Table of Contents

  1. The Concept of Absolute Value Inequalities
  2. Solving Absolute Value Inequalities
  3. Example: Solving $ |x - 3| < 5 $
  4. R Code Example: Visualizing an Absolute Value Inequality
  5. Practice Problems

Absolute value inequalities involve expressions within absolute value symbols and are solved to find the range of values that satisfy the inequality. The absolute value of a number is its distance from zero on the number line, regardless of direction. This concept leads to two different scenarios when solving absolute value inequalities: either the expression inside the absolute value is positive or it's negative.

The Concept of Absolute Value Inequalities

The equation \( |x| = a $\)has two solutions: \( x = a $\)and \( x = -a \). When this is extended to inequalities, we consider two separate cases for \( |x| < a $\)and \( |x| > a \).

  • For \( |x| < a \), the solution is \( -a < x < a \). This represents a range of values.
  • For \( |x| > a \), the solution is divided into two parts: \( x > a $\)or \( x < -a \). This is because the expression inside the absolute value can be either positive or negative.

Solving Absolute Value Inequalities

  1. Isolate the absolute value expression.
  2. Determine the type of inequality: '<' leads to a conjunction (AND), '>' leads to a disjunction (OR).
  3. Solve the inequality for both the positive and negative scenarios.

Example: Solving \( |x - 3| < 5 \)

  • Isolate the absolute value: \( |x - 3| < 5 \).
  • Since it’s a '<' inequality, use conjunction (AND): \( -5 < x - 3 < 5 \).
  • Solve for x: \( -2 < x < 8 \).

R Code Example: Visualizing an Absolute Value Inequality

library(ggplot2)
x_values <- seq(-10, 10, by = 0.1)
y_values <- abs(x_values - 3)
data <- data.frame(x = x_values, y = y_values)
ggplot(data, aes(x = x, y = y)) +
  geom_line() +
  geom_hline(yintercept = 5, linetype="dashed") +
  theme_minimal() +
  ggtitle("Graph of |x - 3| < 5")

Practice Problems

  • Solve and graph \( |2x + 1| > 3 \).
  • Solve \( |x - 4| \leq 6 $\)and represent the solution on a number line.
  • For \( |3x - 5| < 7 \), determine the range of values for x.

This section provides a foundational understanding of absolute value inequalities, crucial for many mathematical applications. The combination of theoretical explanation and practical R examples helps to grasp the concept thoroughly.