Equations and Inequalities

A linear equation is a statement that two expressions are equal, with the unknown letter appearing only to the power $1$ — for example $3x + 5 = 20$. To solve it you find the value of $x$ that makes the statement true.

The big idea is balance. Whatever you do to one side of the equation you must do to the other, otherwise the equals sign no longer holds. Think of it as a pair of scales: take $5$ off the left, you must take $5$ off the right.

You also work in reverse order. To peel away from $x$ you undo the operations one at a time, addition or subtraction first and then multiplication or division.

For an equation with the unknown on both sides, gather the letters on one side and the numbers on the other before you finish. For one with brackets, expand first or divide both sides by the bracket if the multiplier is a whole number. Always check your answer by substituting it back: in $3x + 5 = 20$, $3(5) + 5 = 20$ — perfect.

Some equations look bigger but use the same balancing rules. The trick is to make them look smaller before you solve.

For an equation with letters on both sides, like $5x - 4 = 2x + 11$, subtract the smaller letter term to keep your coefficient positive. Subtracting $2x$ from both sides gives $3x - 4 = 11$, then $3x = 15$, then $x = 5$.

For an equation with brackets, like $2(x + 4) = 18$, you have two clean options:

• Expand the bracket: $2x + 8 = 18$, then $2x = 10$, then $x = 5$.
• Divide by the multiplier first: $x + 4 = 9$, then $x = 5$.

Either path works — choose whichever is cleaner.

For fractional coefficients, clear the fraction by multiplying both sides by the denominator. So $\tfrac{x}{3} + 2 = 7$ becomes $x + 6 = 21$, giving $x = 15$. Or simply subtract first: $\tfrac{x}{3} = 5$, then $x = 15$. The path is up to you — pick the one with fewer slips.

An inequality swaps the equals sign for one of $<$, $>$, $\le$ or $\ge$. The solution is not a single value but a range of values.

You solve inequalities almost exactly like equations: keep them balanced and use inverse operations. So $2x + 3 < 11$ becomes $2x < 8$, then $x < 4$.

There is one new rule, and it is the one to remember: if you multiply or divide both sides by a negative number, you must flip the inequality sign. So $-3x \ge 12$ becomes $x \le -4$ (divide by $-3$ and flip).

When you show an inequality on the number line, the picture tells the story:

• Open circle (hollow) for $<$ or $>$ — the endpoint itself is not included.
• Closed circle (filled) for $\le$ or $\ge$ — the endpoint is included.
• Draw a line or arrow towards the values that satisfy the inequality.

For a double inequality like $-1 < x \le 4$, both endpoints get a circle of their own (open at $-1$, closed at $4$) and the line connects them.

Simultaneous equations are a pair of equations that share the same two unknowns, usually $x$ and $y$. A solution is the pair $(x,y)$ that makes both equations true at the same time.

There are two friendly methods.

1. Elimination — line the two equations up and add or subtract to make one letter disappear. Take $x + y = 10$ and $x - y = 4$. Adding the two equations gives $2x = 14$, so $x = 7$. Substitute back into the first: $7 + y = 10$, so $y = 3$.

2. Substitution — rearrange one equation to make a letter the subject, then slot it into the other. Take $y = 2x + 1$ and $3x + y = 16$. Substitute the first into the second: $3x + (2x + 1) = 16$, so $5x = 15$, giving $x = 3$. Then $y = 2(3) + 1 = 7$.

Two quick tips. Match the coefficients first if a letter does not already line up. For $2x + y = 9$ and $x - y = 3$, the $y$ terms already cancel by adding — no rescaling needed. And always check both equations, not just the one you solved.

Many puzzles and real-world questions become easy once you write down an equation. The recipe is short:

1. Pick a letter for the unknown (usually $x$).
2. Translate the sentence into an equation, one piece at a time.
3. Solve the equation using the balancing rules.
4. Answer the original question in words — not just "$x = \ldots$".

For example: "I think of a number, double it, add $5$, and the result is $19$. What was my number?" Let the number be $x$. The equation is $2x + 5 = 19$, so $2x = 14$ and $x = 7$. The number was $7$.

Word problems often hide inequalities, too. *"A taxi charges \3$plus$$2$per kilometre. How far can I travel for less than$$15$?"* Let the distance be$d$km. Then$3 + 2d < 15$, so$2d < 12$and$d < 6$km. Real situations sometimes need an *integer* answer — here you could travel up to$5$ km if the company only charges by whole kilometres. Always sense-check the answer against the story.

Fluency with equations and inequalities is the gateway to almost every algebra topic that follows:

• Linear graphs treat $y = mx + c$ as a rule and use balancing to find points of intersection.
• Sequences use equations when you solve for the term number from the position-to-term rule.
• Geometry uses equations whenever a side length or angle is given as an algebraic expression.
• In real life, inequalities describe budgets, speeds and limits, and simultaneous equations crop up whenever two quantities interact.

If a later topic feels shaky, ask whether a missed balancing step or a flipped inequality is the real culprit. Come back here for a top-up any time — accurate working beats clever shortcuts.