Solving Equations and Inequalities

Understanding Solving as Answering a Question

In this lesson:

• Test whether a value satisfies an equation
• Use a substitution table to find solutions
• Understand why inequalities have multiple solutions

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Interpret solving an equation as finding which value(s) make it true
2. Use substitution to test whether a number satisfies an equation or inequality
3. Distinguish equations from inequalities: inequalities can have a set of solutions

What Question Are You Answering?

When you solve — what are you looking for?

• A procedure to follow?
• Or: which value makes both sides equal?

Try it: does work? Does work? How do you know?

Solving Means Answering a Question

The question: Which value makes this equation true?

• A value that makes an equation true satisfies it
• That value is called the solution

Does satisfy ? ✓ Yes — is the solution.

The Substitution Table in Action

The only value that satisfies is .

Your Turn: Solve the Second Equation

Which value satisfies ?

Quick Check — Vocabulary and Edge Cases

• In your own words: what does it mean to say " satisfies "?
• What would we say if no candidate made the equation true?

Think before the next slide.

Same Question — Now for Inequalities

We found one value that makes true.

Same question, now for an inequality:

Which values from make true?

Inequalities Can Satisfy Multiple Values

and both satisfy — two solutions.

The Boundary Value Does Not Count

Is 12 < 12? No — 12 equals 12, not less than 12.

• The symbol is , not
• Equal is not less than
• does not satisfy

Solutions Exist Beyond the Candidate Set

Values outside the set can satisfy too:

• Try : ✓
• Try : ✓

The solution is all values that make true — infinitely many.

Quick Check — How Many Solutions?

From , tested against :

• How many values from the set are solutions?
• Is a solution? Why or why not?
• Are there values outside the set that also satisfy ?

Answer all three before moving on.

Equations vs. Inequalities — Key Contrast

Key Takeaways and Warnings to Remember

✓ Solving answers: which values make this true?
✓ Equations: typically one solution from a set
✓ Inequalities: multiple values — collect all true rows

Strict or excludes the boundary value

For inequalities, test every candidate — not just the first

Next Lesson Builds on This Substitution Skill

You can now test whether any value satisfies any equation or inequality.

Next (6.EE.B.7): Find solutions without testing a candidate set — using inverse operations.

Every answer you find by inverse operations can still be verified by substitution.