Equations and Inequalities

An equation has an equals sign. It says the left side and the right side are worth exactly the same. For example, $x + 3 = 7$ says "a number plus 3 equals 7".

The best picture for an equation is a balance scale. The two sides are level because they weigh the same.

This balance idea gives us the golden rule:

• Whatever you do to one side, you must do to the other.

If you take 3 off the left pan, you must take 3 off the right pan too, or the scale tips. This rule is what lets us solve equations safely.

An equation is different from an expression. An expression like $x + 3$ has no equals sign — there is nothing to solve. An equation makes a claim, and our job is to find the value of $x$ that makes that claim true.

To solve an equation we want the letter on its own. We do this by undoing whatever has been done to it, using inverse operations — the opposite operation.

So:

• Addition is undone by subtraction, and the other way round.
• Multiplication is undone by division, and the other way round.

Take $x + 3 = 7$. The letter has had 3 added to it, so we undo that by subtracting 3 — from both sides: $x + 3 - 3 = 7 - 3 \quad\Rightarrow\quad x = 4.$

Take $5x = 20$. The letter has been multiplied by 5, so we undo it by dividing by 5 — both sides: $5x \div 5 = 20 \div 5 \quad\Rightarrow\quad x = 4.$

Always ask yourself: "What has been done to the letter?" Then do the opposite.

A one-step equation has just one operation done to the letter. You solve it with a single inverse operation.

Here are the four types, each solved in one move:

• $x + 8 = 15$. Undo $+8$ by subtracting 8: $x = 15 - 8 = 7$.
• $x - 6 = 10$. Undo $-6$ by adding 6: $x = 10 + 6 = 16$.
• $4x = 28$. Undo $\times 4$ by dividing by 4: $x = 28 \div 4 = 7$.
• $\tfrac{x}{3} = 5$. Undo $\div 3$ by multiplying by 3: $x = 5 \times 3 = 15$.

The trick is simply to spot which operation is attached to the letter, then apply its inverse to both sides. With one-step equations you only ever do this once — which makes them the perfect place to build your confidence before two-step equations.

A two-step equation has two operations done to the letter, such as $2x + 5 = 13$. Here the letter is first multiplied by 2, then 5 is added.

To solve it, undo the operations in reverse order — last on, first off. So undo the $+5$ first, then the $\times 2$.

Step by step for $2x + 5 = 13$:

1. Subtract 5 from both sides: $2x = 8$.
2. Divide both sides by 2: $x = 4$.

Why reverse order? Think about getting dressed: socks go on before shoes, so shoes come off before socks. Equations work the same way — the last operation that was done is the first one you undo.

This same method handles $\tfrac{x}{4} - 1 = 2$: add 1 first to get $\tfrac{x}{4} = 3$, then multiply by 4 to get $x = 12$.

Once you have a solution, you can check it yourself — no need to wait for someone to mark it. Just substitute your answer back into the original equation.

If the left side equals the right side, your answer is correct.

Suppose you solved $2x + 5 = 13$ and got $x = 4$. Put 4 back in: $2 \times 4 + 5 = 8 + 5 = 13. \quad\checkmark$

The left side comes to 13 and the right side is 13 — they match, so $x = 4$ is right.

Now suppose you had made a slip and got $x = 5$. Checking it: $2 \times 5 + 5 = 10 + 5 = 15.$

The left side is 15 but the right side is 13 — they do not match, so $x = 5$ is wrong. Checking has caught the error for you.

This habit is one of the most powerful in maths. It turns "I think it is right" into "I know it is right", and it catches careless mistakes before they cost you. Make checking the natural last step of every equation you solve.

Sometimes two amounts are not equal — one is bigger than the other. We use an inequality to say so.

The four signs are:

• $<$ means "less than"
• $>$ means "greater than"
• $\le$ means "less than or equal to"
• $\ge$ means "greater than or equal to"

A handy tip: the sign always opens its wide mouth towards the bigger amount. So $3 < 7$ and $7 > 3$.

An inequality like $x > 2$ is true for many values — 3, 4, 5, 100, even 2.5. We show all of them on a number line.

The circle tells you about the end value:

• An open (empty) circle means that number is not included — used for $<$ and $>$.
• A filled circle means that number is included — used for $\le$ and $\ge$.

So $x \ge 2$ looks the same but with a filled circle, because 2 is allowed.

Getting confident with equations now pays off across many topics:

• Formulae are rearranged using the very same inverse-operation skills.
• Graphs of straight lines come from equations like $y = 2x + 1$.
• Geometry problems often end with an equation to solve for a missing angle or length.
• In everyday life, equations model questions like "I have $20, items cost $3 each — how many can I buy?", which becomes $3x \le 20$.

Inequalities matter too — they describe limits and ranges, such as a minimum age or a speed limit.

If equation work feels hard later, it is usually the balance rule or an inverse operation that needs a refresh. Come back to this guide whenever you need it — there is no prize for rushing.