Elementary Inequality Rules and Sign Diagram Techniques in Calculus

Objectives

• Learn how to solve inequalities involving linear, quadratic, polynomial, and rational expressions

• Solve an inequality to determine when a function is positive and negative

• Solve an inequality to determine when to rewrite an absolute value

• Solve an inequality to determine the domain of a function

Elementary Inequality Rules

Definition and Basic Concepts

Understanding inequalities is essential for analyzing when functions are positive or negative, determining domains, and solving equations involving absolute values. An inequality consists of two mathematical expressions related by symbols such as greater-than (>), less-than (<), greater-than-or-equal-to (≥), or less-than-or-equal-to (≤).

• Solution set of an inequality: The set of all values that make the inequality true. This is often an interval or union of intervals.

• Example: The solution to $x - 1 > 0$ is $x > 1$, which is the interval $(1, \infty)$.

Elementary Inequality Properties (Theorem 1.8.1)

These rules allow manipulation of inequalities in a manner similar to equations, with some important caveats.

• Addition: For any constant $a$ and real numbers $x, y$, $x < y \iff x + a < y + a$

• Multiplication by a positive constant: If $a > 0$, $x < y \iff ax < ay$

• Multiplication by a negative constant: If $b < 0$, $x < y \iff bx > by$ (the inequality reverses)

• Reciprocal: For $x, y > 0$, $0 < x < y \iff 0 < \frac{1}{y} < \frac{1}{x}$

Visualizing Inequalities

Inequalities can be visualized on the number line, where $x < y$ means $x$ is to the left of $y$, and $x > y$ means $x$ is to the right of $y$.

Solving Elementary Inequalities (Example 1.8.4)

To solve inequalities, use algebraic manipulation and consider the sign of expressions.

• Example: Solve $3 - 2x > 7$ and $\frac{5}{x+1} \leq 8$.

• Solution:

  1. For $3 - 2x > 7$: $-2x > 4 \implies x < -2$

  2. For $\frac{5}{x+1} \leq 8$: $5 \leq 8(x+1) \implies 5 \leq 8x + 8 \implies -3 \leq 8x \implies x \geq -\frac{3}{8}$

  3. Consider the sign of $x+1$ for the reciprocal rule. If $x+1 < 0$, $\frac{5}{x+1} \leq 8 \implies x < -1$

  4. Combine cases: The solution set is $(-\infty, -1) \cup [-\frac{3}{8}, \infty)$

Checking Solutions Graphically (Remark 1.8.5)

Graphing the expressions can help verify the solution set. For example, graph $y = 3 - 2x$ and $y = 7$ to see where $3 - 2x > 7$.

Sign Diagram Technique

Introduction to Sign Diagrams

Sign diagrams are a powerful tool for solving inequalities involving polynomials and rational functions. They help identify intervals where a function is positive or negative by analyzing the signs between zeros and points of discontinuity.

Solving Quadratic Inequalities

• Example: Solve $x^2 - x < 0$.

• Factor: $x(x-1) < 0$

• Find intervals where the product is negative: $(x < 0 \text{ and } x-1 > 0)$ or $(x > 0 \text{ and } x-1 < 0)$

• Solution: $x \in (0, 1)$

General Quadratic and Polynomial Inequalities

• For $ax^2 + bx + c < d$, rewrite as $ax^2 + bx + (c-d) < 0$ and factor if possible.

• Use sign diagrams to analyze intervals between zeros.

Constancy of Sign: Algebraic Functions (Theorem 1.8.11)

If $f$ is an algebraic function, then on any interval between zeros or points where $f$ is undefined, $f$ is either always positive or always negative.

Procedure 1.8.12: Sign Diagram Method

To solve inequalities involving algebraic functions:

1. Standard form: Express the inequality as $f(x) < 0$, $f(x) \leq 0$, $f(x) > 0$, or $f(x) \geq 0$.

2. Number line diagram: Mark all zeros and points where $f$ is undefined.

3. Sign diagram: Determine the sign of $f$ in each interval between marked points.

4. Solution: Identify intervals that satisfy the original inequality.

Example: Rational Function Inequality

• Example: Solve $\frac{5}{x+1} \leq 8$.

• Standard form: $\frac{5}{x+1} - 8 \leq 0 \implies \frac{5 - 8(x+1)}{x+1} \leq 0 \implies \frac{-8x - 3}{x+1} \leq 0$

• Zeros: $x = -\frac{3}{8}$; undefined at $x = -1$

• Intervals: $(-\infty, -1)$, $(-1, -\frac{3}{8})$, $(-\frac{3}{8}, \infty)$

• Test points: $f(-2) = \frac{-13}{-1} = 13 > 0$; $f(0) = \frac{-3}{1} = -3 < 0$

• Solution: $f(x) \leq 0$ on $(-1, -\frac{3}{8})$ and $(-\frac{3}{8}, \infty)$

• Final solution: $(-\infty, -1) \cup [-\frac{3}{8}, \infty)$

Example: Quadratic Inequality

• Example: Solve $x^2 > a$.

• Standard form: $x^2 - a > 0$

• Factor: $(x - \sqrt{a})(x + \sqrt{a}) > 0$

• Zeros: $x = \pm \sqrt{a}$

• Sign diagram: $f(x)$ is positive for $x < -\sqrt{a}$ and $x > \sqrt{a}$

• Solution: $x \in (-\infty, -\sqrt{a}) \cup (\sqrt{a}, \infty)$

Inequalities and Squares (Theorem 1.8.15)

For $x^2 < a$ or $x^2 > a$, where $a > 0$:

• $x^2 < a \iff -\sqrt{a} < x < \sqrt{a}$

• $x^2 \leq a \iff x \in [-\sqrt{a}, \sqrt{a}]$

• $x^2 > a \iff x < -\sqrt{a}$ or $x > \sqrt{a}$

• $x^2 \geq a \iff x \in (-\infty, -\sqrt{a}] \cup [\sqrt{a}, \infty)$

Example: Quartic Inequality

• Example: Solve $x^4 > a$ using sign diagram technique.

• Standard form: $x^4 - a > 0$

• Zeros: $x = \pm \sqrt[4]{a}$

• Sign diagram: $f(x)$ is positive for $x < -\sqrt[4]{a}$ and $x > \sqrt[4]{a}$

• Solution: $x \in (-\infty, -\sqrt[4]{a}) \cup (\sqrt[4]{a}, \infty)$

Summary Table: Elementary Inequality Rules

Operation	Rule	Example
Addition	$x < y \iff x + a < y + a$	$2 < 3 \iff 2 + 5 < 3 + 5$
Multiplication (positive)	$x < y \iff ax < ay$ ($a > 0$)	$2 < 3 \iff 2 \times 4 < 3 \times 4$
Multiplication (negative)	$x < y \iff bx > by$ ($b < 0$)	$2 < 3 \iff 2 \times (-1) > 3 \times (-1)$
Reciprocal	$0 < x < y \iff 0 < \frac{1}{y} < \frac{1}{x}$	$0 < 2 < 3 \iff 0 < \frac{1}{3} < \frac{1}{2}$

Summary Table: Sign Diagram Procedure

Step	Description
1. Standard Form	Express the inequality as $f(x) < 0$, $f(x) \leq 0$, $f(x) > 0$, or $f(x) \geq 0$
2. Number Line Diagram	Mark all zeros and points where $f$ is undefined
3. Sign Diagram	Determine the sign of $f$ in each interval between marked points
4. Solution	Identify intervals that satisfy the original inequality

Key Takeaways

• Elementary inequality rules allow manipulation similar to equations, but with special attention to multiplication by negative numbers and reciprocals.

• Sign diagrams are essential for solving polynomial and rational inequalities, providing a visual and systematic approach.

• Always check solutions graphically when possible to confirm interval solutions.

• For quadratic and higher-degree inequalities, factor and use sign diagrams to identify solution intervals.