Solving One-Step Equations

Many people found themselves enjoying math right up until they encountered algebra. So, if solving equations seems scary to you, know that you are not alone. But here is the thing: you already know how to solve equations. When you figure out how much tip to leave so your total comes to $20, or calculate what time you need to leave to arrive somewhere by 3 PM, you are solving equations. You just did not call it that. Now we are going to take that intuition you already have and turn it into a reliable skill you can use anywhere.

Core Concepts

What is an Equation?

An equation is simply a statement that two things are equal. That is it. The equals sign ($=$) is like a balance scale - whatever is on the left side must have the same value as whatever is on the right side.

$$3 + 5 = 8$$

This is an equation. It says “3 plus 5 equals 8.” Both sides have the same value, so the equation is true.

But equations become really useful when we introduce unknowns - values we do not know yet. We use letters (usually $x$, but any letter works) to represent these mystery numbers:

$$x + 5 = 12$$

This equation is asking: “What number, when you add 5 to it, gives you 12?”

You probably already figured out the answer is 7. Congratulations - you just solved an equation.

What Does It Mean to “Solve” an Equation?

To solve an equation means to find the value (or values) that make the equation true. This value is called the solution.

When we say $x = 7$ is the solution to $x + 5 = 12$, we mean that if you plug 7 in for $x$, both sides become equal:

$$7 + 5 = 12 \quad \checkmark$$

That is the whole game: find the number that makes both sides balance.

The Secret Weapon: Inverse Operations

Here is the key insight that makes equation-solving click: every mathematical operation has an opposite that undoes it.

• Addition and subtraction are opposites
• Multiplication and division are opposites

These opposites are called inverse operations. Add 5 to a number, then subtract 5, and you are back where you started. Multiply by 3, then divide by 3, and you get your original number back.

This is the entire strategy for solving equations: use inverse operations to “undo” whatever is being done to the variable until it stands alone.

The Golden Rule of Equations

Whatever you do to one side of an equation, you must do to the other side too.

Think of that balance scale again. If the scale is balanced and you add weight to one side, it tips. To keep it balanced, you must add the same weight to the other side.

$$x + 5 = 12$$

If we subtract 5 from the left side, we must subtract 5 from the right side:

$$x + 5 - 5 = 12 - 5$$ $$x = 7$$

The equation stays balanced, and we have found our answer.

Notation and Terminology

Term	Meaning	Example
Equation	A statement that two expressions are equal	$x + 3 = 10$
Variable	A letter representing an unknown value	$x$, $y$, $n$
Solution	The value that makes the equation true	For $x + 3 = 10$, the solution is $x = 7$
Inverse operation	An operation that undoes another	Subtraction undoes addition
Isolate	Get the variable alone on one side	From $x + 3 = 10$ to $x = 7$
Coefficient	The number multiplied by a variable	In $3x$, the coefficient is 3

Examples

Key Properties and Rules

Solving Addition Equations

If $x + a = b$, then $x = b - a$

Subtract the same number from both sides to isolate the variable.

Solving Subtraction Equations

If $x - a = b$, then $x = b + a$

Add the same number to both sides to isolate the variable.

Solving Multiplication Equations

If $ax = b$, then $x = \frac{b}{a}$

Divide both sides by the coefficient to isolate the variable.

Solving Division Equations

If $\frac{x}{a} = b$, then $x = ab$

Multiply both sides by the divisor to isolate the variable.

Always Check Your Solution

Plug your answer back into the original equation. If both sides are equal, you are correct. This habit catches mistakes and builds confidence.

Introduction to Inequalities

So far, we have worked with equations where two sides are exactly equal. But what about situations where one side is bigger than the other?

An inequality is a statement that compares two values that are not necessarily equal.

Inequality Symbols

Symbol	Meaning	Example
$<$	less than	$3 < 5$ (3 is less than 5)
$>$	greater than	$7 > 2$ (7 is greater than 2)
$\leq$	less than or equal to	$x \leq 4$ (x is 4 or smaller)
$\geq$	greater than or equal to	$y \geq 10$ (y is 10 or larger)
$\neq$	not equal to	$x \neq 0$ (x is not zero)

Memory trick: The symbol always “points to” the smaller value. Think of it as an alligator mouth that wants to eat the bigger number.

Solving One-Step Inequalities

The good news: solving inequalities works almost exactly like solving equations. Use inverse operations to isolate the variable.

$$x + 3 < 10$$

Subtract 3 from both sides: $$x < 7$$

This means $x$ can be any number less than 7: it could be 6, 5, 0, -100, or even 6.999.

Important exception: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. We will explore this more in Algebra I.

Graphing Inequalities on a Number Line

Unlike equations (which usually have one solution), inequalities often have infinitely many solutions. We show all these solutions by graphing on a number line.

For $x < 7$:

• Draw a number line
• Put an open circle at 7 (open because 7 itself is not included)
• Shade everything to the left (all numbers less than 7)

<-----(-------|-------|-------|---○==========>
      4       5       6       7       8

For $x \geq 3$:

• Put a closed circle at 3 (closed because 3 is included)
• Shade everything to the right (all numbers 3 or greater)

<======|-------|---●========|=======>
       1       2   3        4

Open vs. closed circles:

• Open circle ($\circ$): the endpoint is NOT included ($<$ or $>$)
• Closed circle ($\bullet$): the endpoint IS included ($\leq$ or $\geq$)

Real-World Applications

Shopping and Budgeting

You have $50 and want to buy a shirt that costs $32. How much will you have left?

This is an equation in disguise: $32 + x = 50$, so $x = 18$. You will have $18 left.

Or as an inequality: if you need at least $10 left over for lunch, what is the maximum you can spend on the shirt? $x + 10 \leq 50$, so $x \leq 40$.

Cooking and Recipes

A recipe serves 4 people, but you need to serve 12. If the original recipe calls for some amount of flour $f$ and you need $6$ cups total, how much flour did the original call for?

$$3f = 6$$ $$f = 2 \text{ cups}$$

Sports Statistics

A basketball player needs to average at least 20 points per game. After 4 games with scores of 18, 22, 17, and 23, how many points does she need in game 5?

Total points needed: $20 \times 5 = 100$

Current total: $18 + 22 + 17 + 23 = 80$

Points needed in game 5: $80 + x = 100$, so $x = 20$ points.

Time Management

Your movie starts at 7:30 PM and it takes 45 minutes to get to the theater. What time should you leave?

If $t$ = departure time in minutes after 6:00 PM: $$t + 45 = 90$$ (since 7:30 is 90 minutes after 6:00) $$t = 45$$

You should leave at 6:45 PM.

Self-Test Problems

Summary

• An equation is a statement that two expressions are equal. The goal of solving is to find the value that makes it true.
• Inverse operations undo each other: addition/subtraction are inverses, and multiplication/division are inverses.
• The golden rule: whatever you do to one side of an equation, do to the other side too.
• To solve an equation, use inverse operations to isolate the variable (get it alone on one side).
• Always check your answer by plugging it back into the original equation.
• An inequality compares values using $<$, $>$, $\leq$, or $\geq$ instead of $=$.
• Inequalities are solved the same way as equations (for one-step problems), but they have many solutions instead of just one.
• Graph inequalities on a number line using open circles for $<$ and $>$, or closed circles for $\leq$ and $\geq$.
• You already solve equations in daily life - now you have the tools to write them down and solve them systematically.