The document you've uploaded is a study guide focusing on compound inequalities and absolute value inequalities. It provides explanations, examples, and exercises for solving these inequalities, which are crucial in algebra.

Key Points from the Study Guide:

1. Compound Inequalities:

   • These involve two inequalities joined by "and" or "or".
   • And: The solution is the intersection of the two sets.
   • Or: The solution is the union of the two sets.

   Examples:

   • For "and" compound inequalities: $-4 < x \leq 3$, the solution is where both conditions are true.
   • For "or" compound inequalities: $x < -3$ or $x \geq 1$, the solution includes any values that satisfy either condition.

2. Examples:

   • Example 1 solves the inequality $3x + 5 \leq 24$ and $5x + 19 \geq 14$, solving each part and then graphing the solution.
   • Example 2 addresses solving $3y - 2 \geq 7$ or $2y - 1 \leq 9$, graphing the combined solution set.

3. Exercises: The exercises ask you to solve various inequalities and graph the solutions:

   • Exercise 1: $-10 < 4x + 2 \leq 14$
   • Exercise 2: $3x + 8 \leq 23$ or $4x - 6 \geq 7$
   • And several others involving similar inequalities, requiring both solving and graphing on a number line.

Would you like to go through a specific example or exercise in detail, or have a more thorough explanation of any concept?

5 Related Questions:

1. What are the steps for solving a compound inequality involving "and"?
2. How do you represent the solution to an inequality on a number line?
3. What are absolute value inequalities, and how do they differ from compound inequalities?
4. How do you determine whether to use an "and" or "or" when solving compound inequalities?
5. Can a solution to a compound inequality involve no values, and why?

Tip:

Always isolate the variable in each part of the compound inequality before combining the solution sets for either intersection ("and") or union ("or").