Solving Equations & Inequalities from a Set of Values |...

Section 1

Where Did Equations Come From?

Have you ever wondered where equations started? People have been solving math puzzles for thousands of years — long before calculators or computers existed. The idea of figuring out an unknown number has been around since ancient times, and it all started with real-life problems like sharing food, trading goods, and building structures.

Here's a quick look at how this idea grew over time:

~1800 BCE

Ancient Babylonians carved math problems into clay tablets. They wrote puzzles like "I found a number — when I add 5, I get 12. What is the number?" They tested different values to find answers.

~300 BCE

Greek mathematicians like Euclid organized rules for numbers and shapes. They used logical steps to prove that certain values were correct solutions.

~820 CE

A Persian scholar named al-Khwarizmi wrote one of the first algebra textbooks. The word "algebra" actually comes from the Arabic title of his book! He described methods for finding unknown values.

1600s CE

Mathematicians in Europe started using letters like x and y to stand for unknown numbers. René Descartes helped make this common. This is the same notation (way of writing things) you use today!

Today

In your 6th grade class, you solve equations and inequalities by testing values from a given set — the same basic idea the Babylonians used almost 4,000 years ago!

So when you test whether a number makes an equation true, you're doing something humans have done for thousands of years. The big question this lesson answers is: how do we figure out which numbers from a list are solutions?

Section 2

Core Principles & Definitions

Before we start solving, let's make sure you know the key words and ideas. Think of these as your toolkit — once you understand them, the rest of the lesson will feel much easier.

Equation

A math sentence that uses an equal sign (=) to show that two sides have the same value. Example: x + 3 = 10

Inequality

A math sentence that uses symbols like <, >, ≤, or ≥ to compare two sides. Example: x > 4 means "x is greater than 4."

Variable

A letter (like x, n, or y) that stands for a number we don't know yet. It's like a blank space waiting to be filled.

Solution & Specified Set

A solution is a value that makes the equation or inequality true. A specified set is a list of numbers you're allowed to test, like {1, 2, 3, 4, 5}.

When you solve an equation or inequality from a specified set, you don't guess randomly. Instead, you take each number from the given set and substitute it (plug it in) for the variable. Then you check: does the math sentence come out true or false?

✦ KEY TAKEAWAY

Think of it like trying keys on a lock. You have a ring of keys (the specified set), and you try each one in the lock (the equation or inequality). Some keys won't turn — those numbers don't work. But the key that opens the lock? That's your solution! An equation usually has one key that fits, while an inequality might have several.

Section 3

Visual Explanation

Let's see how substitution works with a picture. Below is a diagram showing the equation x + 4 = 9 and the specified set {3, 4, 5, 6}. Watch what happens when we plug in each value:

As you can see, we tried every number in the set. Only x = 5 made both sides equal, so it's the solution. The other numbers made the left side too small or too big. This "try each one" method is called substitution — you substitute (replace) the variable with a number and check if the result is true.

Section 4

How It Works — Step by Step

Here's the process you'll follow every time. It works for both equations and inequalities!

The Substitution Process

1. Read the equation or inequality 2. Pick a value from the set 3. Replace the variable with that value 4. Simplify both sides 5. Check: Is the statement true?

Repeat for every value in the set. Collect the ones that make it true — those are your solutions.

Let's be clear about what "true" means for equations versus inequalities:

For an Equation (uses =)

Left side must EQUAL the right side

Example: If x + 4 = 9, then plugging in x = 5 gives 9 = 9. That's true! ✓

For an Inequality (uses <, >, ≤, ≥)

Left side must satisfy the comparison

Example: If x > 4, plugging in x = 6 gives 6 > 4. That's true! ✓ But x = 3 gives 3 > 4. That's false. ✗

Here's something important to notice: an equation usually has one solution from the set, but an inequality can have multiple solutions. That's because many different numbers can be greater than, less than, or equal to something.

✦ KEY TAKEAWAY

An equation is like asking "which friend is exactly 5 feet tall?" — probably just one person. An inequality is like asking "which friends are taller than 4 feet?" — that could be several people! Both questions have answers, but inequalities usually have more of them.

Section 5

Equations vs. Inequalities — A Closer Look

Let's compare equations and inequalities side by side. This will help you understand the differences and see which inequality symbols mean what.

Notice something on that number line? For the inequality n ≥ 3, we got four solutions. That's because the symbol ≥ means "greater than or equal to," so 3 itself counts, along with every number bigger than 3 in our set.

If the inequality had been n > 3 (just "greater than," without the equal part), then 3 would not count, and we'd only have {4, 5, 6} as solutions. That little line under the > sign makes a difference!

Feature	Equation (=)	Inequality (<, >, ≤, ≥)
Symbol used	`=`	`<, >, ≤, ≥`
What it asks	Which value makes both sides equal?	Which values satisfy the comparison?
Number of solutions (from a set)	Usually one	Often several
How to check	Both sides must be the same number	The comparison must be true
Example	`x + 2 = 7 → x = 5`	`x + 2 > 4 → x = 3, 4, 5…`

Section 6

Worked Example

Let's walk through a complete problem together. Follow every step so you see exactly how it's done.

Which values from the set {2, 4, 6, 8, 10} make the inequality 3 × n − 2 > 10 true?

Step 1 — Understand the problemWe need to find which numbers from {2, 4, 6, 8, 10} make the expression 3 × n − 2 come out greater than 10. We'll test each one.

Step 2 — Test n = 2Replace n with 2: 3 × 2 − 2 = 6 − 2 = 4

Is 4 > 10? No. So n = 2 is not a solution.

Step 3 — Test n = 4Replace n with 4: 3 × 4 − 2 = 12 − 2 = 10

Is 10 > 10? No — 10 equals 10, but it's not greater than 10. So n = 4 is not a solution.

Step 4 — Test n = 6Replace n with 6: 3 × 6 − 2 = 18 − 2 = 16

Is 16 > 10? Yes! So n = 6 is a solution. ✓

Step 5 — Test n = 8Replace n with 8: 3 × 8 − 2 = 24 − 2 = 22

Is 22 > 10? Yes! So n = 8 is a solution. ✓

Step 6 — Test n = 10Replace n with 10: 3 × 10 − 2 = 30 − 2 = 28

Is 28 > 10? Yes! So n = 10 is a solution. ✓

Final Step — Collect the solutionsThe values from the set that make 3 × n − 2 > 10 true are: {6, 8, 10}. Notice that n = 4 didn't work because the symbol was > (strictly greater than), not ≥. The result of 10 is equal to 10, not greater. Always read the symbol carefully!

Section 7

Strengths, Limitations & Tips

Testing values from a set is a powerful method, but like any tool, it has strengths and limitations. Let's be honest about both.

Aspect	Strength ✓	Limitation ✗
Ease of use	Very straightforward — just plug in and check!	Can be slow if the set has many numbers.
Accuracy	You'll always get the right answer if you check carefully.	If you make an arithmetic mistake on one value, you might miss a solution.
Understanding	Helps you really see why a value works or doesn't.	Doesn't teach you shortcuts for harder equations later on.
Flexibility	Works for any equation or inequality you'll see in 6th grade.	Only works when you have a specific set to test — not for open-ended problems.

Here are some common mistakes to watch out for:

⚠ Common Mistake #1: Confusing > with ≥

If the symbol is > (greater than), a value that makes both sides equal does NOT count. If the symbol is ≥ (greater than or equal to), then equal DOES count. Always look at that tiny line under the symbol.

⚠ Common Mistake #2: Forgetting order of operations

When you substitute, remember to multiply and divide before you add and subtract. For example, in 3 × n + 1, you do the multiplication first, then add 1.

⚠ Common Mistake #3: Stopping too early

For inequalities, don't stop after you find the first solution! There might be more. Always test every value in the set.

✦ KEY TAKEAWAY

Testing values from a set is like being a detective checking alibis. You have a list of suspects (the set), and you question each one (substitute). Some are guilty (they make the statement true), and some are innocent (they make it false). A good detective checks everyone on the list — don't skip any values!

Section 8

What Comes Next?

Right now, you're solving equations and inequalities by testing values from a given set. This is an important skill, but it's also the starting point for something bigger. In 7th and 8th grade, you'll learn how to solve equations algebraically — meaning you'll use math rules to find the answer without testing every number.

For example, instead of testing {1, 2, 3, 4, 5, 6, 7} in the equation x + 3 = 10, you'll learn to subtract 3 from both sides to get x = 7 directly. That's much faster, especially when the numbers get bigger!

Feature	What You Do Now	What You'll Learn Next
Method	Substitute each value and check	Use inverse operations to isolate the variable
Speed	Works great for small sets	Finds the answer in fewer steps
Set needed?	Yes — you need a list of values to test	No — you can solve without a list
Builds toward	Understanding what a solution means	Solving any equation or inequality you encounter

Here's the cool part: even when you learn those faster methods, you can still use substitution to double-check your answer. Got x = 7 by algebra? Plug 7 back in and see if it works! That's why this skill stays useful forever.

Section 9

Practice Problems

Time to try it yourself! Start with the first problem and work your way down. Each one gets a little more challenging. Click "Show Answer" when you're ready to check your work.

PROBLEM 1 — CONCEPTUAL

What does it mean to "solve an equation from a specified set"? In your own words, describe what you would do if someone gave you the equation x + 1 = 6 and the set {3, 4, 5, 6}.

PROBLEM 2 — BASIC

Which value from the set {1, 3, 5, 7, 9} is the solution to the equation 2 × y = 14?

PROBLEM 3 — INTERMEDIATE

Which values from the set {2, 4, 6, 8, 10, 12} make the inequality n + 5 ≤ 11 true?

PROBLEM 4 — APPLIED

You're saving up to buy a book that costs $15. Each week you earn a different amount. Your possible weekly earnings are {$3, $5, $8, $10, $15, $20}. You already have $7 saved. Which weekly earnings would give you enough or more than enough to buy the book after one more week? (In other words, which values of w from the set make 7 + w ≥ 15 true?)

PROBLEM 5 — CHALLENGE

Here's a tricky one! Look at the set {1, 2, 3, 4, 5, 6, 7, 8}. Find all the values that are solutions to both of these at the same time: Equation: 2 × m + 1 = 9 — Inequality: m < 6. Is there a value that satisfies both? If so, what is it?

Summary

Lesson Summary

In this lesson, you learned how to solve equations and inequalities by testing values from a specified set. The key method is substitution: you replace the variable with each number in the set, simplify both sides, and check whether the math sentence comes out true or false. Values that make it true are called solutions.

You also learned that equations (using =) usually have just one solution from a set, while inequalities (using <, >, ≤, or ≥) often have multiple solutions. You practiced reading the inequality symbols carefully — especially the difference between > (strictly greater than) and ≥ (greater than or equal to). This "plug in and check" skill is a foundation you'll build on in every math class going forward. Keep practicing, and remember: every expert was once a beginner!