Table of Contents

1. Compound Inequalities
   1. Definition and Types
   2. Solving "And" Compound Inequalities
   3. Solving "Or" Compound Inequalities
   4. Graphing Compound Inequalities
      1. R Example: Graphing "And" Compound Inequalities
      2. R Example: Graphing "Or" Compound Inequalities

Compound inequalities involve combining two inequalities into one statement using the words "and" or "or". Understanding how to solve and graph these inequalities is crucial in algebra.

Compound Inequalities

Definition and Types

• A compound inequality contains at least two inequalities joined by "and" (intersection) or "or" (union).
• Examples:
  • "and" compound inequality: \(a < x < b\)
  • "or" compound inequality: \(x < a$\)or \(x > b\)

Solving "And" Compound Inequalities

• To solve: Find the set of values that satisfy both inequalities.
• Example: Solve \(3 < x + 2 < 7\)
  • Subtract 2 from all parts: \(1 < x < 5\)

Solving "Or" Compound Inequalities

• To solve: Find values that satisfy at least one of the inequalities.
• Example: Solve \(x \leq -3$\)or \(x \geq 4\)

Graphing Compound Inequalities

• "And" inequalities: Graph the intersection of the two inequalities.
• "Or" inequalities: Graph the union of the two inequalities.

R Example: Graphing "And" Compound Inequalities

plot(NULL, xlim=c(0,10), ylim=c(0,1), xlab="x", ylab="", axes=F)
abline(v=3, col="red")
abline(v=7, col="blue")
segments(3, 0.5, 7, 0.5, col="green", lwd=2)

R Example: Graphing "Or" Compound Inequalities

plot(NULL, xlim=c(-10,10), ylim=c(0,1), xlab="x", ylab="", axes=F)
abline(v=-3, col="red")
abline(v=4, col="blue")
segments(-10, 0.5, -3, 0.5, col="green", lwd=2)
segments(4, 0.5, 10, 0.5, col="green", lwd=2)

Understanding and being able to solve and graph compound inequalities are fundamental skills in algebra. These concepts are not only useful in mathematical problem-solving but also in various applications where conditions can be complex and multi-faceted.