Example Usage

Inequalities are commonly used to express ranges and conditions:

• Range: $2 \leq x < 10$ means $x$ is at least 2 but less than 10.
• Conditional: If $x > 2$, then $x^2 > 4$ (for positive $x$).
• Comparison: $\pi \approx 3.14$, but $\pi \neq 3.14$ (they are approximately equal, not exactly equal).

Problem 1: Solve the inequality $3x + 2 < 11$

Solution:

• $3x + 2 < 11$
• $3x < 9$ (subtract 2 from both sides)
• $x < 3$ (divide both sides by 3)

Answer: $x < 3$ or $x \in (-\infty, 3)$

Problem 2: Find all values of $x$ such that $2 \leq x < 10$

Solution: This means $x$ is at least 2 but less than 10.

Answer: $x \in [2, 10)$ or $\{x \mid 2 \leq x < 10\}$

Problem 3: Compare $\pi$ and $3.14$

Solution:

• $\pi \approx 3.14159$
• Since $\pi > 3.14$, we write: $\pi > 3.14$
• However, $\pi \neq 3.14$ (they are approximately equal, not exactly equal)

Answer: $\pi > 3.14$ and $\pi \approx 3.14$ but $\pi \neq 3.14$

Daily Life Applications

Comparison and inequality symbols help you make decisions, set boundaries, and understand relationships in everyday situations.

Budgeting and Spending Limits

• Less Than ($<$): Set spending limits. If your budget is $500 and you've spent $350, you have $350 < 500$, so you're within budget.
• Less Than or Equal To ($\leq$): Define maximum allowances. If you can spend up to $100 on groceries, you write: $\text{spending} \leq 100$.
• Greater Than ($>$): Compare prices. If Product A costs $50 and Product B costs $75, then $75 > 50$, so Product B is more expensive.

Temperature and Weather

• Range Notation: Express comfortable temperature ranges. If you prefer temperatures between $68°F$ and $72°F$, you write: $68 \leq T \leq 72$.
• Comparison: Compare temperatures. If it's $75°F$ today and $68°F$ yesterday, then $75 > 68$, so today is warmer.

Age Restrictions and Requirements

• Greater Than or Equal To ($\geq$): Express minimum age requirements. If you must be at least $18$ to vote, you write: $\text{age} \geq 18$.
• Less Than ($<$): Define age limits. If children under $12$ get a discount, you write: $\text{age} < 12$.

Health and Fitness Goals

• Ranges: Set target heart rate zones. If your target heart rate is between $120$ and $150$ beats per minute, you write: $120 \leq \text{heart rate} \leq 150$.
• Comparison: Track progress. If your weight was $180$ lbs last month and is $175$ lbs now, then $175 < 180$, showing you've lost weight.

Shopping and Discounts

• Thresholds: Understand discount conditions. "Spend more than $100 and get 10% off" means: $\text{total} > 100$.
• Comparison: Compare deals. If Store A offers 20% off and Store B offers 15% off, then $20 > 15$, so Store A has a better discount.

Time Management

• Deadlines: Express time constraints. If a project is due in less than $5$ days, you write: $\text{days remaining} < 5$.
• Schedules: Define time windows. If meetings should last between $30$ and $60$ minutes, you write: $30 \leq \text{duration} \leq 60$.

Decision Making

When comparing options:

1. Identify what you're comparing (prices, sizes, quantities)
2. Use the right symbol ($<$ for "less than", $\leq$ for "at most")
3. Check if boundaries are included (use $\leq$ or $\geq$ for inclusive, $<$ or $>$ for exclusive)
4. Verify your comparison makes logical sense

These symbols help you set clear boundaries and make informed comparisons in daily life!