Writing and Graphing Inequalities

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Concept Review

Writing and Graphing Inequalities: The Language of Limits

Imagine you're at an amusement park and see a sign that says "You must be at least 48 inches tall to ride this roller coaster." How would a mathematician write this rule? They'd use an inequality — a mathematical sentence that shows when one quantity is greater than or less than another.

Unlike equations that use an equals sign (=) to show exact values, inequalities use special symbols to show relationships between numbers. The two most important symbols are the "greater than" sign (>) and the "less than" sign (<). Think of them as hungry alligator mouths that always want to eat the bigger number!

Writing Inequalities: Translating Words into Math

Let's say the school cafeteria has a rule: "Students can buy fewer than 3 cookies at lunch." In mathematical language, if we let x represent the number of cookies a student can buy, we write this as: x < 3.

Here's how different phrases translate into inequalities:

"More than 15" becomes x > 15
"Less than 8" becomes x < 8
"At least 12" becomes x ≥ 12
"At most 20" becomes x ≤ 20

🔍 The Direction Trick

Here's a secret: the inequality symbol always points toward the smaller number! In x > 7, the symbol points toward 7, meaning x is bigger than 7. In x < 12, the symbol points toward x, meaning x is smaller than 12.

Remember: Big number > small number and small number < big number

Real-World Example: Temperature Alert

Your town issues a heat warning when the temperature goes above 95°F. If t represents the temperature, the condition for a heat warning is t > 95. This means any temperature like 96°F, 100°F, or 110°F would trigger the warning, but 95°F exactly would not (since we need temperatures greater than 95, not equal to 95).

🔑 Key Takeaway

Just like that amusement park sign set a clear boundary for safety, inequalities help us set mathematical boundaries for real situations. Whether it's height restrictions, spending limits, or temperature warnings, inequalities give us the precise language to describe when something is "enough" or "too much" — making the mathematical world as clear as everyday rules.

Sample questions

1. A roller coaster requires riders to be taller than 48 inches. Which inequality represents this?

○ h < 48

✓ h > 48

○ h = 48

○ h >= 48

Answer: h > 48 — "Taller than" means greater than, but not including the value itself.

2. The temperature must stay below 32 degrees. Which inequality fits?

○ t > 32

○ t <= 32

✓ t < 32

○ t = 32

Answer: t < 32 — "Below" means less than.

3. Which inequality represents "a number x is at most 10"?

○ x < 10

○ x > 10

○ x >= 10

✓ x <= 10

Answer: x <= 10 — "At most" means 10 or anything smaller.

Skills in this topic

Write an inequality of the form x > c or x < c to represent a condition
Graph the solutions of inequalities (x > c or x < c) on a number line
Graph inequalities inclusive of the value (x >= c or x <= c)
Recognize that inequalities of this form have infinitely many solutions
Interpret graphed inequalities in a real-world context (e.g., speed limits, height requirements)

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