Equations and Inequalities

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.

• An equation has an equals sign; both sides are equal.
• Picture a balance scale that must stay level.
• Golden rule: do the same thing to both sides.
• An expression has no equals sign and nothing to solve.

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.

• Inverse operations undo each other.
• Add ⇄ subtract; multiply ⇄ divide.
• Ask: what has been done to the letter?
• Do the opposite, to both sides.

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 one-step equation has one operation on the letter.
• Identify the operation, then apply its inverse.
• Always change both sides equally.
• There are four types: $+$, $-$, $\times$ and $\div$.

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 $x = 12$.

• A two-step equation has two operations on the letter.
• Undo operations in reverse order — last on, first off.
• Usually undo $+$ or $-$ first, then $\times$ or $\div$.
• Keep both sides equal at every step.

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.

• Checking means substituting your answer back in.
• Use the original equation, not a rearranged one.
• If both sides match, the solution is correct.
• If they do not match, find and fix the slip.

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.

• An inequality compares unequal amounts.
• $<$, $>$, $\le$, $\ge$ — the mouth opens to the bigger side.
• An open circle excludes the end value ($<$, $>$).
• A filled circle includes the end value ($\le$, $\ge$).

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.

• Rearranging formulae reuses inverse operations.
• Straight-line graphs come from equations.
• Geometry problems often finish with an equation.
• Equations and inequalities model real-life limits.

Solve it like a two-step equation. First subtract 1 from both sides.