on the number line? Remember, an open circle indicates $ < $ or $ > $, while a closed (filled) circle indicates $ \le $ or $ \ge $.
  • What is the only difference between solving equations and inequalities? The big thing to remember is that when multiplying or dividing by a negative number, you have to flip the inequality sign!

Key Concepts:

  • "And" Inequalities: These inequalities require both conditions to be true. For example: $ -2 < x \le 6 $. This means x is greater than -2 AND less than or equal to 6.
  • "Or" Inequalities: These inequalities require at least one of the conditions to be true. For example: $ x < 1 \text{ or } x > 4 $. This means x is either less than 1 OR greater than 4.

Solving Compound Inequalities:

When solving compound inequalities, remember to isolate the variable in both inequalities. If you perform an operation on one part of the inequality, you must perform it on all parts to maintain the balance.

For example, let's solve the following "and" inequality:

$$ -3 \le 2x + 5 < 7 $$
  1. Subtract 5 from all parts: $$ -3 - 5 \le 2x + 5 - 5 < 7 - 5 $$ which simplifies to $$ -8 \le 2x < 2 $$.
  2. Divide all parts by 2: $$ \frac{-8}{2} \le \frac{2x}{2} < \frac{2}{2} $$ which simplifies to $$ -4 \le x < 1 $$.

Graphing Compound Inequalities:

Graphing helps visualize the solution set. Remember:

  • Use open circles for $ < $ and $ > $.
  • Use closed circles for $ \le $ and $ \ge $.
  • For "and" inequalities, shade the region where the two inequalities overlap.
  • For "or" inequalities, shade the regions that satisfy either inequality.

Homework:

Complete the problems on Page 46, numbers 37-48. Practice makes perfect, so don't hesitate to work through all of them.

Keep up the great work, and see you in the next class!