Equations and Inequalities

An equation has an equals sign: it says the left side and the right side are worth exactly the same. The best picture is still a balance scale that must stay level.

The golden rule carries straight through Stage 8: whatever you do to one side, you must do to the other. Add, subtract, multiply or divide — as long as both sides get the same treatment, the scale stays balanced.

What is new at Stage 8 is the kind of equation you face: ones with brackets, like $3(x + 2) = 18$, and ones with the unknown on both sides, like $2x + 1 = x + 5$. The balance rule does not change — you just have a little tidying to do first. Master that tidying and these equations become as routine as the one-step ones you already know.

When an equation contains a bracket, the first move is to expand it. Once the bracket is gone, you have an ordinary equation to solve with inverse operations.

Example: Solve $3(x + 2) = 18$.

Step by step:

1. Expand the bracket: $3(x + 2) = 3x + 6$, so $3x + 6 = 18$.
2. Subtract 6 from both sides: $3x = 12$.
3. Divide both sides by 3: $x = 4$.

Remember to expand the bracket fully — both terms inside. A common slip is $3(x + 2) = 3x + 2$, which loses the $\times 2$ and gives a wrong answer. Expand carefully, and the rest is the balancing you already know.

Sometimes the letter appears on both sides, like $5x + 2 = 2x + 11$. The extra first move is to gather all the letter terms onto one side — using the balance rule.

Step by step for $5x + 2 = 2x + 11$:

1. Subtract $2x$ from both sides — this clears the letter from the right: $3x + 2 = 11$.
2. Subtract 2 from both sides: $3x = 9$.
3. Divide both sides by 3: $x = 3$.

A handy tip: subtract the smaller letter term ($2x$ here, not $5x$). That keeps the letter term positive, which is easier to work with. Once the letters are all on one side, you have an ordinary equation — and you already know how to finish those.

Once you have a solution, check it yourself. Substitute your answer back into the original equation. If the left side equals the right side, your answer is right.

Suppose you solved $5x + 2 = 2x + 11$ and got $x = 3$. Put 3 back in both sides: $\text{Left: } 5 \times 3 + 2 = 17. \qquad \text{Right: } 2 \times 3 + 11 = 17.$

Both sides come to 17 — they match, so $x = 3$ is right.

If they did not match, you would know there was a slip to find — and checking would have caught it before it cost you. This habit turns "I think it is right" into "I know it is right". Make checking the natural last step of every equation you solve.

Sometimes two amounts are not equal — one is bigger or smaller. An inequality says so, using one of four signs:

• $<$ 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 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 the number is not included — used for $<$ and $>$.
• A filled circle means the number is included — used for $\le$ and $\ge$.

So $x \ge 1$ has a filled circle at 1, because 1 is allowed.

Here is the good news: solving an inequality works just like solving an equation. You use the same inverse operations on both sides — you simply keep the inequality sign in place of the equals sign.

Example: Solve $2x + 3 < 11$.

Step by step:

1. Subtract 3 from both sides: $2x < 8$.
2. Divide both sides by 2: $x < 4$.

The solution $x < 4$ means any number below 4 works — you can show it on a number line with an open circle at 4 and an arrow pointing left.

Unlike an equation, an inequality usually has many answers, not just one. That is the only real difference — the solving steps are exactly the same.

Getting confident with equations and inequalities 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, a budget or a speed limit.

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