Why this matters for FAST: Inequalities appear frequently on FAST, often in word problems about limits, minimums, and maximums. Students must translate words to symbols, solve, and graph solutions correctly using open or closed circles.
Students use a closed circle when they should use an open circle, or vice versa. They don't connect the symbol to the graphing convention.
"Open circle = the number is NOT included (< or >). Think: the circle is 'open' because the number escapes! Closed circle = the number IS included (≤ or ≥). The number is 'captured' inside the filled circle."
Students solve correctly but shade the number line in the wrong direction.
"Always read the variable side: 'x > 3' means x is GREATER than 3, so shade to the RIGHT (bigger numbers). 'x < 3' means x is LESS than 3, so shade to the LEFT (smaller numbers)."
Students mix up "at least" (≥) with "at most" (≤) or forget to include the equals part.
"At LEAST 5 means 5 or MORE → x ≥ 5. At MOST 5 means 5 or LESS → x ≤ 5. 'At least' and 'at most' INCLUDE the number (closed circle)!"
Review equations: "If x + 3 = 7, x equals one number: 4. But what if x + 3 could be LESS than 7? That's where inequalities come in - they have MANY solutions!"
The Four Inequality Symbols
| < | Less than | Open circle, shade LEFT |
| > | Greater than | Open circle, shade RIGHT |
| ≤ | Less than or equal to | Closed circle, shade LEFT |
| ≥ | Greater than or equal to | Closed circle, shade RIGHT |
"Think of the symbol as an alligator mouth - it always opens toward the BIGGER side. And remember: if there's a line under the symbol (≤ or ≥), we INCLUDE that number with a closed circle."
Solve: x + 5 > 12
x + 5 > 12
- 5 - 5
x > 7
Solutions: 8, 9, 10, 11... (any number greater than 7)
"Solving inequalities works just like solving equations - use inverse operations! The difference is that inequalities have INFINITELY MANY solutions, not just one."
Graph: x > 7
Open circle at 7 (7 is NOT included), shade right (greater numbers)
Graph: x ≤ 4
Closed circle at 4 (4 IS included), shade left (lesser numbers)
Work through these together:
"Which graph represents x ≤ 3?"
A) Open circle at 3, shade right B) Closed circle at 3, shade left C) Open circle at 3, shade left D) Closed circle at 3, shade right
Correct answer: B) Closed circle at 3, shade left. The ≤ includes 3 (closed circle), and "less than" means shade left toward smaller numbers.
For struggling students: Create a symbol reference card. Use the "alligator eats the bigger number" analogy. Practice with concrete examples (ages, heights, money) before abstract variables.
For advanced students: Introduce compound inequalities (5 < x < 10) or two-step inequalities. Challenge them to write real-world scenarios that match given inequalities.
For home: Send Parent Activity sheet. Families can identify inequalities in daily life: speed limits (x ≤ 55), age requirements (x ≥ 13), budget limits (cost ≤ $50).