Equations and Inequalities

By Stage 9 the toughest single equations combine everything at once: brackets to expand and the unknown on both sides. The balance rule has not changed — do the same thing to both sides — you just follow a clear order.

The order is: expand any brackets, then gather the letter terms onto one side, then solve with inverse operations.

So $3(x + 2) = x + 14$ goes: expand to $3x + 6 = x + 14$; subtract $x$ to get $2x + 6 = 14$; subtract 6 to get $2x = 8$; divide by 2 to reach $x = 4$.

The cure for these longer equations is one move per line, rewriting the whole equation each time. That single habit removes nearly every careless slip.

Every solution deserves a check. Substitute your answer back into the original equation. If the left side equals the right side, you know you are right.

Suppose you solved $3(x + 2) = x + 14$ and got $x = 4$. Put 4 back into both sides of the original equation.

Left side: $3 \times (4 + 2) = 18$. Right side: $4 + 14 = 18$. They match, so $x = 4$ is right.

Always check using the original equation, not a line you have already rearranged — substituting into a rearranged line could hide the very mistake you are hunting for. Checking turns "I think it is right" into "I know it is right".

An inequality compares two amounts that are not equal, using $<$, $>$, $\le$ or $\ge$. The good news: solving an inequality works exactly like solving an equation — the same inverse operations on both sides, just keeping the inequality sign in place of the equals sign.

So $4x - 3 < 17$ goes: add 3 to both sides for $4x < 20$; divide both sides by 4 for $x < 5$.

The one real difference is the answer. An equation usually has a single solution; an inequality has a range of them. $x < 5$ means every number below 5 works — 4, 3, 0, $-10$, even $4.9$. That range is what you show on a number line next.

Because an inequality has many answers, you show the whole range on a number line. Two details carry the meaning.

The circle tells you about the end value: an open (empty) circle means the number is not included — used for $<$ and $>$; a filled circle means the number is included — used for $\le$ and $\ge$. The arrow points the way the solutions continue.

So $x < 5$ has an open circle at 5 with an arrow pointing left. By contrast, $x \ge 2$ has a filled circle at 2 with an arrow pointing right.

Read your solved inequality carefully: the sign chooses the circle, and which side the variable is on chooses the arrow direction.

Stage 9 introduces simultaneous equations: two equations that are true at the same time, sharing two unknowns. The solution is the pair of values that satisfies both.

When the same letter term appears in both equations, a tidy trick works: add or subtract the equations to make that letter disappear.

Take $x + y = 10$ and $x - y = 4$. Adding them, the $+y$ and $-y$ cancel: $2x = 14$, so $x = 7$. Now substitute $x = 7$ into the first equation: $7 + y = 10$, so $y = 3$.

The solution is $x = 7,\ y = 3$ — and you can check it in both equations: $7 + 3 = 10$ and $7 - 3 = 4$. When the matching terms have the same sign, subtract instead of add.

Confidence with equations and inequalities pays off right across later maths:

• Simultaneous equations grow into a major tool, used to solve problems with two unknowns at once.
• Straight-line graphs come directly from equations like $y = 2x + 1$, and a simultaneous pair meets where two lines cross.
• Formulae are rearranged with the very same inverse-operation skills.
• In everyday life, equations and inequalities model real limits — a budget becomes $3x \le 20$, a minimum age becomes $a \ge 16$.

If equation work feels hard later, it is usually the balance rule, expanding a bracket, or choosing add-or-subtract for a simultaneous pair that needs a quick refresh. There is no prize for rushing — accuracy first.