Solving Inequalities

Concept Explanation

1. Single Linear Inequalities

For single inequalities involving one relational operator (such as $ax + b < c$), isolate the variable using standard linear operations while tracking sign flips:

• Addition/Subtraction: Adding or subtracting any real number preserves the inequality direction:
  $$x - 5 \ge 3 \implies x \ge 8 \quad \iff \quad [8, \infty)$$
• Multiplication/Division by Positive Numbers: Preserves direction:
  $$3x < 12 \implies x < 4 \quad \iff \quad (-\infty, 4)$$
• Multiplication/Division by Negative Numbers: Reverses the inequality:
  $$-2x \le 10 \implies x \ge \frac{10}{-2} \implies x \ge -5 \quad \iff \quad [-5, \infty)$$

2. Double (Compound / Sandwich) Inequalities

A double inequality of the form $a < f(x) < b$ represents an intersection ("AND") condition: $f(x) > a$ and $f(x) < b$.

When solving a linear double inequality, apply algebraic operations simultaneously to all three parts:

1. Subtract $4$ across all three segments:
   $$1 - 4 \le 3x < 13 - 4 \implies -3 \le 3x < 9$$
2. Divide all three segments by $3$:
   $$\frac{-3}{3} \le x < \frac{9}{3} \implies -1 \le x < 3 \quad \iff \quad [-1, 3)$$

If dividing all three segments by a negative number, every inequality sign reverses:

3. Non-Linear Inequalities & Sign Charts

For polynomials and rational expressions (e.g., $x^2 - 4 > 0$ or $\frac{x-1}{x+3} \le 0$), solve by factoring and analyzing interval signs across critical boundary points (roots and vertical asymptotes):

1. Set the inequality to zero: $P(x) \ge 0$ or $\frac{P(x)}{Q(x)} \le 0$.
2. Identify all critical points where the expression equals zero or is undefined.
3. Test a sample point in each sub-interval on the real number line to determine whether the expression is positive $(+)$ or negative $(-)$.

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Engineering Applications

Inequalities define operational bounds, structural tolerances, and sensor envelopes:

• Structural Safety Envelopes: Civil engineers specify stress bounds $\sigma_{\text{min}} \le \sigma(t) \le \sigma_{\text{allowable}}$ to prevent bridge beam buckling.
• Control Signal Clamping: Electrical engineers design operational thresholds $V_{\text{low}} < V_{\text{sensor}}(t) < V_{\text{high}}$ to filter voltage noise and trigger emergency shutdown clamps.
• Squeeze Theorem Bounding: When mathematical functions oscillate wildly, bounding them between two solvable envelope curves $g(x) \le f(x) \le h(x)$ allows engineers to prove stability as time $t \to 0$ or $t \to \infty$.

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Example Problems

Example 1: Solving a Double Inequality

Solve the double inequality for $x$ and express the solution in interval notation:

Solution

Step 1: Subtract $2$ from all three parts:

$$-5 - 2 < -3x \le 11 - 2$$

$$-7 < -3x \le 9$$

Step 2: Divide all three parts by $-3$ (and flip the inequality signs):

$$\frac{-7}{-3} > x \ge \frac{9}{-3}$$

$$\frac{7}{3} > x \ge -3 \quad \iff \quad -3 \le x < \frac{7}{3}$$

Step 3: State the interval notation:

$$x \in \left[-3, \frac{7}{3}\right)$$

Example 2: Bounding Inequality for the Squeeze Theorem

Given $-1 \le \sin\left(\frac{1}{x}\right) \le 1$ for all $x \neq 0$, construct a sandwich inequality for $f(x) = x^2 \sin\left(\frac{1}{x}\right)$ and determine whether the inequality direction changes for $x < 0$.

Solution

Step 1: Analyze the multiplier $x^2$: Since $x^2 > 0$ for all real numbers $x \neq 0$, multiplying across the inequality preserves the direction for both positive and negative $x$.

Step 2: Multiply each term by $x^2$:

$$-1 \cdot x^2 \le x^2 \sin\left(\frac{1}{x}\right) \le 1 \cdot x^2$$

$$-x^2 \le x^2 \sin\left(\frac{1}{x}\right) \le x^2$$

Step 3: Conclusion for Squeeze Theorem: Because $\lim_{x \to 0} (-x^2) = 0$ and $\lim_{x \to 0} (x^2) = 0$, the trapped center term $\lim_{x \to 0} x^2 \sin(1/x) = 0$.

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Connections & References

• Parent Concepts: Limits, Piecewise Functions
• Direct Theoretical Applications: Squeeze Theorem, Limits of Composite Functions, Continuity
• Course Links: Calc-1 Session 03 | Calc-1 Session 05