One solution
Solve 2x + 5 = 11
- 2x + 5 = 11
- 2x = 6 subtract 5 from both sides
- x = 3 divide both sides by 2
The solution is 3; the solution set is {3}. Checking gives 2(3) + 5 = 11, a true equation.
Integrated Math 2–3 Honors · Lesson Review
What Is an Equation?
Learn how expressions become equations, what it means to solve, and how solution sets connect equations, graphs, relations, and inequalities.
01 · Watch
Lesson video
0:0001What Is an Equation? — Expressions vs. Equations2:3702True, False, and Open Equations4:1803Solving and Equivalent Equations6:1704Solutions and Solution Sets12:3805No Solutions and All Real Numbers18:1706Equations in Two Variables27:4407Relations and Their Representations31:2908Inequalities36:4309Key Takeaways
Video not loading? Watch directly on YouTube.
02 · Understand
Learning targets
- Distinguish an expression from an equation or inequality.
- Classify equations as true, false, or open.
- Explain solution, solution set, solve, and equivalent equations.
- Represent solutions in one and two variables.
- Connect equations, graphs, and ordered-pair relations.
03 · Vocabulary
Key definitions
These are the definitions to know by heart.
- 01Equation
- A mathematical statement that two expressions are equal.
- 02True equation
- An equation whose statement is true.
- 03False equation
- An equation whose statement is false.
- 04Open equation
- An equation containing a variable whose truth depends on the value used for that variable.
- 05Solve
- Find the solution set.
- 06Solution (one variable)
- A number that makes the equation true when it replaces the variable.
- 07Solution set
- The set of all solutions.
- 08Equivalent equations
- Equations that have the same solution set.
- 09Solution (two variables)
- An ordered pair that makes the equation true when its values replace their respective variables.
- 10Inequality
- A mathematical statement that one expression is <, >, ≤, or ≥ another expression.
- 11Relation
- A set of ordered pairs.
04 · Apply
Worked examples
Follow the solution set—not just the final line.
Fraction solution
Solve 5x − 8 = 4
- 5x − 8 = 4
- 5x = 12 add 8 to both sides
- x = 12/5 divide both sides by 5
Solution set: {12/5}.
Special solution sets
No solution
3x + 4 = 3x + 7
4 = 7
The equivalent equation is always false. There are no solutions, so the solution set is ∅.
All real numbers
4x + 5 = 3x + x + 5
5 = 5
The equivalent equation is always true. Every real number works, so the solution set is ℝ.
Two variables
Solutions to y = 2x + 3
A solution is an ordered pair. Examples include (1, 5), (0, 3), (1/2, 4), and (3, 9). Every point on the line is a solution; the whole line is the solution set.
y = 2x + 3
Inequality
Solve 2x − 3 > −9
- 2x − 3 > −9
- 2x > −6 add 3 to both sides
- x > −3 divide by positive 2; do not flip the sign
The graph uses an open circle at −3 and shades to the right.
05 · Avoid
Common mistakes
An expression is not solved.
Expressions can be simplified, factored, or rewritten. Solving requires a statement with a solution set.
The solution is a value.
For 2x + 5 = 11, the solution is 3 and the solution set is {3}.
∅ is not {∅}.
The empty set contains nothing. A set containing the empty set contains one object.
Use ordered pairs in two variables.
The order tells which value replaces x and which replaces y.
06 · Check yourself
Quick check
Open each answer after you decide.
1 Is 3x + 4 an equation?
No. It is an expression because it does not state that two expressions are equal.
2 Classify 5 + 3 = 8.
True equation. The two numerical expressions have the same value.
3 What is the solution set of 4x + 1 = 4x + 1?
All real numbers, ℝ. Every real number makes the identity true.
4 Is (2, 7) a solution to y = 2x + 3?
Yes. Substitution gives 7 = 2(2) + 3, which is true.
5 Name three ways to represent a relation.
List ordered pairs, graph them, or use an equation whose solution set is the relation.
07 · Revisit
Transcript and vocabulary practice
Use the corrected, timestamped transcript to search for a term or return to an explanation. Then complete the vocabulary worksheet attached to the eKadence activity: write each definition and create your own example.
Download corrected transcript