Lesson 4.4 — Inequalities & Variable Relationships
Not everything in math is "equal" — sometimes one side is bigger, smaller, or "at least" as much, and we have symbols for that.
🎙️ Narration script
Hey! So far we've talked about things being equal. But in real life, lots of things aren't equal. One side is bigger, or smaller, or "at least" as much. That's what today is about: inequalities.
An equation says two things are exactly equal. An inequality says one side is greater than or less than the other. We have four symbols. Greater than. Less than. Greater than or equal to. And less than or equal to. The two with the little line underneath include the "or equal to" part.
Here's a handy trick for which way the symbol points. The wide open side always faces the bigger number, and the little point faces the smaller one. So five is greater than two. Think of it as a hungry mouth that always wants to eat the larger amount.
Now here's something cool. A statement like x is greater than three isn't just one answer. It describes many numbers, every single number bigger than three. So we show them all at once on a number line.
Let's graph x is greater than three. First, find the boundary number, that's three. Next, choose your circle. Open circle for greater than or less than, because the endpoint is not included. Filled-in circle for the "or equal to" versions. Since this is just greater than, we use an open circle. Then pick the direction: greater means bigger numbers, so we shade and arrow to the right. Quick check, is five greater than three? Yes, and five is in our shaded part.
One more idea: independent and dependent variables. If pay equals nine times hours, you choose the hours, so hours is the independent variable. The pay follows from it, so pay is the dependent variable. Just ask, which one causes the other?
Recap! Inequalities compare with greater than, less than, and the "or equal to" versions. Open circle, filled circle, then shade toward the answers. Nicely done!
1 Core idea
An equation says two things are exactly equal (=). An inequality says one side is
greater than or less than the other. We use four symbols: > (greater than),
< (less than), ≥ (greater than or equal to), and ≤ (less than or equal to).
A statement like x > 3 isn't one answer — it describes many numbers (every number bigger than 3).
We can show all of those answers at once on a number line.
x > 3 works the same way, and the number line just shades in every value that "gets on the ride."
x > 3 leaves the boundary out — and real heights stop somewhere, but the shaded ray runs on forever.2 Key terms
- Inequality
- A statement comparing two values that are not necessarily equal, using > < ≥ ≤.
- Greater than (>)
- The left side is bigger:
5 > 2. - Less than (<)
- The left side is smaller:
2 < 5. - At least (≥) / At most (≤)
- "At least 3" means
≥ 3(3 is allowed). "At most 3" means≤ 3. - Independent variable
- The input you choose or change on your own (often
x). - Dependent variable
- The output that depends on the input (often
y).
3 Real-life examples
- Age limit: "You must be at least 13" →
age ≥ 13. - Speed limit: "Drive no faster than 65" →
speed ≤ 65. - Spending: "I have under $20 to spend" →
cost < 20. - Relationship: Hours worked (
h) decides your pay (p) at $9/hour →p = 9h. Herehis independent andpis dependent.
x > 3 and x ≥ 3 shade almost exactly the same numbers. Which single value separates them, and why does its circle switch from open to filled?
Reveal the thinking
4 Common doubts
Which way does > point?
The wide open side faces the bigger number, and the small point faces the smaller one: 5 > 2. Think of it as a hungry mouth that always eats the larger amount.
What's the difference between > and ≥?
The line under ≥ means "or equal to." x > 3 does not include 3, but x ≥ 3 does include 3.
Open circle or closed circle on a number line?
Open circle (○) for > or < — the endpoint is not included. Closed/filled circle (●) for ≥ or ≤ — the endpoint is included.
How do I know which variable is independent?
Ask "which one causes the other?" The cause (input) is independent; the result (output) is dependent. Hours cause pay, so hours is independent.
5 Step-by-step (graph an inequality)
- Find the boundary number — in
x > 3, that number is 3. - Choose the circle: open (○) for > or <, filled (●) for ≥ or ≤. Since this is >, use an open circle.
- Pick the direction: "greater than" means larger numbers, so shade and arrow to the right.
- Check a point: try x = 5 → is 5 > 3? Yes, so the shading covers 5. ✓
📊 See it · graphing x > 3
Open circle at 3 → 3 is not a solution. The shaded arrow points right, so every number bigger than 3 (like 4, 5, 6…) works. If it were x ≥ 3, the circle would be filled in.
Dependent vs. independent. In p = 9h (pay = $9 × hours), you pick the hours, so h is the independent variable; the pay p follows from it, so p is the dependent variable.
Hours h (independent) | Pay p = 9h (dependent) |
|---|---|
| 1 | $9 |
| 2 | $18 |
| 3 | $27 |
- Write "a club can have at most 30 members" as an inequality, and say which circle you'd draw.
answer
"At most" means ≤, somembers ≤ 30. Since ≤ includes 30, draw a filled (closed) circle. - For
p = 9h(pay from hours worked), which variable is independent and which is dependent?answer
You choose the hours, so h is independent; the pay follows from it, so p is dependent.
> < ≥ ≤. The open side faces the bigger
number. On a number line, use an open circle for >/< and a filled circle for ≥/≤, then shade toward the solutions.
In a relationship, the independent variable is the input you choose and the dependent variable is the output that follows.Grounded in CA CCSS-M, Grade 6 · 6.EE.5, 6.EE.8 & 6.EE.9 (inequalities & relationships between dependent and independent variables), California Department of Education. Hero image generated with Gemini Nano Banana Pro.